1 Million+ Step-by-step solutions

According to a Bureau of Labor Statistics release of March 25, 2015, statisticians earn an average of $84,010 a year. Suppose that the current annual earnings of all statisticians have the mean and standard

deviation of $84,010 and $20,000, respectively, and the shape of this distribution is skewed to the right. Let x be the average earnings of a random sample of a certain number of statisticians. Calculate the mean and standard deviation of x and describe the shape of its sampling distribution when the sample size is

a. 20

b. 100

c. 800

According to a Bureau of Labor Statistics release of March 25, 2015, statisticians earn an average of $84,010 a year. Suppose that the current annual earnings of all statisticians have the mean and standard deviation of $84,010 and $20,000, respectively, and the shape of this distribution is skewed to the right. Find the probability that the mean earnings of a random sample of 256 statisticians is

a. Between $81,400 and $82,600

b. Within $1500 of the population mean

c. $85,300 or more

d. Not within $1600 of the population mean

e. Less than $83,000

f. Less than $85,000

g. More than $86,000

h. Between $83,000 and 85,500

According to a Pew Research survey conducted in February 2014, 24% of American adults said they trust the government in Washington, D.C., always or most of the time. Suppose that this result is true for the current population of American adults. Let p̂ be the proportion of American adults in a random sample who hold the aforementioned opinion. Find the mean and standard deviation of the sampling distribution of p̂ and describe its shape when the sample size is:

a. 30

b. 300

c. 3000

According to a Gallup poll conducted August 7–10, 2014, 48% of American workers said that they were completely satisfied with their jobs. Assume that this result is true for the current population of American workers.

a. Find the probability that in a random sample of 1150 American workers, the proportion who will say they are completely satisfied with their jobs is

i. greater than .50

ii. between .46 and .51

iii. less than .45 iv. between .495 and .515

v. less than .49 vi. more than .47

b. What is the probability that in a random sample of 1150 American workers, the proportion who will say they are completely satisfied with their jobs is within .025 of the population proportion?

c. What is the probability that in a random sample of 1150 American workers, the proportion who will say they are completely satisfied with their jobs is not within .03 of the population proportion?

d. What is the probability that in a random sample of 1150 American workers, the proportion who will say they are completely satisfied with their jobs is greater than the population proportion by .02 or more?

Briefly explain the meaning of an estimator and an estimate.

What is the point estimator of the population mean, μ? How would you calculate the margin of error for an estimate of μ?

Briefly explain the difference between a confidence level and a confidence interval.

What is the margin of error of estimate for μ when σ is known? How is it calculated?

How will you interpret a 99% confidence interval for μ? Explain.

According to the 2015 Physician Compensation Report by Medscape (a subsidiary of WebMD), American orthopedists earned an average of $421,000 in 2014. Suppose that this mean is based on a random sample of 200 American orthopaedists, and the standard deviation for this sample is $90,000. Make a 90% confidence interval for the population mean μ.

The following data give the one-way commuting times (in minutes) from home to work for a random sample of 30 workers.

a. Calculate the value of the point estimate of the mean one-way commuting time from home to work for all workers.

b. Construct a 99% confidence interval for the mean one-way commuting time from home to work for all workers.

What assumption(s) must hold true to use the normal distribution to make a confidence interval for the population proportion, p?

What is the point estimator of the population proportion, p?

According to a Gallup poll conducted January 5–8, 2014, 67% of American adults were dissatisfied with the way income and wealth are distributed in America. Assume that this poll is based on a random sample of 1500 American adults.

a. What is the point estimate of the corresponding population proportion?

b. Construct a 98% confidence interval for the proportion of all American adults who are dissatisfied with the way income and wealth are distributed in America. What is the margin of error for this estimate?

