AP Statistics - STA: Statistical Inference

In a random sample of 150 adults over the age of 45, 30 say they have played in a band at least one time in their lives. (a) Construct a 99% confidence interval for the proportion of all adults over the age of 45 who have played in a band at least one time in their lives. (b) Suppose all adults over the age of 45 who have played in a band at least one time in their lives each send in a donation of \$10 to the organization Musicians without Borders, which aims to empower musicians as social activists. Assuming there are 125,000,000 adults over the age of 45 in the U.S., what is a 99% confidence interval for what these donations would total for this worthwhile charity?

There are 12,500 high school students in a large city school district. Administrators and teachers want to determine the extent to which parents do homework for their children. In an anonymous survey, 150 of the 500 students say that someone else has done their homework at least once. Of those 150 students, 90 or \(\dfrac{90}{150} = 0.6\) or 60% say that a parent has done their homework for them at least once. (a) Explain how to pick a simple random sample of 500 students for the anonymous survey. (b) What is wrong with using 0.6 to calculate a confidence interval of the proportion of all high school students in the city for whom a parent has done their homework for them at least once? (c) Given the above sample, what is an estimate of the number of high school students in the city for whom a parent has done their homework for them at least once?

In a random sample of 915 adults, 366 say they believe in ghosts. (a) With what margin of error can we find a 95% confidence interval of the proportion of adults who believe in ghosts? (b) With what confidence can we report a margin of error of ±2 percent in giving a confidence interval of the proportion of adults who believe in ghosts?

Among all the banking firms in a large city, five are randomly selected. From each of these, 10 employees are randomly chosen. (a) Is this a simple random sample (SRS)? Why or why not? Explain why your answer would or would not change if you knew that all banking firms in the city had the same number of employees. (b) Suppose that 38 of the 50 selected employees are banking assistants. Calculate a 95% confidence interval estimate for the proportion of this city's banking firm employees who are banking assistants.

A survey is to be conducted to determine the proportion of young adults who earn more than \$7.25 per hour (minimum wage). Two sampling methods are being considered. Method A involves standing on a downtown street corner and randomly picking several young adults to interview every 10 minutes during a 12-hour period. Method B involves posting the question on several popular young adult websites; the viewer simply has to click on one of two possible answers to participate. (a) State a possible source of bias for Method A, and describe how it may affect the results. (b) State a possible source of bias for Method B, and describe how it may affect the results. (c) How many young adults should be interviewed to estimate the proportion of the young adult population who earn more than \$7.25 to within ±0.05 with 95% confidence? (d) If separate estimates are desired for males and females, identify and describe a proper sampling method.

A senator's approval rating stood at 65 percent before she took a crucial vote. (a) Her staff believes the rating is still around 65 percent. To confirm this, how large of a simple random sample (SRS) should the staff sample to obtain a 94% confidence interval estimate with a margin of error ≤3.5%? (b) The senator's staff randomly samples 700 people and finds 432 people approve of the senator's job performance. Is there evidence that the rating has changed from 65 percent? Perform an appropriate statistical test. (c) In part (b) above, suppose the staff suspects the rating has gone down. Is there evidence that the approval rating has slipped down from 65 percent? Perform an appropriate statistical test. (d) Are the answers in (b) and (c) contradictory? Explain.

Below are all the scores of a school's AP Statistics students on a practice 40 question multiple-choice exam. 33 31 37 39 27 31 40 36 27 27 27 30 34 38 27 29 27 38 37 40 33 36 29 26 34 32 39 32 39 36 32 32 25 31 26 40 33 37 29 26 35 26 37 33 27 28 32 37 33 32 (a) Using the following line from a random number table, explain and carry out a procedure to select a simple random sample of size 10 from the population above. 77219 48190 20235 26836 23590 44492 14607 09431 75299 42662 (b) Using this sample, and assuming all conditions for inference are met, construct a 90 percent confidence interval for the population mean \(\mu\). (c) The true population mean is 32.44. Is it in your interval, and is this unexpected?

