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mathematics
statistics the art and science
Statistics The Art And Science Of Learning From Data 4th Edition Alan Agresti, Christine A. Franklin, Bernhard Klingenberg - Solutions
Which of the following is always true?a. If A and B are independent, then they are also disjoint.b. P(AΙB) + P(AΙBc) = 1c. If P(AΙB) = P(BΙA), then A and B are independent.d. If A and B are disjoint, then A and B cannot occur at the same time.
Choose ALL correct responses. For two events A and B, P(A) = 0.5 and P(B) = 0.2. Then P(A or B) equalsa. 0.10, if A and B are independentb. 0.70, if A and B are independentc. 0.60, if A and B are independentd. 0.70, if A and B are disjoint
Of the participants at a conference, 50% attended breakfast, 90% attended dinner, and 40% attended both breakfast and dinner. Given that a participant attended breakfast, find the probability that she also attended dinner.
A couple has five children, all girls. They are expecting a sixth child. The father tells a friend that by the law of large numbers the chance of a boy is now much greater than 1/2. Comment on the father’s statement.
In Example 16, we estimated the probability of winning the game was 0.65.a. If you conducted another 20 simulations of this game, what probability of winning would you expect to get?b. The simulation in the example consisted of 20 repetitions. Conduct another 80 repetitions, for a total of 100.
A standard deck of poker playing cards contains four suits (clubs, diamonds, hearts, and spades) and 13 different cards of each suit. During a hand of poker, 5 of the 52 cards have been exposed. Of the exposed cards, 3 were diamonds. Tony will have the opportunity to draw two more cards, and he has
Petra Kvitova of the Czech Republic won the 2014 Wimbledon Ladies’ Singles Championship. In the final game against Eugenie Bouchard of Canada she had 41 first serves, of which 28 were good, and three double faults.a. Find the probability that her first serve is good.b. Find the conditional
Are people happy in their romantic relationships? The table shows results from the 2012 General Social Survey for adults classified by gender and happiness.a. Estimate the probability that an adult is very happy in his or her romantic relationship.b. Estimate the probability that an adult is very
A 2014 Gallup poll reported that 27% of people in the United States are obese (having a body mass index score of 30 or more). Blacks have the highest obesity rate at 35%, whereas Asians have the lowest, at 9%.a. Of the three percentages (estimated probabilities) reported, which are conditional?
Table 5.3 on audit status and income follows. Show how to find the probability of:a. Being audited, given that the taxpayer is in the lowest income category.b. Being in the lowest income category, given that the taxpayer is audited. Audited Income No Yes $1mil 0.0003
A 2014 article from Business Insider (http://www.business insider.com/department-of-justice report- shows-high-recidivism-rate-2014-4) discusses recidivism rates in the United States. Recidivism is defined as being reincar cerated within five years of being sent to jail initially. Among the data
You are the director of newspaper sales for the local paper. Each customer has signed up for either weekday delivery or weekend delivery. You record whether he or she received the delivery as Y for yes and N for no. The probabilities of the customer receiving the newspaper are as follows.a. Display
At the local cell phone store, the probability that a customer who walks in will purchase a new cell phone is 0.2. The probability that the customer will purchase a new cell phone protective case is 0.25. Is this information sufficient to determine the probability that a customer will purchase a
Every year the insurance industry spends considerable resources assessing risk probabilities. To accumulate a risk of about one in a million of death, you can drive 100 miles, take a cross country plane flight, work as a police officer for 10 hours, work in a coal mine for 12 hours, smoke two
Some friends are playing a game in which two six-sided dice are rolled (craps, Settlers of Catan, etc.). One of them exclaims, “Since the sum of the dice is between 2 and 12, there is a one in eleven chance that each result will occur.” Do you agree or disagree with his assessment of the
Use the web app Random Numbers (go to the tab that says Coin Flips) on the book’s website or other software (such as random.org/coin) to illustrate the long-run definition of probability by simulating short term and long-term results of flipping a balanced coin. a. Keep the probability of a
You are asked to use your best judgment to estimate the probability that there will be a nuclear war within the next 10 years. Is this an example of relative frequency or subjective definition of probability? Explain.
Before the first human heart transplant, Dr. Christiaan Barnard of South Africa was asked to assess the probability that the operation would be successful. Did he need to rely on the relative frequency definition or the subjective definition of probability? Explain.
