unit statistics homework 1 population and sample

Chapter 1: Sampling and Data

Chapter 1 Homework

Homework from 1.2.

For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate.

A fitness center is interested in the mean amount of time a client exercises in the center each week.

The population is all of the clients of the fitness center. A sample of the clients that use the fitness center for a given week. The average amount of time that all clients exercise in one week. The average amount of time that a sample of clients exercises in one week. The amount of time that a client exercises in one week. Examples are: 2 hours, 5 hours, and 7.5 hours –>

Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.

• all children who take ski or snowboard lessons
• a group of these children
• the population mean age of children who take their first snowboard lesson
• the sample mean age of children who take their first snowboard lesson
• X = the age of one child who takes his or her first ski or snowboard lesson
• values for X , such as 3, 7, and so on

A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.

the cardiologist’s patients a group of the cardiologist’s patients the mean recovery period of all of the cardiologist’s patients the mean recovery period of the group of the cardiologist’s patients X = the mean recovery period of one patient values for X, such as 10 days, 14 days, 20 days, and so on –>

Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.

• the clients of the insurance companies
• a group of the clients
• the mean health costs of the clients
• the mean health costs of the sample
• X = the health costs of one client
• values for X , such as 34, 9, 82, and so on

A politician is interested in the proportion of voters in his district who think he is doing a good job.

all voters in the politician’s district a random selection of voters in the politician’s district the proportion of voters in this district who think this politician is doing a good job the proportion of voters in this district who think this politician is doing a good job in the sample X = the number of voters in the district who think this politician is doing a good job Yes, he is doing a good job. No, he is not doing a good job. –>

A marriage counselor is interested in the proportion of clients she counsels who stay married.

• all the clients of this counselor
• a group of clients of this marriage counselor
• the proportion of all her clients who stay married
• the proportion of the sample of the counselor’s clients who stay married
• X = the number of couples who stay married

Political pollsters may be interested in the proportion of people who will vote for a particular cause.

all voters (in a certain geographic area) a random selection of all the voters the proportion of voters who are interested in this particular cause the proportion of voters who are interested in this particular cause in the sample X = the number of voters who are interested in this particular cause yes, no –>

A marketing company is interested in the proportion of people who will buy a particular product.

• all people (maybe in a certain geographic area, such as the United States)
• a group of the people
• the proportion of all people who will buy the product
• the proportion of the sample who will buy the product
• X = the number of people who will buy it
• buy, not buy

Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.

What is the population she is interested in?

• all Lake Tahoe Community College students
• all Lake Tahoe Community College English students
• all Lake Tahoe Community College students in her classes
• all Lake Tahoe Community College math students

Consider the following:

[latex]X[/latex] = number of days a Lake Tahoe Community College math student is absent

In this case, X is an example of a:

• population.

The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a:

More Homework from 1.2

For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data.

number of tickets sold to a concert

quantitative discrete, 150

percentage of body fat

quantitative continuous, 19.2% –>

favorite baseball team

qualitative, Oakland A’s

time in line to buy groceries

quantitative continuous, 7.2 minutes –>

number of students enrolled at Evergreen Valley College

quantitative discrete, 11,234 students

most-watched television show

qualitative, Dancing with the Stars –>

brand of toothpaste

qualitative, Crest

distance to the closest movie theater

quantitative continuous, 8.32 miles –>

age of executives in Fortune 500 companies

quantitative continuous, 47.3 years

number of competing computer spreadsheet software packages

quantitative discrete, three –>

Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.

“Number of times per week” is what type of data?

• qualitative
• quantitative discrete
• quantitative continuous

“Duration (amount of time)” is what type of data?

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.

• Using complete sentences, list three things wrong with the way the survey was conducted.
• Using complete sentences, list three ways that you would improve the survey if it were to be repeated.

The survey would not be a true representation of the entire population of air travelers.

Conducting the survey on a holiday weekend will not produce representative results.

• Conduct the survey during different times of the year.

Conduct the survey using flights to and from various locations.

Conduct the survey on different days of the week.

Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Answers will vary. Sample Answer: Randomly choose 25 colleges in the state. Use all statistics classes from each of the chosen colleges in the sample. This can be done by listing all the colleges together with a two-digit number starting with 00 then 01, etc. The list of colleges can be found on Wikipedia. http://en.wikipedia.org/wiki/List_of_colleges_and_universities_in_California Use a random number generator to pick 25 colleges. –>

Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon.

List some practical difficulties involved in getting accurate results from a telephone survey.

Answers will vary. Sample Answer: Not all people have a listed phone number. Many people hang up or do not respond to phone surveys. –>

List some practical difficulties involved in getting accurate results from a mailed survey.

Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete.

With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey.

Ask everyone to include their age then take a random sample from the data. Include in the report how the survey was conducted and why the results may not be accurate. –>

The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is

• cluster sampling
• stratified sampling
• simple random sampling
• convenience sampling

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:

• simple random

Name the sampling method used in each of the following situations:

convenience cluster stratified systematic simple random

A “random survey” was conducted of 3,274 people of the “microprocessor generation” (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had 💲2,000 to spend, they would use it for computer equipment. Also, 66% of those surveyed considered themselves relatively savvy computer users.

• Do you consider the sample size large enough for a study of this type? Why or why not?

Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute’s road show called “America’s Smithsonian.”

• With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
• With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.

Yes, in polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. We do not have enough information to decide if this is a random sample from the U.S. population. No, this is a convenience sample taken from individuals who visited an exhibition in the Angeles Convention Center. This sample is not representative of the U.S. population. It is possible that the two sample statistics, 48% and 66% are larger than the true parameters in the population at large. In any event, no conclusion about the population proportions can be inferred from this convenience sample. –>

The Gallup-Healthways Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below.

Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous.

• Do you have any health problems that prevent you from doing any of the things people your age can normally do?
• During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
• In the last seven days, on how many days did you exercise for 30 minutes or more?
• Do you have health insurance coverage?

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent postcards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.

• Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
• What effect does the low response rate have on the reliability of the sample?
• Are these problems examples of sampling error or nonsampling error?
• During the same year, George Gallup conducted his own poll of 30,000 prospective voters. His researchers used a method they called “quota sampling” to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

The country was in the middle of the Great Depression, and many people could not afford these “luxury” items and therefore were not able to be included in the survey. Samples that are too small can lead to sampling bias. sampling error stratified

Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI’s Uniform Crime Report . One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates.

Which of the potential problems with samples discussed in [link] could explain this connection?

Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate.

Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size.

YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks:

“Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?” 1

As of April 25, 11 people responded to this question. Each participant answered “NO!”

Which of the potential problems with samples discussed in this module could explain this connection?

Self-Selected Samples: Only people who are interested in the topic are choosing to respond. Sample Size Issues: A sample with only 11 participants will not accurately represent the opinions of a nation. Undue Influence: The question is wording in a specific way to generate a specific response. Self-Funded or Self-Interest Studies: This question was generated to support one person’s claim and it was designed to get the answer that the person desires. –>

A scholarly article about response rates begins with the following quote:

“Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.” 2

The Pew Research Center for People and the Press admits:

“The percentage of people we interview – out of all we try to interview – has been declining over the past decade or more.” 3

• What are some reasons for the decline in response rate over the past decade?
• Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.
• Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed
• When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the subject of the survey.

Bringing It Together

Seven hundred and seventy-one distance learning students at Long Beach City College responded to surveys in the 2010-11 academic year. Highlights of the summary report are listed in [link] .

• What percentage of the students surveyed do not have a computer at home?
• About how many students in the survey live at least 16 miles from campus?
• If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why?

4% 13% Not necessarily. Long Beach City is the seventh largest college in California, and it has an enrollment of approximately 27,000 students. On the other hand, Great Basin College has its campuses in rural northeastern Nevada, and its enrollment of about 3,500 students. –>

Several online textbook retailers advertise that they have lower prices than on-campus bookstores. However, an important factor is whether the Internet retailers actually have the textbooks that students need in stock. Students need to be able to get textbooks promptly at the beginning of the college term. If the book is not available, then a student would not be able to get the textbook at all, or might get a delayed delivery if the book is back ordered.

