Exam 5: Understanding and Comparing Distributions

The owner of a small clothing store is concerned that only 28% of people who enter her store actually buy something. A marketing salesman suggests that she invest in a new line of celebrity mannequins (think Seth Rogan modeling the latest jeans…). He loans her several different "people" to scatter around the store for a two-week trial period. The owner carefully counts how many shoppers enter the store and how many buy something so that at the end of the trial she can decide if she'll purchase the mannequins. She'll buy the mannequins if there is evidence that the percentage of people that buy something increases. -In this context describe a Type II error and the impact such an error would have on the store.

The countries of Europe report that 46% of the labor force is female. The United Nations wonders if the percentage of females in the labor force is the same in the United States. Representatives from the United States Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate a percentage of females in the United States labor force. -Interpret the confidence interval in this context.

According to the 2010 U.S. Census, 11.7% of the people in the state of Oregon were Hispanic or Latino. A political party wants to know how much impact the Hispanic and Latino vote will have, so they wonder if that percentage has changed since then. They take a random sample of 853 adults in Oregon and ask, among other things, their race. 113 of the people surveyed were Hispanic or Latino. Can the political party conclude that the Hispanic proportion of the population has changed since 2010? -Explain what your P-value means in this context.

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels. -Write the company's null and alternative hypotheses.

Pew Research found that, in 2013, 50% of American adults favored allowing same-sex couples to marry legally. This is up from 48% in 2012. The 2013 estimate was based on a random sample of 1,501 adults. Assume the same sample size was used in 2012. ["Changing Attitudes on Gay Marriage," Pew Internet and American Life Project, June 2013 -What is meant by the phrase "95% confident" in this context?

A report on health care in the US said that 28% of Americans have experienced times when they haven't been able to afford medical care. A news organization randomly sampled 801 black Americans, of whom 38% reported that there had been times in the last year when they had not been able to afford medical care. Does this indicate that this problem is more severe among black Americans? -Test an appropriate hypothesis and state your conclusion. (Make sure to check any necessary conditions and to state a conclusion in the context of the problem.)

Pew Research reports that 63% of the U.S. adult cell phone owners use their phone to go online. A company wants to target 16- to 24-year olds for advertising and they wonder if that age group has a similar pattern of phone use. -The company wants to estimate the true percentage of 16- to 24-year old cell phone owners who use their phone to go online to within ±7.5%, with 95% confidence. How many cell phone owners in this age group should they sample?

To plan the budget for next year a college needs to estimate what impact the current economic downturn might have on student requests for financial aid. Historically, this college has provided aid to 35% of its students. Officials look at a random sample of this year's applications to see what proportion indicate a need for financial aid. Based on these data they create a 90% confidence interval of (32%, 40%). Could this interval be used to test the hypothesis H0: p = 0.35 versus HA: p ×0.35 at the a = 0.10 level of significance?

Pew Research reports that 63% of the U.S. adult cell phone owners use their phone to go online. A company wants to target 16- to 24-year olds for advertising and they wonder if that age group has a similar pattern of phone use. -Should the company conclude that the percentage of cell phone owners in this age group who use their phone to go online is different from 63%? Explain.

According to the 2010 U.S. Census, 11.7% of the people in the state of Oregon were Hispanic or Latino. A political party wants to know how much impact the Hispanic and Latino vote will have, so they wonder if that percentage has changed since then. They take a random sample of 853 adults in Oregon and ask, among other things, their race. 113 of the people surveyed were Hispanic or Latino. Can the political party conclude that the Hispanic proportion of the population has changed since 2010? -Was your test one-tail upper tail, lower tail, or two-tail? Explain why you chose that kind of test in this situation.

Cereal A box of Raspberry Crunch cereal contains a mean of 13 ounces with a standard deviation of 0.5 ounce. The distribution of the contents of cereal boxes is approximately Normal. What is the probability that a case of 12 cereal boxes contains a total of more than 160 ounces?

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels. -What level of significance did the research department use?

The average composite ACT score for Ohio students who took the test in 2003 was 21.4.Assume that the standard deviation is 1.05. In a random sample of 25 students who took the exam in 2003, what is the probability that the average composite ACT score is 22 or more? (Make sure to identify the sampling distribution you use and check all necessary conditions.)

Herpetologists (snake specialists) found that a certain species of reticulated python has an average length of 20.5 feet with a standard deviation of 2.3 feet. The scientists collect a random sample of 30 adult pythons and measure their lengths. In their sample the mean length was 19.5 feet long. One of the herpetologists fears that pollution might be affecting the natural growth of the pythons. Do you think this sample result is unusually small? Explain.

A certain population is strongly skewed to the left. We want to estimate its mean, so we collect a sample. Which should be true if we use a large sample rather than a small one? I. The distribution of our sample data will be more clearly skewed to the left. II. The sampling model of the sample means will be more skewed to the left. III. The variability of the sample means will greater.

The board of directors for Procter and Gamble is concerned that only 19.5% of the people who use toothpaste buy Crest toothpaste. A marketing director suggests that the company invest in a new marketing campaign which will include advertisements and new labeling for the toothpaste. The research department conducts product trials in test markets for one month to determine if the market share increases with new labels. -In this context describe a Type I error and the impact such an error would have on the company.

According to the Centers for Disease Control, about 36% of U.S. adults were obese in 2013. To see if that has changed since the last study, a random sample of 250 U.S. adults will be checked. a. Describe the sampling distribution model for the sample proportion (assuming no change in obesity rates) by naming the model and telling its mean and standard deviation. Justify your answer. b. Sketch and clearly label the model. c. What is the probability that in this group less than 25% of the adults will be found to be obese?

A certain population is bimodal. We want to estimate its mean, so we will collect a sample. Which should be true if we use a large sample rather than a small one? I. The distribution of our sample data will be clearly bimodal. II. The sampling distribution of the sample means will be approximately normal. III. The variability of the sample means will be smaller.

A city planner wants to construct a confidence interval for the proportion of workers who would take public transportation to work if it were available. What sample size should he use to estimate the proportion to within 4 percentage points at a 90% level of confidence?

We have calculated a confidence interval based on a sample of n = 180. Now we want to get a better estimate with a margin of error only one third as large. We need a new sample with n at least…