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Statistics The Art And Science Of Learning From Data 4th Global Edition Alan Agresti, Christine A. Franklin, Bernhard Klingenberg - Solutions
7.61 Standard deviation of a proportion Suppose x = 1 with probability p, and x = 0 with probability 11 - p2. Then, x is the special case of a binomial random variable with n = 1, so that = np(1 p) = p(1 - p). With n trials, using the formula /Vn for a standard deviation of a sample mean, explain
7.52 Simulating pizza preference with p 0.5 Access the Sampling Distribution of the Sample Proportion web app.a. For p = 0.5 and a sample of size n = 3, the graph of the sampling distribution is given below. (This is the graph corresponding to the table in Exercise 7.49, part b.) Compute the mean
7.51 Pizza preference with p 0.6 Refer to the previous two exercises but now suppose the population proportion in Boston preferring pizza A is p = 0.6 instead of p = 0.5. (Then, the eight possible outcomes (AAA, AAD, . . . , DDA, DDD) of a sample of size n = 3 are not equally likely, as in part b
7.49 Sampling distribution for small and large n The owners of Aunt Erma’s Restaurant in Boston plan an advertising campaign with the claim that more people prefer the taste of their pizza (which we’ll denote by A) than the current leading fast-food chain selling pizza (which we’ll denote by
7.10 Effect of n on sample proportion The figure illustrates two sampling distributions for sample proportions when the population proportion p = 0.50.a. Find the standard deviation for the sampling distribution of the sample proportion with (i) n = 100 and (ii) n = 1000.b. Explain why the sample
9.130 Refer to Exercise 8.128, “Randomized response,” in Chapter 8. Before carrying out the method described there, the class was asked to hypothesize or predict what they believe is the value for the population proportion of students who have had alcohol at a party. Use the class estimate for
9.128 Rejecting true H0? A medical researcher conducts a significance test whenever she analyzes a new data set.Over time, she conducts 100 independent tests.a. Suppose the null hypothesis is true in every case.What is the distribution of the number of times she rejects the null hypothesis at the
9.127 Standard error formulas Suppose you wanted to test H0: p = 0.50, but you had 0 successes in n trials. If you had found the test statistic by using the se = 2pn 11 - pn 2 >n designed for confidence intervals, show what happens to the test statistic. Explain why se0 = 2p011 - p02 >n is a more
9.126 True or false The power of a test = 1 - P1Type II error2.
9.125 True or false The P-value is the probability that H0 is true.
9.124 True or false When testing H0: m = 100 against Ha: m • 100, the probability of a Type II error decreases the further the true population mean m is from 100.
9.123 True or false For a fixed significance levela, the probability of a Type II error increases when the sample size increases.
9.122 True or false A 95% confidence interval for m = population mean IQ is (96, 110). So, in the test of H0: m = 100 against Ha: m • 100, the P@value 7 0.05.
9.121 True or false A study about the change in weight on a new diet reports P@value = 0.043 for testing H0: m = 0 against Ha: m • 0. If the authors had instead reported a 95% confidence interval for m, then the interval would have contained 0.
9.120 True or false If we reject H0: m = 0 in a study about change in weight on a new diet using a = 0.01, then we also reject it using a = 0.05.
9.119 True or false It is always the case that P1Type II error2 = 1 - P1Type I error2.
9.118 Multiple choice: Interpret P(Type II error) For a test of H0: m = 0 against Ha: m 7 0 based on n = 30 observations and using a = 0.05 significance level, P1Type II error2 = 0.36 at m = 4. Identify the response that is incorrect.a. At m = 5, P1Type II error2 6 0.36.b. If a = 0.01, then at m =
9.117 Multiple choice: Pollution Exercise 9.34 concerned an industrial plant that may be exceeding pollution limits.An environmental action group took four readings to analyze whether the true mean discharge of wastewater per hour exceeded the company claim of 1000 gallons.When we make a decision
9.116 Multiple choice: Probability of P-value When H0 is true in a t test with significance level 0.05, the probability that the P-value falls … 0.05a. equals 0.05.b. equals 0.95.c. equals 0.05 for a one-sided test and 0.10 for a twosided test.d. can’t be specified because it depends also on
9.115 Multiple choice: Small P-value The P-value for testing H0: m = 100 against Ha: m • 100 is 0.001. This indicates thata. There is strong evidence that m = 100.b. There is strong evidence that m • 100, since if m were equal to 100, it would be unusual to obtain data such as those observed.c.