According to a Gallup poll conducted April 3–6, 2014, 21% of Americans aged 18 to 29 said that college loans and/or expenses were the top financial problem facing their families. Suppose that this poll was based on a random sample of 1450 Americans aged 18 to 29.

a. What is the point estimate of the corresponding population proportion?

b. Construct a 95% confidence interval for the proportion of all Americans aged 18 to 29 who will say that college loans and/or expenses were the top financial problem facing their families. What is the margin of error for this estimate?

In a January 2014 survey conducted by the Associated Press- We TV, 68% of American adults said that owning a home is the most important thing or a very important but not the most important thing (opportunityagenda.org). Assume that this survey was based on a random sample of 900 American adults.

a. Construct a 95% confidence interval for the proportion of all American adults who will say that owning a home is the most important thing or a very important but not the most important thing.

b. Explain why we need to construct a confidence interval. Why can we not simply say that 68% of all American adults would say that owning a home is the most important thing or a very important but not the most important thing?

In an online poll conducted by the St. Louis Post-Dispatch during September 2014, people were asked about their favorite sports to watch on television. Of the respondents, 42% selected baseball, 18% mentioned hockey, 36% liked football, and 4% selected basketball (www.stltoday.com). Using these results, find a 98% confidence interval for the population percentage that corresponds to each response. Write a one-page report to present your results to a group of college students who have not taken statistics. Your report should answer questions such as the following: (1) What is a confidence interval? (2) Why is a range of values (interval) more informative than a single percentage (point estimate)? (3) What does 98% confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?

Harris Interactive conducted an online poll of 2097 Americanadults between July 17 and 21, 2014, on the topic “Who are we lying to?” In response to one of the questions, 37% of American adults said that they have lied to get out of work (www.harrisinteractive.com).

a. What is the point estimate of the corresponding population proportion?

b. Construct a 99% confidence interval for the proportion of all American adults who will say that they have lied to get out of work.

Which of the two hypotheses (null and alternative) is initially assumed to be true in a test of hypothesis?

Consider H_{0}: μ = 55 versus H_{1}: μ ≠ 55.

a. What type of error would you make if the null hypothesis is actually false and you fail to reject it?

b. What type of error would you make if the null hypothesis is actually true and you reject it?

What does the level of significance represent in a test of hypothesis? Explain.

By rejecting the null hypothesis in a test of hypothesis example, are you stating that the alternative hypothesis is true?

Consider the following null and alternative hypotheses:

H_{0}: μ = 25 versus H_{1}: μ ≠ 25

Suppose you perform this test at α = .05 and reject the null hypothesis.

Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is “statistically significant” or would you state that this difference is “statistically not significant?” Explain.

Consider the following null and alternative hypotheses:

H_{0}: μ = 60 versus H_{1}: μ > 60

Suppose you perform this test at α = .01 and fail to reject the null hypothesis.

Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is “statistically significant” or would you state that this difference is “statistically not significant?” Explain.

For each of the following significance levels, what is the probability of making a Type I error?

a. α = .025

b. α = .05

c. α = .01

According to the U.S. Bureau of Labor Statistics, all workers in America who had a bachelor’s degree and were employed earned an average of $1224 a week in 2014. A recent sample of 400 American workers who have a bachelor’s degree showed that they earn an average of $1260 per week. Suppose that the population standard deviation of such earnings is $160.

a. Find the p-value for the test of hypothesis with the alternative hypothesis that the current mean weekly earning of American workers who have a bachelor’s degree is higher than $1224. Will you reject the null hypothesis at α = .025?

b. Test the hypothesis of part a using the critical-value approach and α = .025.

According to the National Association of Colleges and Employers, the average starting salary of 2014 college graduates with a bachelor’s degree was $45,473 (www.naceweb.org). A random sample of 1000 recent college graduates from a large city showed that their average starting salary was $44,930. Suppose that the population standard deviation for the starting salaries of all recent college graduates from this city is $7820.

a. Find the p-value for the test of hypothesis with the alternative hypothesis that the average starting salary of recent college graduates from this city is less than $45,473. Will you reject the null hypothesis at α = .01? Explain. What if α = .025?

b. Test the hypothesis of part a using the critical-value approach. Will you reject the null hypothesis at α = .01? What if α = .025?