A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook for each of the following seven subjects: calculus, biology, chemistry, physics, statistics, geology, and general engineering. He consults textbook industry sales data and selects the most popular nationally used textbook in each of these subjects. He visits websites for a random sample of major online textbook sellers and looks up each of these seven textbooks to see if they are available in stock for quick delivery through these retailers. Based on his investigation, he writes an article in which he draws conclusions about the overall availability of all college textbooks through online textbook retailers.

Write an analysis of his study that addresses the following issues: Is his sample representative of the population of all college textbooks? Explain why or why not. Describe some possible sources of bias in this study, and how it might affect the results of the study. Give some suggestions about what could be done to improve the study.

Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences; there are many subjects in the humanities, social sciences, and other subject areas (for example: literature, art, history, psychology, sociology, business) that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results.

He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways:

• the most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies, OR
• the most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly.

In reality, many college students do not use the most popular textbooks in their subject, and this study gives no useful information about the situation for those less popular textbooks.

He could improve this study by:

• expanding the selection of subjects he investigates so that it is more representative of all subjects studied by college students, and
• expanding the selection of textbooks he investigates within each subject to include a mixed representation of both the most popular and less popular textbooks.

HOMEWORK from 1.3

Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

• Fill in the blanks in [link] .
• What percent of students take exactly two courses?
• What percent of students take one or two courses?

Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in [link] .

• What percent of adults flossed six times per week?
• What percentage flossed at most three times per week?

Nineteen immigrants to the U.S were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2 5 7 2 2 10 20 15 0 7 0 20 5 12 15 12 4 5 10 .

[link] was produced.

• Fix the errors in [link] . Also, explain how someone might have arrived at the incorrect number(s).
• Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the U.S. for 5 years.”
• Fix the statement in b to make it correct.
• What fraction of the people surveyed have lived in the U.S. five or seven years?
• What fraction of the people surveyed have lived in the U.S. at most 12 years?
• What fraction of the people surveyed have lived in the U.S. fewer than 12 years?
• What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

The Frequencies for 15 and 20 should both be two and the Relative Frequencies should both be

The mistake could be due to copying the data down wrong. The Cumulative Relative Frequency for five years should be 0.4737. The mistake is due to calculating the Relative Frequency instead of the Cumulative Relative Frequency. The Cumulative Relative Frequency for 15 years should be 0.8947 The 47% is the Cumulative Relative Frequency, not the Relative Frequency. 47% of the people surveyed have lived in the U.S. for five years or less.

How much time does it take to travel to work? [link] shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

The sum of the travel times is 1,173.1. Divide the sum by 50 to calculate the mean value: 23.462. Because each state’s travel time was measured to the nearest tenth, round this calculation to the nearest hundredth: 23.46.

Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least 💲5 per share, and have reported annual revenue between 💲5 million and 💲1 billion. [link] shows the ages of the chief executive officers for the first 60 ranked firms.

• What is the frequency for CEO ages between 54 and 65?
• What percentage of CEOs are 65 years or older?
• What is the relative frequency of ages under 50?
• What is the cumulative relative frequency for CEOs younger than 55?
• Which graph shows the relative frequency and which shows the cumulative relative frequency?

26 (This is the count of CEOs in the 55 to 59 and 60 to 64 categories.) 12% (number of CEOs age 65 or older ÷ total number of CEOs) 14/60; 0.23; 23% 0.45 Graph A represents the cumulative relative frequency, and Graph B shows the relative frequency. –>

Use the following information to answer the next two exercises: [link] contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

What is the relative frequency of direct hits that were category 4 hurricanes?

• Not enough information to calculate

What is the relative frequency of direct hits that were AT MOST a category 3 storm?

HOMEWORK from 1.4

How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation.

Use key terms from this module to describe the design of this experiment.

Explanatory variable: amount of sleep

Response variable: performance measured in assigned tasks

Treatments: normal sleep and 27 hours of total sleep deprivation

Experimental Units: 19 professional drivers

Lurking variables: none – all drivers participated in both treatments

Random assignment: treatments were assigned in random order; this eliminated the effect of any “learning” that may take place during the first experimental session

Control/Placebo: completing the experimental session under normal sleep conditions

Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time

An advertisement for Acme Investments displays the two graphs in [link] to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected?