9.114 Incorrectly posed hypotheses What is wrong with expressing hypotheses about proportions and means in a form such as H0: pn = 0.50 and H0: x = 0?
9.113 Interpret P-value One interpretation for the P-value is that it is the smallest value for the significance level a for which we can reject H0. Illustrate using the P-value of 0.057 from the previous exercise.
9.112 Bad P-value interpretations A random sample of size 1000 has x = 104. The significance level a is set at 0.05.The P-value for testing H0: m = 100 against Ha: m • 100 is 0.057. Explain what is incorrect about each of the following interpretations of this P-value and provide a proper
9.111 False-positive biopsy According to the October 15, 2015 release of the American Cancer Society, for women with a first mammography screening starting at 40 years, the estimated 10-year cumulative risk for a false-positive biopsy result with annual screening was (7.0% [95% CI, 6.1%–7.8%]).
9.110 More doctors recommend An advertisement by Company A says that three of every four doctors recommend pain reliever A over all other brands combined.a. If the company based this claim on interviewing a random sample of doctors, explain how it could use a significance test to back up the
9.109 Significance Explain the difference between statistical significance and practical significance. Make up an example to illustrate your reasoning.
9.108 Report P-value It is more informative and potentially less misleading to conclude a test by reporting and interpreting the P-value rather than by merely indicating whether you reject H0 at the 0.05 significance level. One reason is that a reader can then tell whether the result is significant
9.107 Why not accept H0? Explain why the terminology “do not reject H0” is preferable to “accept H0.”
9.106 Choosing a An alternative hypothesis states that a newly developed drug is better than the one currently used to treat a serious illness. If we reject H0, the new drug will be prescribed instead of the current one.a. Why might we prefer to use a smaller significance level than 0.05, such as
9.105 Overestimated effect When medical stories in the mass media report dangers of certain agents (e.g., coffee drinking), later research often suggests that the effects may not exist or are weaker than first believed. Explain how this could happen if some journals tend to publish only
9.104 Curcumin and muscle soreness Turmeric (Curcuma longa) is a spice used in many systems of medicine as an anti-inflammatory agent to treat a wide variety of conditions.Studies have shown curcumin from turmeric can reduce muscle soreness after exercising. However, a more recent study (Source:
9.103 Subgroup lack of significance A crossover study on comparing a magnetic device to placebo for reducing pain in 54 people suffering from low back or knee pain(neuromagnetics.mc.vanderbilt.edu/publications) reported a significant result overall, the magnetic device being preferred to placebo.
9.102 No significant change and P-value An article in a journal states that “no statistical significant change was found in the mean glucose level of the brain before and after making a cell phone call for 50 minutes 1P@value = 0.632.” In practical terms, how would you explain to someone who
9.101 Two-sided or one-sided? A medical researcher gets a P-value of 0.056 for testing H0: m = 0 against Ha: m • 0.Since he believes that the true mean is positive and is worried that his favorite journal will not publish the results because they are not “significant at the 0.05 level,”he
9.100 Statistics and scientific objectivity The president of the American Statistical Association stated, “Statistics has become the modern-day enforcer of scientific objectivity.Terms like randomization, blinding, and 0.05 significance wield a no-doubt effective objectivity nightstick.” He
9.99 NBA home court advantage In 1976–1977, the home team won 68.5% of all the games played. In 2002–03, it was 62.8%. Since then, the home team’s win percentage has fallen steadily to 53.7 in 2014–15 where 1230 games had been played during the regular season (www.espn.com).Analyze the data
9.98 Gender of best friend A GSS question asked the gender of your best friend. Of 1381 people interviewed, 147 said their best friend had the opposite gender, and 1234 said their best friend had the same gender. Prepare a short report in which you analyze these data using a confidence interval and
9.97 Class data Refer to the data file your class created in Activity 3 at the end of Chapter 1. For a variable chosen by your instructor, conduct inferential statistical analyses.Prepare a report, summarizing and interpreting your findings. In this report, also use graphical and numerical methods
9.96 Student data Refer to the FL Student Survey data file on the text’s website. Test whether the (a) population mean political ideology (on a scale of 1 to 7, where 4 = moderate) equals or differs from 4.0 and (b) population proportion favoring affirmative action equals or differs from 0.50.