In a Gallup Annual Economy and Personal Finance poll, conducted April 3–6, 2014, 21% of adults aged 18 to 29 said that college costs and loans were the biggest financial problem their families were dealing with. Suppose two adults aged 18 to 29 are selected. Find the following probabilities.

a. Both adults will say that college costs and loans are the biggest financial problem their families are dealing with.

b. Exactly one adult will say that college costs and loans are the biggest financial problem their families are dealing with.

Terry & Sons makes bearings for autos. The production system involves two independent processing machines so that each bearing passes through these two processes. The probability that the first processing machine is not working properly at any time is .08, and the probability that the second machine is not working properly at any time is .06. Find the probability that both machines will not be working properly at any given time.

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 69, and then picks a Powerball number from 1 through 26. The lottery organization randomly draws 5 different white balls from 69 balls numbered 1 through 69, and then randomly picks a red Powerball number from 1 through 26. Note that it is possible for the Powerball number to be the same as one of the first five numbers.

a. If a player’s first five numbers match the numbers on the five white balls drawn by the lottery organization and the player’s red Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.)

b. If a player’s first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about $1,000,000. Find the probability that a player who buys one ticket will win this prize.

According to Moebs Services Inc., the average cost of an individual checking account to major U.S. banks was $380 in 2013 (www.moebs.com). A bank consultant wants to determine whether the current mean cost of such checking accounts at major U.S. banks is more than $380 a year. A recent random sample of 150 such checking accounts taken from major U.S. banks produced a mean annual cost to

them of $390. Assume that the standard deviation of annual costs to major banks of all such checking accounts is $60.

a. Find the p-value for this test of hypothesis. Based on this p-value, would you reject the null hypothesis if the maximum probability of Type I error is to be .05? What if the maximum probability of Type I error is to be .01?

b. Test the hypothesis of part a using the critical-value approach and α = .05. Would you reject the null hypothesis? What if α = .01? What if α = 0?

According to a survey by the College Board, undergraduate students at private nonprofit four-year colleges spent an average of $1244 on books and supplies in 2014–2015 (www.collegeboard.org). A recent random sample of 200 undergraduate college students from a large private nonprofit four-year college showed that they spent an average of $1204 on books and supplies during the last academic year. Assume that the standard deviation of annual expenditures on books and supplies by all such students at this college is $200.

a. Find the p-value for the test of hypothesis with the alternative hypothesis that the annual mean expenditure by all such students at this college is less than $1244. Based on this p-value, would you reject the null hypothesis if the significance level is .025?

b. Test the hypothesis of part a using the critical-value approach and α = .025. Would you reject the null hypothesis? Explain.

The XO Group Inc., released the results of its annual Real Weddings Study on March 27, 2014 (www.theknot.com). According to this study, the average cost of a wedding in America was $29,858 in 2013. A recent sample of 100 American couples who got married this year produced a mean wedding cost of $32,084 with a standard deviation of $9275. Using a 2.5% significance level and the criticalvalue approach, can you conclude that the current mean cost of a wedding in America is higher than $29,858? Find the range for the p-value for this test. What will your conclusion be using this p-value range and α = .05?

According to ValuePenguin, the average annual cost of automobile insurance was $1388 in the state of Nevada in 2014 (www. valuepenguin.com). An insurance broker is interested to find if the current mean annual rate of automobile insurance in Nevada is more than $1388. She took a random sample of 100 insured automobiles from the state of Nevada and found the mean annual automobile insurance rate of $1413 with a standard deviation of $122.

a. Using a 1% significance level and the critical-value approach, can you conclude that the current mean annual automobile insurance rate in Nevada is higher than $1388?

b. Find the range for the p-value for this test. What will your conclusion be using this p-value range and α = .01?