The goals scored per game during a 21-game span for each of two hockey teams is shown below. (a) The standard deviation of goals scored for Team A is 1.5. Explain what this says about variability. (b) Looking at the dotplots, what can be said about the difference between goals scored by the two teams? Explain. (c) A 95% confidence interval estimate for the difference in mean goals scored per game is (–0.17, 1.88). From this calculation, is there evidence of a difference? Explain.

In 2015, processed meat was classified as a carcinogen by the World Health Organization (WHO). In a representative sample of eight months, a student counts how many times a month he eats processed meat at school. In an independent representative sample of eight months, he counts how many times he eats processed meat at home. The data are shown in the following table. School: 4 5 5 3 6 4 5 7 Home: 4 3 4 4 3 2 5 4 Calculate a 90% confidence interval estimate for the difference between the mean number of times a month this student eats processed meat at school and at home.

In a random sample of eight students taking a college writing class, the weights in ounces of their term papers and their resulting grades are tabulated below. Assume all conditions for inference are satisfied. Weight (oz): 12.1 11.1 11.1 6.5 4.7 10.7 5.9 14.4 Grade: 78 79 76 65 67 79 77 94 (a) Predict the mean grade for all students who turn in term papers weighing 1 pound (16 ounces). (b) Find a 95% confidence interval for the average increase in grade for each additional ounce in weight.

Data on the number of ice cream cones sold on a weekday night and outside temperature for a simple random sample (SRS) of 10 summer days are gathered. A scatterplot and regression output follow. (a) From the above output, what is the equation of the regression line? (b) Interpret the slope of the regression line in context. (c) Assuming all conditions for inference are satisfied, compute the margin of error for a 95% confidence interval for the slope.

A dispatcher for a trucking firm believes the greater the number of hours slept before a trucker begins a long haul, the higher the trucker will score on an alertness test required at the halfway distance. The dispatcher looks up the records of a random sample of 18 long-haul truckers and has them fill out a questionnaire about hours of sleep. Some computer graphical output is shown below. (a) Is this an observational study or an experiment? Explain. (b) Comment on the design of the study. (c) The dispatcher took a statistics course long ago and plans to do a test of significance for evidence of an association between score and hours sleep. Comment on whether the necessary assumptions are met for inference on slope for a least squares regression line.

A random sample of adults with various THC levels (a urine test for cannabis) were given a concentration test consisting of a series of small objects, each of which could be fit into an appropriate shaped hole. Each person had five minutes to find the proper holes for as many objects as possible. Computer regression output of number of objects successfully placed plotted against urine THC level (100 ng/mL) is as follows. (a) What is the equation of the regression line? (b) Find a 95% confidence interval for the slope, and interpret in context. Assume all conditions for inference are satisfied. (c) Find a 95% confidence interval for the y-intercept, and interpret in context. Assume all conditions for inference are satisfied.

A D1 university recruiter claims that 10 percent of its baseball players go on to play professionally after graduation. A reporter contacts a simple random sample (SRS) of baseball players who graduated during the past 20 years and finds that only 32 out of 450 went on to play professionally. Is there sufficient evidence to write an article disputing the university's claim? Give statistical justification for your conclusion.

In past years, 3 percent of all job applicants lied about their education. The HR division of a major company believes the true figure is now higher and plans to investigate a simple random sample (SRS) of applicants to test the hypothesis. (a) Explain why it would not be appropriate to run a one-proportion z-test on an SRS of 150 applicants. (b) What is the minimum sample size necessary to run this hypothesis test? (c) Suppose the HR division uses your result from part (b) and finds that 16 of the applicants lied about their salary. Is this sufficient evidence to say that the percentage of applicants lying about their salary is now over 3 percent?

(a) In a random survey of 500 men who asked a woman's father for permission before asking the woman for her hand in marriage, 143 said the marriage ended in divorce. In a random survey of 500 men who did not first ask a woman's father for permission, 169 of the marriages ended in divorce. Is there convincing statistical evidence that men who ask a woman's father for permission before asking the woman have marriages with lower divorce rates? (b) Based on the above study, a marriage counselor tells young men that if they want to lower the probability of divorce, they should ask a father's permission before asking a woman for her hand in marriage. Is the counselor's advice justified based on this study? Explain.