9.95 P(Type II error) with smaller n Consider Example 12 about testing H0: p = 1>3 against Ha: p 7 1>3 for the astrology study, with n = 116. Find P(Type II error) for testing H0: p = 1>3 against Ha: p 7 1>3 when actually p = 0.50, if the sample size is 60 instead of 116. Do this by showing thata.
9.94 Legal trial errors Consider the analogy discussed in Section 9.4 between making a decision about a null hypothesis in a significance test and making a decision about the innocence or guilt of a defendant in a criminal trial.a. Explain the difference between Type I and Type II errors in the
9.93 How to reduce chance of error? In making a decision in a significance test, a researcher worries about rejecting H0 when it may actually be true.a. Explain how the researcher can control the probability of this type of error.b. Why should the researcher probably not set this probability equal
9.92 Net migration of EU citizens The Office for National Statistics is the United Kingdom’s largest independent producer of official statistics. According to its findings, the net migration of EU citizens to the UK was estimated to be 184,000 in the year ending December 2015 as compared to
9.91 CI and test connection The P-value for testing H0: m = 100 against Ha: m • 100 is 0.043.a. What decision is made using a 0.05 significance level?b. If the decision in part a is in error, what type of error is it?c. Does a 95% confidence interval for m contain 100?Explain.
9.90 CI and test Refer to the previous exercise.a. For which significance levels can you reject H0?(i) 0.10, (ii) 0.05, or (iii) 0.01.b. Based on the answers in parta, for which confidence levels would the confidence interval contain 500?(i) 0.90, (ii) 0.95, or (iii) 0.99.c. Use part a and part b
9.89 Wage claim false? Management claims that the mean income for all senior-level assembly-line workers in a large company equals $500 per week. An employee decides to test this claim, believing that it is actually different from $500. For a random sample of nine employees, the incomes are 430,
9.88 Catalog sales A company that sells products through mail-order catalogs wants to evaluate whether the mean sales for its most recent catalog were different from the mean of $15 from past catalogs. For a random sample of 100 customers, the mean sales were $10, with a standard deviation of $10.
9.87 Tennis balls in control? When it is operating correctly, a machine for manufacturing tennis balls produces balls with a mean weight of 57.6 grams. The last eight balls manufactured had weights 57.3, 57.4, 57.2, 57.5, 57.4, 57.1, 57.3, 57.0a. Using a calculator or software, find the test
9.86 Increasing blood pressure In the previous exercise, suppose you had predicted that if the mean changed, it would have increased above the control value. State the alternative hypothesis for this prediction and report and interpret the P-value.
9.85 Blood pressure When Vincenzo Baranello’s blood pressure is in control, his systolic blood pressure reading has a mean of 130. For the last six times he has monitored his blood pressure, he has obtained the values 140, 150, 155, 155, 160, 140.a. Does this provide strong evidence that his true
9.84 Females liberal or conservative? Example 10 compared mean political beliefs (on a 1 to 7 point scale) to the moderate value of 4.0, using GSS data. Test whether the population mean equals 4.00 for females, for whom the sample mean was 4.06 and standard deviation was 1.37 for a sample of size
9.81 Practice steps of test for mean For a quantitative variable, you want to test H0: m = 0 against Ha: m • 0. The 10 observations are 3, 7, 3, 3, 0, 8, 1, 12, 5, 8.a. Show that (i) x = 5.0, (ii) s = 3.71, (iii) standard error = 1.17, (iv) test statistic = 4.26, and (v) df = 9.b. The P-value is
9.79 Type I and Type II errors Refer to the previous exercise.a. Explain what Type I and Type II errors mean in the context of that exercise.b. If you made an error with the decision in partd, is it a Type I or a Type II error?