According to the analysis of Federal Reserve statistics and other government data, American households with credit card debts owed an average of $15,706 on their credit cards in August 2015 (www.nerdwallet.com). A recent random sample of 500 American households with credit card debts produced a mean credit card debt of $16,377 with a standard deviation of $3800. Do these data provide significant evidence at a 1% significance level to conclude that the current mean credit card debt of American households with credit card debts is higher than $15,706? Use both the p-value approach and the critical-value approach.

According to a study conducted in 2015, 18% of shoppers said that they prefer to buy generic instead of name-brand products. Suppose that in a recent sample of 1500 shoppers, 315 stated that they prefer to buy generic instead of name-brand products. At a 5% significance level, can you conclude that the proportion of all shoppers who currently prefer to buy generic instead of name-brand products is higher than .18? Use both the p-value and the critical-value approaches.

According to the U.S. Census Bureau, in 2014, 62% of Americans age 18 and older were married. A recent sample of 2000 Americans age 18 and older showed that 58% of them are married. Can you reject the null hypothesis at a 1% significance level in favor of the alternative that the percentage of current population of Americans age 18 and older who are married is lower than 62%? Use both the p-value and the critical-value approaches.

According to a 2014 CIRP Your First College Year Survey, 88% of the first-year college students said that their college experience exposed them to diverse opinions, cultures, and values (www.heri.ucla.edu). Suppose in a recent poll of 1800 first-year college students, 91% said that their college experience exposed them to diverse opinions, cultures, and values. Perform a hypothesis test to determine if it is reasonable to conclude that the current percentage of all first year college students who will say that their college experience exposed them to diverse opinions, cultures, and values is higher than 88%. Use a 2% significance level, and use both the p-value and the critical-value approaches.

According to an article in Forbes magazine of April 3, 2014, 57% of students said that they did not attend the college of their first choice due to financial concerns (www.forbes.com). In a recent poll of 1600 students, 864 said that they did not attend the college of their first choice due to financial concerns. Using a 1% significance level, perform a test of hypothesis to determine whether the current percentage of students who did not attend the college of their first choice due to financial concerns is lower than 57%. Use both the p-value and the critical-value approaches.

In a Gallup poll conducted July 7–10, 2014, 45% of Americans said that they actively try to include organic foods into their diets (www.gallup.com). In a recent sample of 2100 Americans, 1071 said that they actively try to include organic foods into their diets. Is there significant evidence at a 1% significance level to conclude that the current percentage of all Americans who will say that they actively try to include organic foods into their diets is different from 45%? Use both the p-value and the critical-value approaches.

A two-tailed test is a test with

a. Two rejection regions

b. Two non-rejection regions

c. Two test statistics

A one-tailed test

a. Has one rejection region

b. Has one non-rejection region

c. Both a and b

According to the Kaiser Family Foundation, U.S. workers who had employer-provided health insurance paid an average premium of $1170 for single (one person) health insurance coverage during 2013 (www.kff.org). Suppose that a recent random sample of 100 workers with employer-provided health insurance selected from a large city paid an average premium of $1198 for single health insurance coverage. Assume that such premiums paid by all such workers in this city have a standard deviation of $125.

a. Using the critical-value approach and a 1% significance level, can you conclude that the current average such premium paid by all such workers in this city is different from $1170?

b. Using the critical-value approach and a 2.5% significance level, can you conclude that the current average such premium paid by all such workers in this city is higher than $1170?

c. What is the Type I error in parts a and b? What is the probability of making this error in each of parts a and b?

d. Calculate the p-value for the test of part a. What is your conclusion if α = .01?

e. Calculate the p-value for the test of part b. What is your conclusion if α = .025?

The standard recommendation for automobile oil changes is once every 5000 miles. A local mechanic is interested in determining whether people who drive more expensive cars are more likely to follow the recommendation. Independent random samples of 45 customers who drive luxury cars and 40 customers who drive compact lower price cars were selected. The average distance driven between oil changes was 5187 miles for the luxury car owners and 5214 miles for the compact lower-price cars. The sample standard deviations were 424 and 507 miles for the luxury and compact groups, respectively. Assume that the two population distributions of the distances between oil changes have the same standard deviation.

a. Construct a 95% confidence interval for the difference in the mean distances between oil changes for all luxury cars and all compact lower-price cars.

b. Using a 1% significance level, can you conclude that the mean distance between oil changes is less for all luxury cars than that for all compact lower-price cars?