It is difficult to distinguish between marshmallows and mushrooms by taste alone if one is not allowed to see or smell. A person claims he can distinguish between these, and the following test is designed. He will be given a sample of each in random order to taste while blindfolded with his nose pinched. This will be repeated 16 times. Let \(p\) be the proportion of times the person answers correctly. (a) What are the null and alternative hypotheses? (b) Suppose he correctly answers in 12 out of the 16 trials. What is the probability of answering exactly 12 of 16 if he is simply guessing? (c) What is the P-value if he answers 12 of 16, and interpret this in context. (d) Is there sufficient evidence to reject the null hypothesis? Give an answer in context.

A company advertises it has a process that can extract a mean of 35 grams of dissolved salts from 1 liter of seawater. A geologist believes the true figure is lower. Using this process, a sample of fifteen 1-liter containers of seawater from 15 random locations yields a mean of 34.82 grams of dissolved salts with a standard deviation of 0.65 grams. Assume the sample distribution is symmetric and unimodal with no outliers. (a) Is there sufficient evidence for the geologist to dispute the advertisement? Justify your answer. (b) A large-scale test of a second company's process shows yields of dissolved salts that are roughly normally distributed with a mean of 34.75 grams and a standard deviation of 0.83 grams. What is the probability that using this second process, a 1-liter container of seawater will yield at least 35 grams of dissolved salts? (c) What is the probability that using this second process on 10 randomly selected 1-liter containers of seawater, at least 2 of them yield at least 35 grams of dissolved salts?

The cash register totals are noted at a random sampling of 100 sales at each of 2 competing hardware stores.

Store	$5	$10	$25	$50	Mean	SD
A	10	35	48	7	$19.50	$11.56
B	40	30	15	15	$16.25	$15.72

(a) Is the mean sale in Store A significantly greater than the mean sale in Store B? Explain. (b) Is the proportion of \$50 sales in Store A significantly less than the proportion of \$50 sales in Store B? Explain. (c) Using the above results, what should each hardware store emphasize to its stockholders?

To test whether the company's secretaries type letters quicker from listening to someone speak live or from recorded dictation, an office manager randomly selects five secretaries. He instructs them to time themselves (in seconds) typing a long letter while someone reads the letter to them and then time themselves again typing the same letter from a recording of someone reading the letter. The results are summarized in the following table.

Secretary	1	2	3	4	5
Live Dictation Time (in seconds)	241	247	217	255	255
Recorded Dictation Time (in seconds)	239	248	217	257	259

(a) Is 167 pounds in the 95% confidence interval for the mean weight? (b) Is there evidence at the 5% significance level that the mean weight is less than 167 pounds? Perform an appropriate hypothesis test. (c) Is there a contradiction between the answers in (a) and (b) above? Explain.

Each of 300 students was randomly (coin toss) placed into either Professor A's or Professor B's section of a required college writing course. At the end of the semester, students gave a score from 1 to 5 with "poor" = 1 and "great" = 5 for their professor. The results are shown in the following table.

Score	1	2	3	4	5
Professor A Frequency	16	41	53	38	13
Professor B Frequency	45	50	29	11	4

(a) Use a graphical display to compare the scores received. Write a few sentences, based on your graphs, to compare Professor A and Professor B. (b) Does there appear to be significantly greater satisfaction with Professor A as compared to Professor B? Give a statistical justification for your answer.

A handyman wishes to test which radial arm saw can cut through lumber more quickly. He picks out a random sampling of five pieces of lumber of various species and thicknesses. The cutting times (in seconds) are summarized in the following table.

