examples of null hypothesis and alternative hypothesis statements

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• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis (H 0 ): There’s no effect in the population .
• Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
• Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Statistics By Jim

Making statistics intuitive

Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

• Null Hypothesis H 0 : No effect exists in the population.
• Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
• Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
• Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

• Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
• Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933).  On the Problem of the most Efficient Tests of Statistical Hypotheses .  Philosophical Transactions of the Royal Society A .  231  (694–706): 289–337.

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Reader Interactions

January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

Comments and Questions Cancel reply

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

The actual test begins by considering two  hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

In a  hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

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Hypothesis Testing with One Sample

Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p = 0.40
• H a : p > 0.40

<!– ??? –>

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

Chapter Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

Formula Review

H 0 and H a are contradictory.

If α ≤ p -value, then do not reject H 0 .

If α > p -value, then reject H 0 .

α is preconceived. Its value is set before the hypothesis test starts. The p -value is calculated from the data.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

The mean entry level salary of an employee at a company is 💲58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

• H 0 : __________
• H a : __________
• H 0 : μ = 15
• H a : μ ≠ 15

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, H 0 , and the alternative hypothesis. H a , in terms of the appropriate parameter ( μ or p ).

• The mean number of years Americans work before retiring is 34.
• At most 60% of Americans vote in presidential elections.
• The mean starting salary for San Jose State University graduates is at least 💲100,000 per year.
• Twenty-nine percent of high school seniors get drunk each month.
• Fewer than 5% of adults ride the bus to work in Los Angeles.
• The mean number of cars a person owns in her lifetime is not more than ten.
• About half of Americans prefer to live away from cities, given the choice.
• Europeans have a mean paid vacation each year of six weeks.
• The chance of developing breast cancer is under 11% for women.
• Private universities’ mean tuition cost is more than 💲20,000 per year.
• H 0 : μ = 34; H a : μ ≠ 34
• H 0 : p ≤ 0.60; H a : p > 0.60
• H 0 : μ ≥ 100,000; H a : μ < 100,000
• H 0 : p = 0.29; H a : p ≠ 0.29
• H 0 : p = 0.05; H a : p < 0.05
• H 0 : μ ≤ 10; H a : μ > 10
• H 0 : p = 0.50; H a : p ≠ 0.50
• H 0 : μ = 6; H a : μ ≠ 6
• H 0 : p ≥ 0.11; H a : p < 0.11
• H 0 : μ ≤ 20,000; H a : μ > 20,000

Over the past few decades, public health officials have examined the link between weight concerns and teen girls’ smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

• p < 0.30
• p > 0.30

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

• p > 0.20
• p < 0.20

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

• H o : \(\overline{x}\) = 4.5, H a : \(\overline{x}\) > 4.5
• H o : μ ≥ 4.5, H a : μ < 4.5
• H o : μ = 4.75, H a : μ > 4.75
• H o : μ = 4.5, H a : μ > 4.5

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm.

Null and Alternative Hypotheses Copyright © 2013 by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

Example 10.2: Hypotheses with One Sample of One Categorical Variable Section  

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

• Research Question : Are artists more likely to be left-handed than people found in the general population?
• Response Variable : Classification of the student as either right-handed or left-handed

State Null and Alternative Hypotheses

• Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
• Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

Example 10.3: Hypotheses with One Sample of One Measurement Variable Section  

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

• Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
• Response Variable : dosage of the active ingredient found by a chemical assay.
• Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
• Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section  

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

• Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
• Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
• Explanatory (Grouping) Variable: Sex
• Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
• Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section  

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

• Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
• Response Variable : Weight loss (pounds)
• Explanatory (Grouping) Variable : Type of diet
• Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
• Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section  

• Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
• Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
• Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
• Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section  

• Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
• Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
• Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
• Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section  

• Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
• Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
• Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
• Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

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• Statistical power
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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Hypothesis Testing: Null Hypothesis and Alternative Hypothesis

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Figuring out exactly what the null hypothesis and the alternative hypotheses are is not a walk in the park. Hypothesis testing is based on the knowledge that you can acquire by going over what we have previously covered about statistics in our blog.

So, if you don’t want to have a hard time keeping up, make sure you have read all the tutorials about confidence intervals , distributions , z-tables and t-tables .

We've also made a video on null hypothesis vs alternative hypothesis - you can watch it below or just scroll down if you prefer reading.

Confidence intervals provide us with an estimation of where the parameters are located. You can obtain them with our confidence interval calculator and learn more about them in the related article.

However, when we are making a decision, we need a yes or no answer. The correct approach, in this case, is to use a test .

Here we will start learning about one of the fundamental tasks in statistics - hypothesis testing !

The Hypothesis Testing Process

  First off, let’s talk about data-driven decision-making. It consists of the following steps:

• First, we must formulate a hypothesis .
• After doing that, we have to find the right test for our hypothesis .
• Then, we execute the test.
• Finally, we make a decision based on the result.

Let’s start from the beginning.

What is a Hypothesis?

Though there are many ways to define it, the most intuitive must be:

“A hypothesis is an idea that can be tested.”

This is not the formal definition, but it explains the point very well.

So, if we say that apples in New York are expensive, this is an idea or a statement. However, it is not testable, until we have something to compare it with.

For instance, if we define expensive as: any price higher than $1.75 dollars per pound, then it immediately becomes a hypothesis .

What Cannot Be a Hypothesis?