The following information was obtained from two independent samples selected from two normally distributed populations with unequal and unknown population standard deviations.

n_{1} = 14 x̅_{1} = 109.43 s_{1} = 2.26

n_{2} = 15 x̅_{2} = 113.88 s_{2} = 5.84

Test at a 5% significance level if the two population means are different.

The following information was obtained from two independent samples selected from two populations with unequal and unknown population standard deviations.

n_{1} = 48 x̅_{1} = .863 s_{1} = .176

n_{2} = 46 x̅_{2} = .796 s_{2} = .068

Test at a 1% significance level if the two population means are different.

The following information was obtained from two independent samples selected from two normally distributed populations with unequal and unknown population standard deviations.

n_{1} = 14 x̅_{1} = .109.43 s_{1} = 2.26

n_{2} = 15 x̅_{2} = .113.88 s_{2} = 5.84

Test at a 1% significance level if μ_{1} is less than μ_{2}.

The following information was obtained from two independent samples selected from two populations with unequal and unknown population standard deviations.

n_{1} = 48 x̅_{1} = .863 s_{1} = .176

n_{2} = 46 x̅_{2} = .796 s_{2} = .068

Test at a 2.5% significance level if μ_{1} is greater than μ_{2}.

According to a Bureau of Labor Statistics release of March 25, 2015, financial analysts earned an average of $110,510 in 2014. Suppose that the 2014 earnings of all financial analysts had a mean of $110,510. A recent sample of 400 financial analysts showed that they earn an average of $114,630 a year. Assume that the standard deviation of the annual earnings of all financial analysts is $30,570.

a. Using the critical-value approach, can you conclude that the current average annual earnings of financial analysts is higher than $110,510? Use α = .01.

b. What is the Type I error in part a? Explain. What is the probability of making this error in part a?

c. Will your conclusion of part a change if the probability of making a Type I error is zero?

d. Calculate the p-value for the test of part a. What is your conclusion if α = .01?

According to a Bureau of Labor Statistics release of February 20, 2015, 79% of American children under age 18 lived with at least one other sibling in 2014. Suppose that in a recent sample of 2000 American children under age 18, 1620 were living with at least one other sibling.

a. Using the critical-value approach and α = .05, test if the current percentage of all American children under age 18 who live with at least one other sibling is different from 79%.

b. How do you explain the Type I error in part a? What is the probability of making this error in part a?

c. Calculate the p-value for the test of part a. What is your conclusion if α = .05?

PolicyInteractive of Eugene, Oregon conducted a study of American adults in April 2014 for the Center for a New American Dream. Seventy-five percent of the adults included in this study said that having basic needs met is very or extremely important in their vision of the American dream (www.newdream.org). A recent sample of 1500 American adults were asked the same question and 72% of them said that having basic needs met is very or extremely important in their vision of the American dream.

a. Using the critical-value approach and α = .01, test if the current percentage of American adults who hold the above mentioned opinion is less than 75%.

b. How do you explain the Type I error in part a? What is the probability of making this error in part a?

c. Calculate the p-value for the test of part a. What is your conclusion if α = .01?

According to an estimate, 75% of cell phone owners in a large city had smart phones in 2014. In a recent sample of 1000 cell phone owners selected from this city, 790 had smart phones. At a 2% significance level, can you conclude that the current proportion of cell phone owners in this city who have smart phones is different from .75?

To test the hypothesis that the mean blood pressure of university professors is lower than that of company executives, which of the following would you use?

a. A left-tailed test

b. A two-tailed test

c. A right-tailed test

What is a goodness-of-fit test and when is it applied? Explain.