9.78 Jurors and gender A jury list contains the names of all individuals who may be called for jury duty. The proportion of the available jurors on the list who are women is 0.53.If 40 people are selected to serve as candidates to serve on the jury, show all steps of a significance test of the
9.76 Start a hockey team A fraternity at a university lobbies the administration to start a hockey team. To bolster its case, it reports that of a simple random sample of 100 students, 83% support starting the team. Upon further investigation, their sample has 80 males and 20 females.Should you be
9.75 Zika virus According to a study, 358 travel-associated cases of Zika virus were reported in the United States between January 1, 2015 and April 13, 2016. During January 3–March 5, 2016, 4534 persons who traveled to or moved from areas with active Zika virus transmission were tested for
9.74 Plant inheritance In an experiment on chlorophyll inheritance in maize (corn), of the 1103 seedlings of self-fertilized green plants, 854 seedlings were green and 249 were yellow.Theory predicts the ratio of green to yellow is 3 to 1. Show all five steps of a test of the hypothesis that 3 to 1
9.73 Majority supports gay marriage An Associated Press article appearing in the New York Times in March 2015 mentioned that “In the late 1980s, support for gay marriage was essentially unheard of in America. Just a quarter century later, it’s now favored by a clear majority of Americans.”
9.71 2016 Irish Exit Poll The 2016 General Election Exit Poll was conducted exclusively by Ipsos MRBI, on behalf of The Irish Times. It took place among a national sample of 5260 voters across 200 polling stations throughout all constituencies in the Republic of Ireland. For two candidates, Fine
9.70 Free-throw accuracy Consider all cases in which a pro basketball player shoots two free throws and makes one and misses one. Which do you think is more common:making the first and missing the second, or missing the first and making the second? One of the best shooters was Larry Bird of the
9.69 ESP A person who claims to possess extrasensory perception(ESP) says she can guess more often than not the outcome of a flip of a balanced coin. Out of 30 flips, she guesses correctly 18 times. Would you conclude that she truly has ESP? Answer by reporting all five steps of a significance test
9.68 Write H0 and Ha For each of the following scenarios, define the parameter of interest and appropriate null and alternative hypotheses.a. To motivate more fathers to take a parental leave, the government of Austria extended child care benefits for two more months if the father stays home for
9.67 H0 or Ha? For each of the following hypotheses, explain whether it is a null hypothesis or an alternative hypothesis:a. For females, the population mean on the political ideology scale is equal to 4.0.b. For males, the population proportion who support the death penalty is larger than 0.50.c.
9.66 Exploring Type II errors Refer to the web app from Activity 2 at the end of this section, now assuming that we are using the two-sided test H0: p = 0.33 against Ha: p • 0.33. (Select “not equal” for the type of the alternative hypothesis in the web app.)a. Explain the effect of
9.65 Power for knee osteoarthritis treatment An ankle-foot orthosis (AFO) is a specially designed brace to support and improve the function of the foot and ankle. A 2016 study on the treatment of knee osteoarthritis investigated the biomechanical effects of the Agilium Freestep AFO on the lever arm
9.64 Type II error with two-sided Ha In Example 12 for testing H0: p = 1>3 (astrologers randomly guessing) with n = 116 when actually p = 0.50, suppose we used Ha: p • 1>3.Then show that:a. A Type II error occurs if 0.248 6 pn 6 0.419.b. The probability is 0.00 that pn 6 0.248 and 0.96 that pn 7
9.63 P (Type II error) increase Consider the hypothesis H0: p = 0.3 against Ha: p 7 0.3 for testing at a 5%significance level.a. Assume n = 200, find P(Type II error) when (i)p = 0.4 and (ii) p = 0.35b. Assume n = 100, find P(Type II error) when (i)p = 0.4 and (ii) p = 0.35c. Explain intuitively
9.62 Balancing Type I and Type II errors Recall that for the same sample size, the smaller the probability of a Type I error,a, the larger the P(Type II error). Let’s check this for Example 12. There we found P(Type II error) for testing H0: p = 1>3 (astrologers randomly guessing) against Ha: p 7
9.61 Gender bias in selecting managers Exercise 9.20 tested the claim that female employees were passed over for management training in favor of their male colleagues.Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is 40% female and 60% male.