The graphs do not show scales of values. We do not know the period of time each graph represents; they may show data from different years. We also do not know if the vertical scales on each graph are equivalent. The scales may have been adjusted to exaggerate or minimize trends. There is no reliable information to be gleaned from these graphs, and setting them up as examples of performance is misleading. –>

The graph in [link] shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints?

You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading.

Introductory Statistics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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UNIT 1: SAMPLING AND DATA

STATISTICS is used to make informed decisions based on data analysis. This process starts with the evaluation and implementation of appropriate data gathering techniques. This unit introduces the concept of data and sampling methods commonly used in statistical research and studies.

UNIT CONTENTS

• 1.1 Definitions of Statistics, Probability, and Key Terms
• 1.2 Data, Sampling, and Variation in Data and Sampling
• 1.3 Frequency, Frequency Tables, and Levels of Measurement
• 1.4 Experimental Design and Ethics

TOPICS COVERED

• Definitions of Statistics, Probability, and Key Terms
• Data, Sampling, and Variation in Data and Sampling
• Frequency, Frequency Tables, and Levels of Measurement
• Experimental Design and Ethics
• Qualitative and quantitative variables
• Frequency tables, histograms, and Interpretations

LEARNING OUTCOMES

• Distinguish between qualitative and quantitative variables
• Compare and contrast population parameters with sample statistics
• Classify data according to their level of measurement
• Apply various types of sampling methods to data collection
• Create and interpret frequency tables, histograms
• Distinguish between observational studies and experiments

Statistics Study Guide Copyright © by Ram Subedi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Unit 13 – Statistics

Variability and Sampling

LESSON/HOMEWORK

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Population Parameters

The Normal Distribution

Normal Distribution Z-Scores

Sample Means

Sample Proportions

The Difference in Samples Means

The Distribution of Sample Means

The Distribution of Sample Proportions

Margin of Error

Linear Regression and Lines of Best Fit

Other Types of Regression

Unit Review

Unit 13 Review – Statistics

UNIT REVIEW

EDITABLE REVIEW

Unit 13 Assessment Form A

EDITABLE ASSESSMENT

Unit 13 Assessment Form B

Unit 13 Assessment Form C

Unit 13 Assessment Form D

Unit 13 Exit Tickets

Unit 13 Mid-Unit Quiz (Through Lesson 6) – Form A

Unit 13 Mid-Unit Quiz (Through Lesson 6) – Form B

Unit 13 Mid-Unit Quiz (Through Lesson 6) – Form C

Unit 13 Mid-Unit Quiz (Through Lesson 6) – Form D

U13.AO.01 – Lesson 4.5.Sampling a Population

EDITABLE RESOURCE

U13.AO.02 – Statistical Simulation Packet

U13.AO.03 – Normal Distributions Practice

U13.AO.04 – Practice with Margin of Error and 95% Confidence Intervals

U13.AO.05 – Practice with Linear and Exponential Regressions

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N ( 244 , 15 50 ) N ( 244 , 15 50 )

As the sample size increases, there will be less variability in the mean, so the interval size decreases.

X is the time in minutes it takes to complete the U.S. Census short form. X ¯ X ¯ is the mean time it took a sample of 200 people to complete the U.S. Census short form.

CI: (7.9441, 8.4559)

The level of confidence would decrease, because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases.

• x ¯ x ¯ = 2.2

X ¯ X ¯ is the mean weight of a sample of 20 heads of lettuce.

EBM = 0.07 CI: (2.1264, 2.2736)

The interval is greater, because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.

The confidence level would increase.

(24.52,36.28)

We are 95 percent confident that the true mean age for winter Foothill College students is between 24.52 and 36.28.

The error bound for the mean would decrease, because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

X is the number of hours a patient waits in the emergency room before being called back to be examined. X ¯ X ¯ is the mean wait time of 70 patients in the emergency room.

CI: (1.3808, 1.6192)

• x ¯ x ¯ = 151
• s x s x = 32
• n – 1 = 107

X ¯ X ¯ is the mean number of hours spent watching television per month from a sample of 108 Americans.