Board	A	B	C	D	E
Saw S	3.1	6.8	1.5	4.2	2.8
Saw T	3.3	6.7	1.8	4.5	3.0

(a) What is the mean cutting time for each saw? (b) The handyman performs a two-sample t-test with \(H_0: \mu_S = \mu_T\) and \(H_a: \mu_S \neq \mu_T\), which results in a P-value of 0.886. He concludes that there is not significant evidence of a difference in cutting times. Comment on his choice of test. (c) Perform the proper test to decide if there is significant evidence of a difference in cutting times. Use a significance level of 0.10.

At schools using an innovative math program, a simple random sample (SRS) of 100 students results in an average score of 178 with a standard deviation of 27 on a state test. At schools using a traditional approach, an SRS of 150 students results in an average score of 171 with a standard deviation of 31 on the same state test. (a) Is there evidence that students using the innovative approach have a higher average score than students using the traditional approach? Give statistical justification for your answer. (b) Suppose a study using this design resulted in a P-value less than 0.01. Would it be reasonable for all school boards to push for adoption of the innovative approach? Explain. (c) Assuming standard deviations of 27 and 31 as listed above, how large a sample (same number for both) should be used to be 95 percent sure of knowing the difference in scores to within 5 points?

An athletic trainer wishes to determine if a newly designed gym shoe enhances athletic performance. Ten professional high jumpers competitively jump on two successive days. For each jumper, a coin toss determines whether he uses his old gym shoes or the new pair on the first day. Each jumper then switches shoes on the next day. Their jump heights (feet) are as follows. (a) To determine if there is sufficient statistical evidence on whether professional high jumpers improve their jumps when wearing the new shoes, what statistical test should be used, what are the hypotheses, and are the conditions for inference met? (You are not asked to perform the test.) (b) Does knowing a jumper's jump height with the old shoes help predict his jump height using the new shoes? Perform an appropriate statistical test.

In a baseball game, the slugging average is the mean number of bases per hit. It can be calculated using the following formula. \(\text{Slugging average} = \dfrac{(\text{Singles}) + (\text{Doubles} \times 2) + (\text{Triples} \times 3) + (\text{Home runs} \times 4)}{\text{Total at bats}}\) In a random sampling of 75 hits from each of two major league teams, the distributions of singles (1B), doubles (2B), triples (3B), and homers (HR) are shown in the following table.

	1B	2B	3B	HR
Team A	44	17	10	4
Team B	34	22	12	7

Is there evidence that the two teams have a different slugging average? Give statistical justification for your answer.

Can a particular video game improve a batter's reaction time? Batters' reaction times (fraction of a second between the ball leaving a pitcher's hand and the start of a swing) are measured before and after playing the video game for 25 hours. (a) What is the appropriate hypothesis test, the hypotheses, and the conditions to check? (b) Suppose the test is run and no statistically significant improvement is detected in batter reaction times after the video game training. If the researcher plans a second test, name two specific changes that can be made to increase the power of the test. Explain your choices.

A study is performed on 20 different models of luxury SUVs to see if a new gas additive is effective in reducing carbon dioxide (\(\text{CO}_2\)) emissions. Data on \(\text{CO}_2\) emissions (pounds per gallon of gas) are collected before and after input of the additive. A partial computer output follows.

	N	Mean	StDev	SE Mean
Before	20	20.60	7.35	1.64
After	20	20.05	6.84	1.53
Difference	20	0.55	1.90	0.42

(a) Assuming that graphical displays indicate that assumptions of normality are reasonable, construct a 95% confidence interval for the mean difference in \(\text{CO}_2\) emissions before and after the additive. Interpret your interval in context. (b) Does this confidence interval give sufficient evidence that this additive reduces mean \(\text{CO}_2\) emissions for luxury SUVs?

A World Health Organization (WHO) report gives average life expectancies for 10 randomly chosen countries in each of two regions of the world: Region 1: 63, 72, 48, 67, 68, 70, 59, 51, 63, 52 Region 2: 78, 73, 57, 69, 74, 86, 70, 66, 91, 77 (a) Create a back-to-back stemplot to display this data. (b) Compare the two distributions. (c) A two-sample t-test was performed on the above data with \(H_0: \mu_1 - \mu_2 = 0\) and \(H_a: \mu_1 - \mu_2 < 0\). Conditions for inference were met, the test statistic was \(t = -3.134\), and the P-value was 0.0029. Write a proper conclusion.