A pollster agency wants to estimate the proportion of citizens of the European Union who support same-sex unions. She claims that if the sample size is large enough, she does not need to worry about the method of selecting the sample. Do you agree with the pollster’s comments? Explain.
Consider a random number generator designed for equally likely outcomes. Which of the following is not correct, and why?a. For each random digit generated, each integer between 0 and 9 has probability 0.10 of being selected.b. If you generate 10 random digits, each integer between 0 and 9 must
The powerrank.com website (http:// thepowerrank.com/2014/06/06/world-cup-2014-winprobabilities- from-the-power-rank/) listed the probability of each team to win the 2014 World Cup in soccer as follows:1. Brazil, 35.9%.2. Argentina, 10.0%.3. Spain, 8.9%.4. Germany, 7.4%.5. Netherlands, 5.7%.6.
The American Heart Association reports that a total cholesterol score of 240 or higher represents high risk of heart disease. A study of postmenopausal women reported a mean of 220 and standard deviation of 40. If the total cholesterol scores have a normal distribution, what proportion of the women
Exercise 6.28 mentioned that in the United States, birth weight of boys is approximately normal, with mean 3.41 kg and standard deviation 0.55 kg. For girls, the birth weight is also approximately normal with mean 3.29 kg and standard deviation 0.52 kg.a. A weight below 2.5 kg is considered low
A collector is interested in two paintings by the same artist available at an auction. She plans to bid $3,000 for the first painting auctioned off and $2,000 for the second. She estimates that the probability she will win the bid for the first painting (and spend $3,000) is 30%. If she gets that
Four of the 20 students (20%) in a class are fraternity or sorority members. Five students are picked at random. Does X = the number of students in the sample who are fraternity or sorority members have the binomial distribution with n = 5 and p = 0.20? Explain why or why not.
For the binomial distribution, the number of trials n is a fixed number. Let X denote the number of girls in a randomly selected family in Canada that has three children. Let Y denote the number of girls in a randomly selected family in Canada (that is, the number of children could be any number).
A study at the Harvard School of Public Health found that 44% of 10,000 sampled college students were binge drinkers. A student at the University of Minnesota plans to estimate the proportion of college students at that school who are binge drinkers. How large a random sample would she need to
Go to the Inference for a Mean web app, accessible on the book’s website. Select the Explore Coverage tab and keep the default skewed shape for the population distribution as well as the default confidence level of 95%. Set the sample size to 30 and generate 100 random samples by selecting 100
Find the margin of error for estimating the population mean when the sample standard deviation equals 100 for a sample size of 25, using confidence levels (i) 95% (ii) 99%. What is the effect of the choice of confidence level? (You can use Table B in the back to find the appropriate
Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100, with a sample size of a. 25b. 100. What is the effect of increasing the sample size? (You can use Table B in the back to find the appropriate t-scores.)
The General Social Survey asks respondents to rate their political views on a seven-point scale, where 1 = extremely liberal, 4 = moderate, and 7 = extremely conservative. A researcher analyzing data from the 2012 GSS obtains MINITAB output:a. Show how to construct the confidence interval from the
For the FL Student Survey data file on the book€™s website, software reports the results for responses on the number of times a week the subject reads a newspaper:a. Is it plausible that μ = 7, where μ is the population mean for all Florida students? Explain.b.
For the question about e-mail in the previous exercise, the 14 females in the GSS of age at least 80 had the responses 0, 0, 0, 0, 1, 1, 1, 2, 2, 6, 6, 7, 7, 10.a. Using the web app, software or a calculator, find the sample mean and standard deviation and the standard error of the sample mean.b.