CI: (142.92, 159.08)

(2.93, 3.59)

We are 95 percent confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.

The error bound would become EBM = 0.245. This error bound decreases, because as sample sizes increase, variability decreases, and we need less interval length to capture the true mean.

It would decrease, because the z -score would decrease, which would reduce the numerator and lower the number.

X is the number of successes where the woman makes the majority of the purchasing decisions for the household. P ′ is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.

CI: (0.5321, 0.6679)

EBM : 0.0679

X is the number of successes where an executive prefers a truck. P ′ is the percentage of executives sampled who prefer a truck.

CI: (0.19432, 0.33068)

EBM : 0.0707

The sampling error means that the true mean can be 2 percent above or below the sample mean.

P ′ is the proportion of voters sampled who said the economy is the most important issue in the upcoming election.

CI: (0.62735, 0.67265);

EBM: 0.02265

the number of girls, ages 8 to 12, in the 5 p.m. Monday night beginning ice-skating class

P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 ) P ′ ~ N ( 0.8 , ( 0.8 ) ( 0.2 ) 80 )

CI = (0.72171, 0.87829).

(0.72; 0.88)

With 92 percent confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72 percent and 88 percent.

The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.

• X is the height of a Swedish male, and is the mean height from a sample of 48 Swedish males.
• Normal. We know the standard deviation for the population, and the sample size is greater than 30.
• CI: (70.151, 71.49)
• EBM = 0.849
• The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.
• x ¯ x ¯ = 23.6
• X is the time needed to complete an individual tax form. X ¯ X ¯ is the mean time to complete tax forms from a sample of 100 customers.
• N ( 23.6 , 7 100 ) N ( 23.6 , 7 100 ) because we know sigma.
• (22.228, 24.972)
• EBM = 1.372
• It will need to change the sample size. The firm needs to determine what the confidence level should be and then apply the error bound formula to determine the necessary sample size.
• The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.
• According to the error bound formula, the firm needs to survey 206 people. Because we increase the confidence level, we need to increase either our error bound or the sample size.
• X is the number of letters a single camper will send home. X ¯ X ¯ is the mean number of letters sent home from a sample of 20 campers.

N 7.9 ( 2.5 20 ) 7.9 ( 2.5 20 )

• CI: (6.98, 8.82)
• The error bound and confidence interval will decrease.
• x ¯ x ¯ = $568,873
• CL = 0.95, α = 1 – 0.95 = 0.05, z α 2 z α 2 = 1.96 EBM = z 0.025 σ n z 0.025 σ n = 1.96 909200 40 909200 40 = $281,764

Alternate solution:

Using the TI-83, 83+, 84, 84+ Calculator

• Press STAT and arrow over to TESTS .
• Arrow down to 7:ZInterval .
• Press ENTER .
• Arrow to Stats and press ENTER .
• σ : 909,200
• x ¯ x ¯ : 568,873
• Arrow down to Calculate and press ENTER .
• The confidence interval is ($287,114, $850,632).
• Notice the small difference between the two solutions—these differences are simply due to rounding error in the hand calculations.
• We estimate with 95 percent confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637.

Use the formula for EBM , solved for n : n =   z 2 σ 2 E B M 2 n =   z 2 σ 2 E B M 2

From the statement of the problem, you know that σ = 2.5, and you need EBM = 1.

z = z 0.035 = 1.812.

(This is the value of z for which the area under the density curve to the right of z is 0.035.)

n =   z 2 σ 2 E B M 2 = 1.812 2 2.5 2 1 2   ≈   20.52 . n =   z 2 σ 2 E B M 2 = 1.812 2 2.5 2 1 2   ≈   20.52 .

You need to measure at least 21 male students to achieve your goal.

• CI: (6244, 11,014)
• It will become smaller.
• x ¯ x ¯ = 2.51
• s x s x = 0.318
• The effective length of time for a tranquilizer
• The mean effective length of time of tranquilizers from a sample of nine patients
• We need to use a Student’s t -distribution, because we do not know the population standard deviation.
• CI: (2.27, 2.76)
• Check student's solution.
• If we were to sample many groups of nine patients, 95 percent of the samples would contain the true population mean length of time.

x ¯ = $ 251 , 854.23 ; x ¯ = $ 251 , 854.23 ;

s =   $ 521 , 130.41 . s =   $ 521 , 130.41 .