An example may be: would the USA do better or worse under a Clinton administration, compared to a Trump administration? Statistically speaking, this is an idea , but there is no data to test it. Therefore, it cannot be a hypothesis of a statistical test.

Actually, it is more likely to be a topic of another discipline.

Conversely, in statistics, we may compare different US presidencies that have already been completed. For example, the Obama administration and the Bush administration, as we have data on both.

A Two-Sided Test

Alright, let’s get out of politics and get into hypotheses . Here’s a simple topic that CAN be tested.

According to Glassdoor (the popular salary information website), the mean data scientist salary in the US is 113,000 dollars.

So, we want to test if their estimate is correct.

The Null and Alternative Hypotheses

There are two hypotheses that are made: the null hypothesis , denoted H 0 , and the alternative hypothesis , denoted H 1 or H A .

The null hypothesis is the one to be tested and the alternative is everything else. In our example:

The null hypothesis would be: The mean data scientist salary is 113,000 dollars.

While the alternative : The mean data scientist salary is not 113,000 dollars.

Author's note: If you're interested in a data scientist career, check out our articles Data Scientist Career Path , 5 Business Basics for Data Scientists , Data Science Interview Questions , and 15 Data Science Consulting Companies Hiring Now .

An Example of a One-Sided Test

You can also form one-sided or one-tailed tests.

Say your friend, Paul, told you that he thinks data scientists earn more than 125,000 dollars per year. You doubt him, so you design a test to see who’s right.

The null hypothesis of this test would be: The mean data scientist salary is more than 125,000 dollars.

The alternative will cover everything else, thus: The mean data scientist salary is less than or equal to 125,000 dollars.

Important: The outcomes of tests refer to the population parameter rather than the sample statistic! So, the result that we get is for the population.

Important: Another crucial consideration is that, generally, the researcher is trying to reject the null hypothesis . Think about the null hypothesis as the status quo and the alternative as the change or innovation that challenges that status quo. In our example, Paul was representing the status quo, which we were challenging.

Let’s go over it once more. In statistics, the null hypothesis is the statement we are trying to reject. Therefore, the null hypothesis is the present state of affairs, while the alternative is our personal opinion.

Why Hypothesis Testing Works

Right now, you may be feeling a little puzzled. This is normal because this whole concept is counter-intuitive at the beginning. However, there is an extremely easy way to continue your journey of exploring it. By diving into the linked tutorial, you will find out why hypothesis testing actually works.

Interested in learning more? You can take your skills from good to great with our statistics course!

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Next Tutorial:  Hypothesis Testing: Significance Level and Rejection Region

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Null Hypothesis Definition and Examples, How to State

What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.

Why is it Called the “Null”?

The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).

Why Do I need to Test it? Why not just prove an alternate one?

The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.

• Null hypothesis : H 0 : The world is flat.
• Alternate hypothesis: The world is round.

Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .

How to State the Null Hypothesis from a Word Problem

You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.

Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.

H 1 : μ > 8.2

Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2

Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.

H 0 : μ ≤ 8.2

Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

How to State the Null Hypothesis: Part Two

But what if the researcher doesn’t have any idea what will happen.

Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks. 

Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

H 0 : μ = 8.2

Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

H 1 : μ ≠ 8.2

In English again, that’s H 1 (The  alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2

That’s How to State the Null Hypothesis!

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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

• Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
• Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis:   H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis :  H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis :  H 0 : There is no relationship between height and shoe size. Alternative Hypothesis :  H a : There is a positive relationship between height and shoe size.

Null Hypothesis :  H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis :  H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing
• Examples of null and alternative hypotheses

Writing null and alternative hypotheses

• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

• (Choice A)   H 0 : p = 0.1 H a : p ≠ 0.1 ‍   A H 0 : p = 0.1 H a : p ≠ 0.1 ‍  
• (Choice B)   H 0 : p ≠ 0.1 H a : p = 0.1 ‍   B H 0 : p ≠ 0.1 H a : p = 0.1 ‍  
• (Choice C)   H 0 : p = 0.1 H a : p > 0.1 ‍   C H 0 : p = 0.1 H a : p > 0.1 ‍  
• (Choice D)   H 0 : p = 0.1 H a : p < 0.1 ‍   D H 0 : p = 0.1 H a : p < 0.1 ‍  

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H 0 " if the sample information favors the alternative hypothesis or "do not reject H 0 " or "decline to reject H 0 " if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ .30 H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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Null Hypothesis and Alternative Hypothesis

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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.

How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.

The Null Hypothesis

The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .

The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.

If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.

For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.

If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.

The Alternative Hypothesis

The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .

The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”

If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.

The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
• What 'Fail to Reject' Means in a Hypothesis Test
• Type I and Type II Errors in Statistics
• An Example of a Hypothesis Test
• The Runs Test for Random Sequences
• An Example of Chi-Square Test for a Multinomial Experiment
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• What Level of Alpha Determines Statistical Significance?
• What Is the Difference Between Alpha and P-Values?
• What Is ANOVA?
• How to Find Critical Values with a Chi-Square Table
• Example of a Permutation Test
• Degrees of Freedom for Independence of Variables in Two-Way Table
• Example of an ANOVA Calculation
• How to Find Degrees of Freedom in Statistics
• How to Construct a Confidence Interval for a Population Proportion
• Degrees of Freedom in Statistics and Mathematics

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