When the GSS asked n = 1050 people in 2012, €œAbout how many hours per week do you spend sending and answering e-mail?€ (EMAILHR), the summary statistics were xÌ… = 6.89 and s = 13.05. TI output with these data (available on the book€™s website) is shown
Example 6 mentioned closing prices for listings of the iPhone 5s on eBay. If you don€™t feel comfortable bidding (or can€™t wait until a listing has ended), you can often purchase the item right away at the indicated Buy It Now price. A random sample of Buy It Now prices for an
Exercise 8.7 reported heights (in mm) of 55.5, 60.3, 60.6, 62.1, 65.5, and 69.2 for six seedlings fourteen days after germination.a. Using the web app, software or a calculator, verify that the 95% confidence interval for the population mean is (57.3, 67.1).b. Name two things you could do to get a
One feature smartphone manufacturers use in advertising is the amount of time one can talk before recharging the battery. Below are 13 values from a random sample of the talk-time (in minutes) of smartphones running on lithium-ion batteries. The summary statistics are x̅ = 553, s = 227, Q1 = 420,
Using Table B, the web app, software or a calculator, report the t-score that you multiply by the standard error to form the margin of error for aa. 95% confidence interval for a mean with 5 observations.b. 95% confidence interval for a mean with 15 observations.c. 99% confidence interval for a
Refer to the previous exercise. For the 530 males in the sample who gave a numeric response from 0 to 6, the mean was 2.51 and the standard deviation was 0.87.a. Find the point estimate of the population mean and show that its standard error is 0.038.b. The 95% confidence interval is 2.43 to 2.59.
The 2012 General Social Survey asked, “What do you think is the ideal number of children for a family to have?” The 590 females who gave a numeric response from 0 to 6 had a median of 2, mean of 2.56, and standard deviation of 0.84.a. What is the point estimate of the population mean?b. Find
Repeat the simulation from Activity 2, but this time simulate 1000 confidence intervals (repeatedly press the Draw Sample(s) button after selecting 100 as the number of samples to generate until you get 1000 simulations) with p = 0.5 instead of p = 0.3. Do this for a confidence level of 95% and
A national survey by the Pew Research Center (www.people-press.org), conducted May 1-5, 2013, among 1,504 adults, finds that support for same-sex marriage continues to grow. It states, “For the first time in Pew Research Center polling, just over half (51%) of Americans favor allowing gays and
In a 2014 Consumer Reports article titled, “The High Cost of Cheap Chicken,” the magazine reported that out of 316 chicken breasts bought in retail stores throughout the United States, 207 contained E. coli bacteria. a. Find and interpret a 99% confidence interval for the population
In Exercise 8.18, it was stated that 72% of the 2113 adults surveyed in a Harris poll believed that stem cell research has merit. In the same study, only 58% of those describing themselves as Republicans believed that it has merit. There were 1010 people who described themselves as Republican. The
A Harris poll of a random sample of 2250 adults in the United States in 2013 reported that 74% believe in God, 68% believe in heaven, 58% believe in hell, and 42% believe in ghosts.4 (Interestingly, except for the belief in ghosts, these numbers are all substantially smaller than the corresponding
Which z-score is used in a(a) 90%, (b) 98%,(c) 99.9% confidence interval for a population proportion?
In the 2012 General Social Survey, respondents were asked whether they favored or opposed the death penalty for people convicted of murder. Software shows results Here, X refers to the number of the respondents who were in favor.a. Show how to obtain the value reported under “Sample p.”b.
In a clinical study (the same as mentioned in Example 4), 3900 subjects were vaccinated with a vaccine manufactured by growing cells in fertilized chicken eggs. Over a period of roughly 28 weeks, 24 of these subjects developed the flu.a. Find the point estimate of the population proportion that
A Gallup poll of 2000 Russians taken between April and June 2014 (after the Olympic games in Russia and the annexation of the Crimean peninsula) showed that 83% approved of President Putin’s performance. If random sampling were used for this survey, what is the margin of error for this estimate
Conduct a search to find a newspaper or magazine article that reported a margin of error. If you prefer, use the Internet for the search.a. Specify the population parameter, the value of the sample statistic, the point estimate, and the size of the margin of error.b. Explain how to interpret the
The Google Play app store for smartphones offers hundreds of games to download for free or for a small fee. The ones for which a fee is charged are called paid games. For a random sample of five paid games taken in July 2014 on the Google platform, the following fees were charged: $1.09, $4.99,
In 2014, news reports worldwide alleged that the U.S. government had hacked German chancellor Angela Merkel’s cell phone. A Pew Research Center survey of German citizens at about that time asked whether they find it acceptable or unacceptable for the U.S. government to monitor communications from