9.60 Find P(Type II error) A study is designed to test H0: p = 0.50 against Ha: p 7 0.50, taking a random sample of size n = 100, using significance level 0.05.a. Show that the rejection region consists of values of pn 7 0.582.b. Sketch a single picture that shows (i) the sampling distribution of
9.59 Selective reporting and p-hacking In the webcomic on the link http://xkcd.com/882/, a girl claims that jelly beans cause acne. Scientists investigate and find no link between the two (p > 0.05). They are asked to check if jelly beans of a particular color cause acne. They test 20 different
9.58 How many medical discoveries are Type I errors? Refer to Example 11. Using a tree diagram, given that H0 is rejected, approximate P(Type I error) under the assumption that a true effect exists 20% of the time and that there’s a 30%chance of a Type II error.
9.57 Selective reporting In 2004, New York Attorney General Eliot Spitzer filed a lawsuit against GlaxoSmithKline pharmaceutical company, claiming that the company failed to publish results of one of its studies that showed that an antidepressant drug (Paxil) may make adolescents more likely to
9.56 Fishing for significance A marketing study conducts 60 significance tests about means and proportions for several groups. Of them, 3 tests are statistically significant at the 0.05 level. The study’s final report stresses only the tests with significant results, not mentioning the other 57
9.55 Effect of n Example 10 analyzed political conservatism and liberalism in the United States. Suppose that the sample mean of 4.09 and sample standard deviation of 1.43 were from a sample size of only 25, rather than 2449.a. Find the test statistic.b. Find the P-value for testing H0: m = 4.0
9.54 Practical significance A study considers whether the mean score on a college entrance exam for students in 2010 is any different from the mean score of 500 for students who took the same exam in 1985. Let m represent the mean score for all students who took the exam in 2010. For a random
9.53 Misleading summaries? Two researchers conduct separate studies to test H0: p = 0.50 against Ha: p • 0.50, each with n = 400.a. Researcher A gets 220 observations in the category of interest, and pn = 220>400 = 0.550 and test statistic z = 2.00. Show that the P@value = 0.046 for Researcher
9.52 Which error is worse? Which error, Type I or Type II, would usually be considered more serious for decisions in the following tests? Explain why.a. A trial to test a murder defendant’s claimed innocence, when conviction results in the death penalty.b. A medical diagnostic procedure, such as
9.51 Detecting pregnancy Home pregnancy tests claim an accuracy rate of over 99%. A positive test does turn out later to be a false positive. There are several reasons why a woman may obtain a positive result in a home pregnancy test when she is not actually pregnant.a. For the home pregnancy test,
9.49 Errors in medicine Consider the test of H0: The new drug is safe against Ha: the new drug is not safe.a. Explain, in context, the conclusion of the test if H0 is rejected.b. Describe, in context, a Type I error.c. Explain, in context, the conclusion of the test if you fail to reject H0.d.
9.48 Errors in the courtroom Consider the test of H0: The defendant is not guilty against Ha: The defendant is guilty.a. Explain, in context, the conclusion of the test if H0 is rejected.b. Describe, in context, a Type I error.c. Explain, in context, the conclusion of the test if you fail to reject
9.47 Anorexia decision Refer to the previous exercise. When we test H0: m = 0 against Ha: m 7 0, we get a P-value of 0.02.a. What would the decision be for a significance level of 0.05? Interpret in context.b. If the decision in part a is in error, what type of error is it?c. Suppose the
9.46 Anorexia errors Example 8 tested a therapy for anorexia, using hypotheses H0: m = 0 and Ha: m • 0 about the population mean weight change m. In the words of that example, what would be (a) a Type I error and(b) a Type II error?
