Exam 6: The Standard Deviation As a Ruler and the Normal Model

A survey asked people "On what percent of days do you get more than 30 minutes of vigorous exercise?" Using their responses we want to estimate the difference in exercise frequency between men and women. We should use a

Test identification Suppose you were asked to analyze each of the situations described below. (NOTE: Do not do these problems!) For each, indicate which procedure you would use (pick the appropriate number from the list), the test statistic (z or t), and, if t, the number of degrees of freedom. A procedure may be used more than once. Type z / t ? d f a. b. c. d. e. f. 1. proportion - 1 sample 2. difference of proportions - 2 samples 3. mean - 1 sample 4. difference of means - independent samples 5. mean of differences - matched pairs a. A personal trainer would like to know if a newly designed bootcamp regimen will significantly build body mass index (BMI). In an effort to test this, the trainer recorded the BMI for 15 different clients prior to the bootcamp and after the bootcamp. The trainer assumes the BMI's are approximately normal. Does the bootcamp regimen as advertised? b. In a study to determine whether there is a difference between the average jail time black and white offenders of minor drug possession are sentenced to, the law students randomly selected 25 cases of each type that resulted in jail sentences during the previous year. A 90% confidence interval was created from the results. c. A bank branch manager is interested in estimating the average wait time for customers in the teller line. The manager records the times for 40 randomly chosen customers. Estimate the wait time with a 95% confidence interval. d. A New York City mayoral candidate wants to assess his constituent's opinions on the controversial "Stop and Frisk" police tactics. A sample of voters from 2 boroughs (Queens and Brooklyn) is selected and asked if they approve of this policy. Do the approval rates vary from each other? e. Is there more gun violence in the summer heat than the winter cold? We get records of the number of gunshot wounds in January and July in a random sample of 50 emergency rooms. f. A board of directors of a local homeowner's association union organization wishes to amend the bylaws. A sample of the residents revealed 310 of 430 were in favor of the amendment. Does the board of directors have the required 75% majority?

Housing costs A government report on housing costs says that single-family home prices nationwide are skewed to the right, with a mean of $235,700. a. We collect price data from a random sample of 50 homes in Orange County, California. Why is it okay to use these data for inference even though the population is skewed? b. The standard deviation of the 50 homes in our sample was $25,500. Specify the sampling model (shape, center, spread) for the mean price of such samples. c. This sample of randomly chosen homes produced a 90% confidence interval for the mean price in Orange County of ($233,954, $246,046). Does this interval provide evidence that single-family home prices are unusually high in this county? Explain briefly. d. Suppose we hope to improve our estimate by choosing a new sample. How many home prices must we survey to have 90% confidence of estimating the mean local price to within $2000?

Game Wizard A friend of yours is constantly bragging that he performs exceptionally well at a popular video game since he scores an average of 1400 points. You think he is not all that hot. Having taken a statistics class, you know that you can actually test to see whether or not your friend scores higher than other players of the game. You collect a random sample of scores from other players at the local arcade and get the following data: 1375 1420 1250 1310 1450 1380 1390 1410 1290 1360 Is your friend right? That is, is there evidence that other players average less than 1400 points per game? Conduct a hypothesis test, making sure to state your conclusion in context of the problem.

Trainers need to estimate the level of fat in athletes to ensure good health. Initial tests were based on a small sample but now the trainers double the sample size for a follow-up test. The main purpose of the larger sample is to…

Suppose that our editor was hoping that the book would have a mean word length of 6.5 letters. Does this sample indicate that the authors failed to meet this goal? Test an appropriate hypothesis and state your conclusion.

A coffee house owner knows that customers pour different amounts of coffee into their cups. She samples cups from 10 customers she believes to be representative of the customers and weighs the cups, finding a mean of 12.5 ounces and standard deviation of 0.5 ounces. Assuming these cups of coffee can be considered a random sample of all cups of coffee which of the following formulas gives a 95% confidence interval for the mean weight of all cups of coffee?

Auto repairs An insurance company hopes to save money on repairs to autos involved in accidents. Two body shops in town seem to do most of the repairs, and the company wonders whether one of them is generally cheaper than the other. From their files of payments made during the past year they select a random sample of ten bills they paid at each repair shop. The data are shown in the table. Bodies Velleman's by Bock Automagic 2130 2570 980 1120 3400 2950 2190 1880 1100 1660 1450 1700 4590 4030 3090 3970 1050 1130 2530 3660 Indicate what inference procedure you would use to see if there is a significant difference in the costs of repairs done at these two body shops, then decide if it is okay to actually perform that inference procedure. (Check the appropriate assumptions and conditions and indicate whether you could or could not proceed. You do not have to do the actual test.)

One common method of evaluating the performance of a mutual fund is to compare its returns to those of a recognized benchmark such as an index of the returns on all securities of the type that the fund accumulates. The Janus Worldwide Fund considers its benchmark to be the MSCI World IndexSM. The table below depicts the annual returns (percent) for a recent ten-year period. Is this fund a good investment? That is, does this fund significantly outperform its benchmark? Janus MSCI Year Worldwide Index 24.23 33.11 5.53 14.72 5.76 9.49 17.84 20.07 9.18 9.04 -45.04 -40.71 37.49 29.99 17.00 11.76 -13.95 -5.54 19.64 15.83 -Explain clearly whether this data should be analyzed using a 2-sample t test approach or a match pairs t-test method.