Note that we are not given the population standard deviation, only the standard deviation of the sample.

There are 30 measures in the sample, so n = 30, and df = 30 - 1 = 29.

CL = 0.96, so α = 1 - CL = 1 - 0.96 = 0.04.

α 2 = 0.02 t α 2 = t 0.02 α 2 = 0.02 t α 2 = t 0.02 = 2.150.

E B M = t α 2 ( s n ) = 2.150 ( 521 , 130.41 30 )   ~   $ 204 , 561.66 . E B M = t α 2 ( s n ) = 2.150 ( 521 , 130.41 30 )   ~   $ 204 , 561.66 .

x ¯ x ¯ - EBM = $251,854.23 - $204,561.66 = $47,292.57.

x ¯ x ¯ + EBM = $251,854.23 + $204,561.66 = $456,415.89.

We estimate with 96 percent confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between $47,292.57 and $456,415.89.

Alternate Solution

The difference between solutions arises from rounding differences.

• X is the number of unoccupied seats on a single flight. X ¯ X ¯ is the mean number of unoccupied seats from a sample of 225 flights.
• We will use a Student’s t-distribution, because we do not know the population standard deviation.
• CI: (11.12 , 12.08)
• CI: (7.64, 9.36)
• The sample should have been increased.
• Answers will vary.
• The sample size would need to be increased, because the critical value increases as the confidence level increases.

X = the number of people who believe that the president is doing an acceptable job;

P ′ = the proportion of people in a sample who believe that the president is doing an acceptable job.

• N ( 0.61 , ( 0.61 ) ( 0.39 ) 1200 ) N ( 0.61 , ( 0.61 ) ( 0.39 ) 1200 )
• CI: (0.59, 0.63)
• Check student’s solution.
• (0.72, 0.82)
• (0.65, 0.76)
• (0.60, 0.72)
• Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.
• We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.
• We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a white person into their families and the proportion of Asian adults who say that their families would welcome a black person into their families.
• X = the number of adult Americans who believe that crime is the main problem; P′ = the proportion of adult Americans who believe that crime is the main problem.
• Because we are estimating a proportion, that P′ = 0.2 and n = 1,000, the distribution we should use is N ( 0.2 , ( 0.2 ) ( 0.8 ) 1000 ) N ( 0.2 , ( 0.2 ) ( 0.8 ) 1000 ) .
• CI: (0.18, 0.22)
• One way to lower the sampling error is to increase the sample size.
• The stated ± 3 percent represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3 percent. Thus, they estimate the percentage of adult Americans who the percentage of adult Americans who that crime is the main problem to be between 18 percent and 22 percent.
• p′ = (0 .55 + 0 .49) 2 (0 .55 + 0 .49) 2 = 0.52; EBP = 0.55 – 0.52 = 0.03
• No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.

STAT TESTS A: 1-PropZinterval with x = (0.52)(1,000), n = 1,000, CL = 0.75.

Answer is (0.502, 0.538).

• Yes, this interval does not fall below 0.50, so we can conclude that at least half of all American adults believe that major sports programs corrupt education – but we do so with only 75 percent confidence.

CL = 0.95; α = 1 – 0.95 = 0.05; α 2 α 2 = 0.025; z α 2 z α 2 = 1.96. Use p ′ = q ′ = 0.5.

n =   z α 2 2 p ′ q ′ E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.05 2 = 384.16 . n =   z α 2 2 p ′ q ′ E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.05 2 = 384.16 .

You need to interview at least 385 students to estimate the proportion to within 5 percent at 95 percent confidence.

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• Authors: Barbara Illowsky, Susan Dean
• Publisher/website: OpenStax
• Book title: Statistics
• Publication date: Mar 27, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/statistics/pages/1-introduction
• Section URL: https://openstax.org/books/statistics/pages/8-solutions

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