Explain the difference between the observed and expected frequencies for a goodness-of-fit test.

To make a goodness-of-fit test, what should be the minimum expected frequency for each category? What are the alternatives if this condition is not satisfied?

A finger-tapping experiment was conducted by a doctoral neuropsychology student. The purpose of this experiment was to determine bilateral nervous system integrity, which provides data about the neuromuscular system and motor control. The experiment requires the subjects to place the palm of the hand on a table and then to tap the index finger on the surface of the table. Ten subjects who had suffered a mild concussion were given a finger tapping test using the right index finger and then the left index finger. The total number of taps in 30 seconds by each subject are listed in the following table. All subject in the data set are right-handed. Assume that the population of paired differences is approximately normally distributed.

a. Make a 95% confidence interval for the mean of the population of paired differences, where a paired difference is equal to the number of taps for the right index finger minus the number of taps for the left index finger.

b. Using a 5% significance level, can you conclude that the average number of taps is different for the right and left index fingers?

When are the samples considered large enough for the sampling distribution of the difference between two sample proportions to be (approximately) normal?

In a survey of American drivers, 79% of women drivers and 85% of men drivers said that they exceeded the speed limit at least once in the past week. Suppose that these percentages are based on random samples of 600 women and 700 men drivers.

a. Let p_{1} and p_{2} be the proportion of all women and men American drivers, respectively, who will say that they exceeded the speed limit at least once in the past week.

Construct a 98% confidence interval for p_{1} − p_{2}.

b. Using a 1% significance level, can you conclude that p_{1} is lower than p_{2}? Use both the critical-value and the p-value approaches.

According to a Bureau of Labor Statistics report released on March 25, 2015, statisticians earn an average of $84,010 a year and accountants and auditors earn an average of $73,670 a year (www.bls. gov). Suppose that these estimates are based on random samples of 2000 statisticians and 1800 accountants and auditors. Further, assume that the sample standard deviations of the annual earnings of these two groups are $15,200 and $14,500, respectively, and the population standard deviations are unknown but equal for the two groups.

a. Construct a 98% confidence interval for the difference in the mean annual earnings of the two groups, statisticians and accountants and auditors.

b. Using a 1% significance level, can you conclude that the average annual earnings of statisticians is higher than that of accountants and auditors?

According to a Bureau of Labor Statistics report released on March 25, 2015, statisticians earn an average of $84,010 a year and accountants and auditors earn an average of $73,670 a year (www.bls.gov). Suppose that these estimates are based on random samples of 2000 statisticians and 1800 accountants and auditors. Further assume that the sample standard deviations of the annual earnings of these two groups are $15,200 and $14,500, respectively, and the population standard deviations are unknown and unequal for the two groups.

a. Construct a 98% confidence interval for the difference in the mean annual earnings of the two groups, statisticians and accountants and auditors.

b. Using a 1% significance level, can you conclude that the average annual earnings of statisticians is higher than that of accountants and auditors?

The chi-square goodness-of-fit test is always

a. Two-tailed

b. Left-tailed

c. Right-tailed

In a Gallup poll conducted August 7–10, 2014, American adults aged 18 and older were asked, “If you were taking a new job and had your choice of a boss, would you prefer to work for a man or a woman?” Of the respondents, 33% said that they would prefer a male boss, 20% said a female boss, 46% said it would not make a difference to them, and 1% had no opinion (www.gallup.com). Suppose these results are true for the 2014 population. Recently, 1500 randomly selected American adults were asked the same question. The following table contains the frequency distribution that resulted from this survey

Test at the 1% significance level whether the current distribution of opinions is different from the 2014 distribution.