A random sample of five men were measured for waist sizes (in inches) and percent body fat, giving the following data.

Waist (inches)	32	35	36	39	44
Body Fat (%)	12	23	24	30	40

(a) Is there evidence of a positive linear relationship? You may refer to the following computer regression printouts. (b) Predict the percent body fat for a man with a 37-inch waist. (c) Give an approximate range for percent body fat for a man with a 37-inch waist.

Doctors are planning a clinical trial of a new "molecularly targeted therapy" for the treatment of leukemia. A random sample of children with leukemia will be treated. After a suitable time period, their disease remission rate will be compared with children being treated with standard chemotherapy for leukemia. (a) What are the null and alternative hypotheses for this significance test? (b) What is a Type I error in this context, and what is the consequence of committing this error? (c) What is a Type II error in this context, and what is the consequence of committing this error?

Executives at a company that manufactures a cooler are thinking of moving to the manufacture of a better cooler but only if a test shows that the new cooler will keep drinks cold for over 24 hours. The company's research department proposes a study with \(H_0: \mu = 24\) and \(H_a: \mu > 24\). (a) Describe a Type II error in context of this study, and describe a possible consequence. (b) Which significance level, \(\alpha = 0.05\) or \(\alpha = 0.10\), would result in a smaller probability of a Type II error? Explain. (c) The statistician in the research department estimates the power of the proposed test to be 0.85 if the true mean is \(\mu = 25\) hours. What does this mean in the context of this study?

In each of the following examples, explain why it is or is not proper to run a chi-square goodness-of-fit test. (a) The ingredient list on a jar of fancy mixed nuts indicates 10% Brazil nuts, 10% cashews, 20% almonds, and the rest peanuts. Suppose you separate and weigh the contents of a 24-ounce jar and find 3 ounces of Brazil nuts, 2 ounces of cashews, 4 ounces of almonds, and 15 ounces of peanuts. Does the jar follow the advertised pattern? (b) A company website claims that in a bag of its large candies, 10% are red, 20% are blue, 40% are yellow, and 30% are orange. Suppose you open a bag with 3 red, 4 blue, 2 yellow, and 10 orange candies. Does the bag follow the advertised pattern? (c) Genetic theory predicts that offspring in a certain experiment should appear in the ratio 1:3:9. Suppose the offspring number 66, 175, and 980, respectively. Is there sufficient evidence that the number of the dominant offspring (980) is greater than expected by the genetic model?

In a survey of all 880 students at a college, the number of times during the academic semester in which a student pulled an all-nighter is tabulated and shown in the following table. All-Nighters Pulled 0 1 2 3 4 5 Number of Students 280 210 175 150 50 15 (a) What are \(\mu\) and \(\sigma\) for all-nighters pulled during a semester for this population? (b) If a simple random sample (SRS) of size \(n = 50\) is taken from this population, describe the sampling distribution of \(\overline{x}\). (c) A college counselor believes that over the years, the number of all-nighters pulled by students during a semester are 0, 1, 2, 3, 4, 5 in the ratio 15:10:10:10:4:1. Perform an appropriate hypothesis test.

A random survey of 250 high school teachers classified job satisfaction versus subject taught with the following results. Happiness Level

Subject Taught	Unhappy	Mildly Happy	Very Happy
Math	10	25	30
English	20	55	45
History	20	25	20

(a) What is the probability that a randomly selected teacher from this sample teaches history and is very happy? (b) What is the probability that a person selected at random from this sample is mildly happy given that he/she teaches math? (c) What is the probability that a person selected at random from this sample teaches English given that he/she is unhappy? (d) Is there evidence of a relationship between happiness level and subject taught? Explain.

Random samples of high school students in 2014, 2016, and 2018 were anonymously surveyed about whether or not they had ever cheated on a quiz or test. The resulting counts are as shown in the table.