9.45 Fracking errors Example 6, in testing H0: p = 0.5 against Ha: p 6 0.5, analyzed whether those opposing increased use of fracking are in the minority. In the words of that example, what would be (a) a Type I error and (b) a Type II error?
9.44 Error probability A significance test about a mean is conducted using a significance level of 0.05. The test statistic equals 10.52. The P-value is 0.003.a. If H0 was true, for what probability of a Type I error was the test designed?b. If the P-value was 0.3 and the test resulted in a
9.43 Dr. Dog In the experiment in Example 4, we got a P@value 6 0.001 for testing H0: p = 1>7 about dogs’ ability to diagnose urine from bladder cancer patients.a. For the significance level 0.05, what decision would you make?b. If you made an error in parta, what type of error was it? Explain
9.42 Test and CI Results of 99% confidence intervals are consistent with results of two-sided tests with which significance level? Explain the connection.
9.40 Anorexia in teenage girls Example 8 described a study about various therapies for teenage girls suffering from anorexia. For each of 17 girls who received the family therapy, the changes in weight were 11, 11, 6, 9, 14, -3, 0, 7, 22, -5, -4, 13, 13, 9, 4, 6, 11.a. Plot these data with a dot
9.39 Assumptions important? Refer to the previous exercise.a. Explain how the result of the 95% confidence interval shown in the table agrees with the test decision using the 0.05 significance level.b. Suppose you instead wanted to perform a one-sided test because the study predicted that the
9.37 Too little or too much wine? Wine-pouring vending machines, previously available in Europe and international airports, have become popular in the past few years in the United States. They are even approved to dispense wine in some Walmart stores. The available pouring options are a 5-ounce
9.35 Weight change for controls A disadvantage of the experimental design in Example 8 on weight change in anorexic girls is that girls could change weight merely from participating in a study. In fact, girls were randomly assigned to receive a therapy or to serve in a control group, so it was
9.34 Lake pollution An industrial plant claims to discharge no more than 1000 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental action group decides to monitor the plant in case this limit is being exceeded. Doing so is expensive, and only a small sample is
9.33 StatCrunch for statistics For effective learning, an instructor advised his students to practice solving statistical problems on StatCrunch, a web-based software, at least 7 hours per week. In an in-class activity, a student surveyed 15 of her 35 classmates to assess the number of hours they
9.32 Female work week When the 583 female workers in the 2012 GSS were asked how many hours they worked in the previous week, the mean was 37.0 hours, with a standard deviation of 15.1 hours. Does this suggest that the population mean work week for females is significantly different from 40 hours?
9.30 Effect of n Refer to the previous exercise. If the same sample mean and standard deviation had been based on n = 5 instead of n = 20, the test statistic would have been t = 1.20. Would the P-value for Ha: m • 100 be larger, or smaller, than when t = 2.40?Why?
9.29 Practice mechanics of a t test A study has a random sample of 20 subjects. The test statistic for testing H0: m=100 is t = 2.40. Find the approximate P-value for the alternative,(a) Ha: m • 100, (b) Ha: m 7 100, and (c) Ha: m 6 100.(Among others, you can use the t Distribution web app to
9.28 Which t has P@value • 0.05? A t test for a mean uses a sample of 15 observations. Find the t test statistic value that has a P-value of 0.05 when the alternative hypothesis is (a)Ha: m • 0, (b) Ha: m 7 0, and (c) Ha: m 6 0. (Among others, you can use the t Distribution web app to find the
9.27 P-value for small samples Example 4, on whether dogs can detect bladder cancer by selecting the correct urine specimen (out of seven), used the normal sampling distribution to find the P-value. The normal distribution P-value approximates a P-value using the binomial distribution.That binomial
9.26 A binomial headache A null hypothesis states that the population proportion p of headache sufferers who have better pain relief with aspirin than with another pain reliever equals 0.50. For a crossover study with 10 subjects, all 10 have better relief with aspirin. If the null hypothesis were
9.25 How to sell a burger A fast-food chain wants to compare two ways of promoting a new burger (a turkey burger). One way uses a coupon available in the store.The other way uses a poster display outside the store.Before the promotion, its marketing research group matches 50 pairs of stores. Each
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