Find the critical value of F for the following.

a. df = (3, 3) and area in the right tail = .05

b. df = (3, 10) and area in the right tail = .05

c. df = (3, 30) and area in the right tail = .05

Find the critical value of F for the following.

a. df = (2, 6) and area in the right tail = .025

b. df = (6, 6) and area in the right tail = .025

c. df = (15, 6) and area in the right tail = .025

Determine the critical value of F for the following.

a. df = (6, 12) and area in the right tail = .01

b. df = (6, 40) and area in the right tail = .01

c. df = (6, 100) and area in the right tail = .01

Find the critical value of F for an F distribution with df = (3, 12) and

a. Area in the right tail = .05

b. Area in the right tail = .10

Find the critical value of F for an F distribution with .025 area in the right tail and

a. df = (4, 11)

b. df = (15, 3)

To make a test of independence or homogeneity, what should be the minimum expected frequency for each cell? What are the alternatives if this condition is not satisfied?

A random sample of 1000 Americans was taken, and these adults were asked if experience in politics was necessary for a candidate to be president of America. The following table presents the results of the survey.

Test at a 1% significance level whether gender and opinions are related.

In a Pew Research Center poll conducted December 3–8, 2013, American adults age 18 and older were asked if Christmas is more a religious or a cultural holiday for them. Of the respondents, 51% said Christmas is a religious holiday for them, 32% said it is a cultural holiday, and 17% gave other answers (www.pewforum.org). Assume that these results are true for the 2013 population of adults. Recently, a random sample of 1200 American adults age 18 and older was taken, and these adults were asked the same question. Their responses are presented in the following table.

Test at a 2.5% significance level whether the distribution of recent opinions is significantly different from that of the 2013 opinions.

In a Harris Poll conducted October 15–20, 2014, American adults were asked “to think ahead 2 to 5 years and assess if they feel solar energy will contribute to meeting our energy needs.” Of the respondents, 31% said solar energy will make a major contribution to meeting our energy needs within the next 2 to 5 years, 53% felt it will make a minor contribution, and 16% expected that it will make hardly any contribution at all (www.harrisinteractive.com). Assume that these results are true for the 2014 population of adults. Recently a random sample of 2000 American adults was selected and these adults were asked the same question. The results of the poll are presented in the following table.

Test at a 2.5% significance level whether the current distribution of opinions to the said question is significantly different from that for the 2014 opinions.

In a recent poll, American adults were asked, if they have a choice, would they prefer to live in a city, suburb, or countryside. The following table shows the frequencies for the three choices.

Test at a 1% significance level if these three places are equally preferred by American adults.

According to a Gallup poll whose results were reported on October 22, 2013, American’s views on legalizing marijuana are changing. In that survey, American adults were asked whether marijuana should be legalized in America. Suppose in a recent survey, 600 Americans were randomly selected from each of the four age groups listed in the table below. The frequencies of the responses for various age groups are listed in this table assuming that every person included in the survey responded yes or no.

Test at a 1% significance level whether the proportion of Americans who support legalizing marijuana is the same for each of the age groups.

A sample of 21 units selected from a normally distributed population produced a variance of 9.2. Test at a 5% significance level if the population variance is different from 6.5.

A sample of 10 units selected from a normally distributed population produced a variance of 7.2. Test at a 1% significance level if the population variance is greater than 4.2.

In 2014, the variance of the ages of all workers at a large company that has more than 30,000 workers was 133. A recent random sample of 25 workers selected from this company showed that the variance of their ages is 112.

a. Using a 2.5% significance level, can you conclude that the current variance of the ages of workers at this company is lower than 133? Assume that the ages of all current workers at this company are (approximately) normally distributed.

b. Construct a 98% confidence intervals for the variance and the standard deviation of the ages of all current workers at this company.

The one-way ANOVA test is always

a. Right-tailed

b. Left-tailed

c. Two-tailed

Explain the meaning of independent and dependent variables for a regression model.

Why is the random error term included in a regression model?

Explain the least squares method and least squares regression line. Why are they called by these names?

Explain the difference between y and ŷ.

Two variables x and y have a positive linear relationship. Explain what happens to the value of y when x increases. Give one example of a positive relationship between two variables.