2014	2016	2018
Admitted to Cheating	78	67	51
Claimed to Never Having Cheated	44	40	59

What does a chi-square test of homogeneity indicate?

During the past year, the proportions of teens eating at Starbucks, Chick-fil-A, and Chipotle have been 0.3, 0.5, and 0.2, respectively. A random survey of 250 teens this past month indicates 71, 136, and 43 visits to Starbucks, Chick-fil-A, and Chipotle, respectively. (a) Is there evidence that the overall pattern of restaurant visits has changed? Give statistical justification for your answer. (b) A sales manager at Starbucks would like to know her company's proportion of this past month's restaurant visits with 95% confidence. What is the margin of error? (c) A sales manager at Chick-fil-A would like to know his company's proportion of last month's restaurant visits within a margin of error of ±0.03. What would be the underlying level of confidence?

A social studies teacher believes that high school students taking social studies classes are more likely than students taking other classes to read newspapers. She randomly picks 40 of her social studies students and 50 students from her other classes, questions them, and gathers the following data.

Read Newspapers	Don't Read Newspapers
Her Social Studies	37	13
Students
Her Other Students	15	35

(a) Are the intended and sampled populations the same or different, and how does this affect the study? (b) Discuss the appropriateness of using a chi-square test of homogeneity. (c) Discuss the appropriateness of using a z-test for differences of proportions. (d) Which would be most appropriate for a quick impression of the data: segmented bar charts, back-to-back stemplots, or parallel boxplots? Draw this visual display.

A random sample of 500 medical records of people who recently had major heart attacks is analyzed for numbers of died versus survived and for education level of the subjects. Some computer output is as follows. Observed (and Expected) Counts

No High School Degree	High School Degree	College Degree
Died	18 (9.9)	38 (42.1)	24 (28)
Survived	44 (52.1)	225 (220.9)	151 (147)

(a) Is there an association between whether or not people survive major heart attacks and their education levels? Do an appropriate statistical test. (b) Do the segmented bar charts give any additional insights into this study? Explain.

A random sampling of a professor's grades (A or B, C or D, F) in both intro and advanced courses yields the following data.

A, B	C, D	F
Intro	35	55	10
Advanced	22	15	3

(a) A chi-square test of independence gives \(\chi^2 = 4.749\). What is a proper conclusion? (b) Is there evidence that this professor gives a higher proportion of A, B grades in his advanced courses than in his intro courses? Give statistical justification for your answer.

500 patients took part in a clinical study using T-cell therapy to treat a particular lymphatic cancer. The following data show the number of these patients who survived for each given number of years.

Survival Years	0	1	2	3	4 or more
Observed Number of Patients	69	106	153	98	74

It is hypothesized that the distribution follows the Poisson distribution: \(P(x) = \dfrac{2^x e^{-2}}{x!}\) where \(e = 2.71828\) and \(x\) is the number of survival years. (a) Using this Poisson distribution, calculate the probabilities of 0, 1, 2, 3, and 4 or more survival years. (b) Using this Poisson distribution, calculate the expected number of patients out of 500 who will survive 0, 1, 2, 3, and 4 or more years. (c) Perform a goodness-of-fit test for this data and the given Poisson distribution.

In an educational study, there is interest in whether or not there is more variability in the mean verbal scores of 8-year-olds or of 3-year-olds. Summary statistics are given below.

Sample Size	Mean	StDev
Age 8	60	93.5	5.4
Age 3	60	78.6	2.7

To test \(H_0: \sigma_8 = \sigma_3\) versus \(H_a: \sigma_8 > \sigma_3\), the researchers will use the ratio of the sample standard deviations, \(\dfrac{s_8}{s_3}\), as a test statistic. (a) What is the observed test statistic from the data of this study? (b) A simulation was conducted to study the sampling distribution of this test statistic. For each repetition of the simulation, two random samples were generated using appropriate population means with identical standard deviations, and the ratio of the sample standard deviations was noted. A histogram of the simulated ratios is given below. Is there statistically significant evidence that there is more variability in the mean verbal scores of 8-year-olds than of 3-year-olds?