A professor at a large university believes that students take an average of 15 credit hours per term. A random sample of 24 students in her class of 250 students reported the following number of credit hours that they were taking: 12 13 14 14 15 15 15 16 16 16 16 16 17 17 17 18 18 18 18 19 19 19 20 21 -Does this sample indicate that students are taking more credit hours than the professor believes? Test an appropriate hypothesis and state your conclusion.

A teacher wants to see if two different forms of an exam are equivalent or if one of the exams is more difficult than the other. She has 120 students, which she randomly sorts into two groups of 60. The group that that takes exam A has a mean score of 78.1% with a standard deviation of 5.6%. Exam B scores an average of 74.8% with a standard deviation of 8.7%. -Improving productivity A packing company considers hiring a national training consultant in hopes of improving productivity on the packing line. The national consultant agrees to work with 18 employees for one week as part of a trial before the packing company makes a decision about the training program. The training program will be implemented if the average product packed increases by more than 10 cases per day per employee. The packing company manager will test a hypothesis using a = 0.05. a. Write appropriate hypotheses (in words and in symbols). b. In this context, which do you consider to be more serious - a Type I or a Type II error? Explain briefly. c. After this trial produced inconclusive results the manager decided to test the training program again with another group of employees. Describe two changes he could make in the trial to increase the power of the test, and explain the disadvantages of each.

A survey question asked students "How many hours of TV do you watch per week?" Using their responses, we want to estimate the difference in mean hours between high school and middle school students. We should use a:

Every year favorite songs compete to be on a Top 200 list based upon sales and rankings by the experts in the music industry. These songs have many characteristics, such as song length and beats per minute, which vary from category to category in the music industry. A disc jockey wondered if the number of beats per minute in songs classified as dance music were lower than the beats per minute in the songs that are ranked on a Top 200 list from 2001. A random sample of songs from each group was selected and the beats per minute are listed in the chart at the right. Does this sample indicate that songs classified as dance music have lower beats per minute than the songs ranked on a Top 200 list? $\quad$ $\quad$ $\quad$$\text { Beats per Minute }$ Dance Songs Top 200 Songs 121119 122120 122121 121118 117122 121121 120119 122123 120119 121118 121118 119120 118120 120124 120123 119 117118 -Test an appropriate hypothesis and state your conclusion.

Feral cats tend to be lighter than domestic cats, mainly due to a lack of regular access to food. An animal shelter in Boston weighed a random sample of newly admitted stray cats over several weeks. They found that a 95% confidence interval for the mean weight (in pounds) of the cats was (8.6014, 9.9986). Which is true?

Based on data from two very large independent samples, two students tested a hypothesis about equality of population means using a = 0.02. One student used a one-tail test and rejected the null hypothesis, but the other used a two-tail test and failed to reject the null hypothesis. Which of these might have been their calculated value of t?

A company checking the productivity of its assembly line monitored a random sample of workers for several days. They found that a 95% confidence interval for the mean number of items produced daily by each worker was (23,27). Which is true?

The two samples whose statistics are given in the table are thought to come from populations with equal variances. What is the pooled estimate of the population standard deviation? Mean SD 25 32 5 20 30 6

Blood pressure Researchers developing new drugs must be concerned about possible side effects. They must check a new medication for arthritis to be sure that it does not cause an unsafe increase in blood pressure. They measure the blood pressures of a group of 12 subjects, then administer the drug and recheck the blood pressures one hour later. The drug will be approved for use unless there is evidence that blood pressure has increased an average of more than 20 points. They will test a hypothesis using a = 0.05. a. Write appropriate hypotheses (in words and in symbols). b. In this context, which do you consider to be more serious - a Type I or a Type II error? Explain briefly. c. After this experiment produced inconclusive results the researchers decided to test the drug again on another group of patients. Describe two changes they could make in their experiment to increase the power of their test, and explain the disadvantages of each.

An old myth claims that boys are better at math than girls. A high school guidance counselor collects the math grades from a random sample of 50 boys and another random sample of 50 girls. The guidance counselor then tested the hypothesis H₀: µ₁ - µ₂ = 0 against the one-tail alternative and found P = 0.35.Which is true?

Family Incomes A government report on standard of living says that family incomes nationwide are skewed to the right, with a mean of $33,400. a. We collect income data from a random sample of 50 local families. Why is it okay to use these data for inference even though the population is skewed? b. The standard deviation of the 50 incomes in our sample was $25,530. Specify the sampling model (shape, center, spread) for the mean income of such samples. c. This sample of randomly chosen families produced a 90% confidence interval for the local mean family income of (32,882, 44,761). Does this interval provide evidence that family incomes are unusually high here? Explain briefly. d. Suppose we hope to improve our estimate by choosing a new sample. How many families must we survey to have 90% confidence of estimating the mean local family income to within $2000?