Two variables x and y have a negative linear relationship. Explain what happens to the value of y when x increases. Give one example of a negative relationship between two variables.

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams.

a. Find the least squares regression line with total payroll as the independent variable and percentage of games won as the dependent variable.

b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the y-intercept and the slope of the regression line give A and B or a and b?

c. Give a brief interpretation of the values of the y-intercept and the slope obtained in part a.

d. Predict the percentage of games won by a team with a total payroll of $150 million.

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the American League baseball teams.

a. Find the least squares regression line with total payroll as the independent variable and percentage of games won as the dependent variable.

b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the y-intercept and the slope of the regression line give A and B or a and b?

c. Give a brief interpretation of the values of the y-intercept and the slope obtained in part a.

d. Predict the percentage of games won by a team with a total payroll of $150 million.

What are the degrees of freedom for a simple linear regression model?

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams.

a. Find the standard deviation of errors, σ_{ε} with percentage of games won as the dependent variable. (Note that this data set belongs to a population.)

b. Compute the coefficient of determination, ????^{2}.

Describe the mean, standard deviation, and shape of the sampling distribution of the slope b of the simple linear regression model.

Explain why the random error term ε is added to the regression model.

Explain the difference between A and a and between B and b for a regression model.

What does a linear correlation coefficient tell about the relationship between two variables? Within what range can a correlation coefficient assume a value?

Can the values of B and ρ calculated for the same population data have different signs? Explain.

For a sample data set, the linear correlation coefficient r has a positive value. Which of the following is true about the slope b of the regression line estimated for the same sample data?

a. The value of b will be positive.

b. The value of b will be negative.

c. The value of b can be positive or negative.

For a sample data set, the slope b of the regression line has a negative value. Which of the following is true about the linear correlation coefficient r calculated for the same sample data?

a. The value of r will be positive.

b. The value of r will be negative.

c. The value of r can be positive or negative.

For a sample data set on two variables, the value of the linear correlation coefficient is (close to) zero. Does this mean that these variables are not related? Explain.

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams.

Compute the linear correlation coefficient, ρ. Does it make sense to make a confidence interval and to test a hypothesis about ρ here? Explain.

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the American League baseball teams.

Compute the linear correlation coefficient, ρ. Does it make sense to make a confidence interval and to test a hypothesis about ρ here? Explain.

The CTO Corporation has a large number of chain restaurants throughout the United States. The research department at the company wanted to find if the restaurants’ sales depend on the mean income of households in the related areas. The company collected information on these two variables for 10 restaurants randomly selected from different areas. The following table gives information on the weekly sales (in thousands of dollars) of these restaurants and the mean annual incomes (in thousands of dollars) of the households for those areas.

a. Taking income as an independent variable and sales as a dependent variable, compute SSxx, SSyy, and SSxy.

b. Find the least squares regression line.

c. Briefly explain the meaning of the values of a and b calculated in part b.

d Calculate r and r2 and briefly explain what they mean.

e. Compute the standard deviation of errors.

f. Construct a 95% confidence interval for B.

g. Test at a 2.5% significance level whether B is positive.

h. Using a 2.5% significance level, test whether ρ is positive.

The following table gives the average weekly retail price of a gallon of regular gasoline in the eastern United States over a 9-week period from December 1, 2014, through January 26, 2015. Consider these 9 weeks as a random sample.

a. Assign a value of 0 to 12/1/14, 1 to 12/8/14, 2 to 12/15/14, and so on. Call this new variable Time. Make a new table with the variables Time and Price.

b. With time as an independent variable and price as the dependent variable, compute SSxx, SSyy, and SSxy.

c. Construct a scatter diagram for these data. Does the scatter diagram exhibit a negative linear relationship between time and price?

d. Find the least squares regression line ŷ = a + bx.

e. Give a brief interpretation of the values of a and b calculated in part d.

f. Compute the correlation coefficient r.

g. Predict the average price of a gallon of regular gasoline in the eastern United States for Time = 26. Comment on this prediction.

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