A recent study from Nielsen (available from nielsen.com; search for “The Cross-Platform Report”) took a sample to investigate how video on demand and other subscription-based services (such as Netflix, Amazon Prime, and Hulu) change TV viewing behavior of U.S. adults. Among others, the study
The formula σ/ˆšn for the standard deviation of x actually is an approximation that treats the population size as infinitely large relative to the sample size n. The exact formula for a finite population size N isThe term ˆš(N - n)/(N - 1) is called the finite
Suppose x = 1 with probability p, and x = 0 with probability (1 - p). Then, x is the special case of a binomial random variable with n = 1, so that With n trials, using the formula s>1n for a standard deviation of a sample mean, explain why the standard deviation of a sample proportion
Under the null hypothesis H0: p1= p2of equal population proportions, the standard error for the difference in the two sample proportions mentioned on page 494 reduces to se0= ˆš(b + c)/n2, where b and c are the off-diagonal elements and n is the number of subjects.a. Show that the null
Refer to Example 12, which compared two groups of seven dogs each in terms of their time interacting with their owners. Figure 10.10 showed the sampling distribution by considering all possible permutations. When n1 = n2, as here, explain why the permutation distribution based on all possible
Suppose there is a higher percentage of successes with Treatment A than with Treatment B at a clinic in Rochester, and there is a higher percentage of successes with Treatment A than with Treatment B at a clinic in Syracuse. For the overall sample (combining results for the two cities), there must
A 2013 survey by the Pew Research Center asked whether there have been times in the past year the respondent has been unable to afford food. Of advanced economies, the country with the second highest response was the United States, 24%. Worldwide, the highest response was in Uganda, 70%. Because
If a 95% confidence interval for (μ1 - μ) contains only positive numbers, then we can conclude that both μ1 and μ2 are positive.
If the sample proportions in Example 4 comparing cancer death rates for aspirin and placebo had sample sizes of only 1000 each, rather than about 11,000 each, then the 95% confidence interval for (p1 - p2) would be (-0.007, 0.021) rather than (0.003, 0.011). This reflects thata. When an effect is
A study compares the population mean annual incomes for Hispanics (μ1) and for whites (μ2) having jobs in construction, using a 95% confidence interval for μ1 - μ2. Choose the best correct choice.a. If the confidence interval is (-6000, -3000), then at this confidence level we conclude that the
Suppose that a 99% confidence interval for the difference p1 - p2 between the proportions of men and women in California who are alcoholics equals (0.02, 0.09). Choose the best correct choice.a. We are 99% confident that the proportion of alcoholics is between 0.02 and 0.09.b. We are 99% confident
In Western Australia, handheld cell phone use while driving has been banned since 2001, but hands-free devices are legal. A study (published in the British Medical Journal in 2005) of 456 drivers in Perth who had been in a crash observed if they were using a cell phone before the crash and if they
The following data refer to a random sample of prize money earned by male and female skiers racing in the 2014/2015 FIS world cup season (in Swiss Franc).Males: 89000, 179000, 8820, 12000, 10750, 66000, 6700, 3300, 74000, 56800Females: 73000, 95000, 32400, 4000, 2000, 57100, 4500Enter the
In the previous exercise, explain how the result of the 95% confidence interval in part b corresponds to the result of the decision using significance level 0.05 in part c.Previous exerciseIn 2014, the General Social Survey asked about the number of hours a week spent on the World Wide Web
Refer to the previous exercise.a. Run a t test and report the P-value for the two-sided alternative hypothesis.b. Using a significance level of 0.10, are the decisions based on the t test and based on the permutation test comparable? Which one would you trust more?c. Remove the largest observations
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: μ ≠ 0, b) Ha: μ > 0, c) Ha: μ < 0.Among others, you can use the t Distribution web app to find the answer.
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? Answer by:a.
Many students brag that they have more than 100 friends on Facebook. For a class project in 2014, a group of students asked a random sample of 13 students at Williams College who used Facebook about their number of friends and got the following data:55, 130, 210, 75, 80, 205, 20, 130, 150, 50, 270,
In Exercise 9.25, a fast-food chain compared two ways of promoting a turkey burger. In a separate experiment with 10 pairs of stores, the difference in the month€™s increased sales between the store that used coupons and the store with the outside poster had a mean of $3000. Does this
A significance test about a proportion is conducted using a significance level of 0.05. The test statistic equals 2.58. The P-value is 0.01.a. If H0 were true, for what probability of a Type I error was the test designed?b. If this test resulted in a decision error, what type of error was it?
Example 6, in testing H0: p = 0.5 against Ha: p < 0.5, analyzed whether those opposing increased use of fracking are in the minority. In the words of thatexample, what would be a) A Type I errorb) A Type II error?
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 H0.d. Describe, in context,
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. Describe, in context, a Type
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 tests. What is misleading about
For testing H0: p = 1/3 (astrologers randomly guessing) against Ha: p > 1/3 with n = 116, Example 12 showed that P(Type II error) = 0.02 when p = 0.50. Now suppose that p = 0.35. Recall that P(Type I error) = 0.05.a. Show that P(Type II error) = 0.89.b. Explain intuitively why P(Type II error)
Consider Example 13 about the power of the test of H0: p = 0.1 against Ha: p > 0.1 used to test the effectiveness of a manual therapy in opening blocked tubes, where p is the probability of at least one unblocked tube after therapy. Another statement in that grant proposal read as follows:
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 increasing the sample size on the
When the 2010 General Social Survey asked, €œWould you be willing to pay much higher taxes in order to protect the environment?€ (variable GRNTAXES), 459 people answered yes and 626 answered no. (We exclude those who made other responses.) Let p denote the population proportion
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.” The article supports this conclusion
Is the ideal number of children equal or different from 2? For testing that the mean response from the 2014 GSS equals 2.0 for the question, €œWhat do you think is the ideal number of children to have?€ software shows results:a. Report the test statistic value and show how it
When all subjects in the 2012 GSS who were working full- or part-time were asked how many hours they worked in the previous week at all jobs (variable HRS1), software produced the following analyses:For this printout,a. State the hypotheses.b. Explain how to interpret the values of (i) SE
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 1345. Carry out the five steps of a
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. However, when the analysis was done
What is wrong with expressing hypotheses about proportions and means in a form such as H0: p̂ = 0.50 and H0: x̅ = 0?
The P-value for testing H0: μ = 100 against Ha: μ ≠ 100 is 0.001. This indicates thata. There is strong evidence that μ = 100.b. There is strong evidence that μ ≠ 100, since if μ were equal to 100, it would be unusual to obtain data such as those observed.c. The probability that μ = 100
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 two sided test.d. Can’t be specified because it depends also on P(Type II error).
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 in the one-sided test by using
For a test of H0: μ = 0 against Ha: μ > 0 based on n = 30 observations and using a = 0.05 significance level, P(Type II error) = 0.36 at μ = 4. Identify the response that is incorrect.a. At μ = 5, P(Type II error) < 0.36.b. If α = 0.01, then at μ = 4, P(Type II error) > 0.36.c. If n
True or False. If we reject H0: μ = 0 in a study about change in weight on a new diet using α = 0.01, then we also reject it using α = 0.05.
True or False. A 95% confidence interval for μ = population mean IQ is (96, 110). So, in the test of H0: μ = 100 against Ha: μ ≠ 100, the P-value > 0.05.
For a fixed significance level a, the probability of a Type II error increases when the sample size increases.
When testing H0: μ = 100 against Ha: μ ≠ 100, the probability of a Type II error decreases the further the true population mean m is from 100.
The P-value is the probability that H0 is true.
According to the Bureau of Labor Statistics, the official unemployment rate was 10.4% among blacks and 4.7% among whites as of February 2015. (www.bls.gov/).a. Identify the response variable and the explanatory variable.b. Identify the two groups that are the categories of the explanatory
The 2011 Bedroom Sleep poll of a random sample of 1500 adults reported that respondents slept an average of 6.5 hours on weekdays and 7.2 hours on weekends, and that 21% of respondents got eight or more hours of sleep on weekdays, whereas 44% got eight or more hours of sleep on weekends
A National Center for Health Statistics data brief published in 2015 (Nr. 181) looked at the association between lung obstruction and smoking status in adults 40 to 79 years old. In a random sample of 6927 adults without any lung obstruction, 54.1% never smoked. In a random sample of 1146 adults
To increase Barack Obama€™s visibility and to raise money for the campaign leading up to the 2008 presidential election, Obama€™s analytics team conducted an A/B test with his website. In the original version, the button to join the campaign read €œSign Up€.
A recent survey asked each participant whether he or she believes in heaven and in hell. Of the 1314 participants, a proportion of 0.85 believed in heaven and a proportion of 0.73 believed in hell. Can we use the inferential methods (confidence interval, significance test) in this section to
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