Exam 9: Understanding Sampling Distributions: Statistics As Random Variables

Use the following information for questions: A large car insurance company is conducting a study on behavior behind the wheel to explain the fact that male drivers (who tend to be thought of as more aggressive drivers) have more accidents, on average, per year, than female drivers. The average number of accidents per year for male drivers is 2.3 with a standard deviation of 0.8 and for female drivers the mean is 1.7 with a standard deviation of 0.6. Independent random samples of 25 male and 20 female drivers are to be selected to take part in the behavior study. -Suppose we can assume that the number of accidents per year is normally distributed, both for males and for females. What is the probability that the average number of accidents in the sample of 25 male drivers is less than the average number of accidents in the sample of 20 female drivers?

In a lottery, a real number from 0 to 100 is picked at random. If X is the number picked, then the mean of X is 50 and the standard deviation of X is 28.9. For a random sample of 60 numbers, use the Empirical Rule to find a range that will include the sample mean, $\overline{X}$ , approximately 95% of the time.

Use the following information for questions: At a small college somewhere on the East coast, 20% of the girls and 30% of the boys smoke at least once a week. Independent random samples of 75 girls and 100 boys are to be selected and the proportions of smokers in the samples are to be calculated. -What is probability that the sample proportion of girl smokers is greater than the sample proportion of boy smokers?

In the population of male students, motivation scores follow a normal distribution with an average of 32.4 and a standard deviation of 7. A random sample of 16 male students is to be taken. They are all asked to answer the motivation questionnaire so that their motivation scores can be determined. What is the probability that the sample mean score will exceed 35?

Heights for women are bell-shaped with a mean of 65 inches and standard deviation of 2.5 inches. If a random sample of 25 women is taken, what is the standard deviation of the sampling distribution of $\overline{x}$ ?

Use the following information for questions: A gasoline tank for a certain car is designed to hold 15 gallons of gas. The manufacturer of this gasoline tank claims that the actual capacity of a randomly selected tank has a distribution that is approximately normal with a mean of 15.0 gallons. A random sample of 25 tanks results in an average of 14.8 gallons with a standard deviation of 0.42 gallons. -What are the degrees of freedom for the t-distribution?

Use the following information for questions: Exam scores for a large introductory statistics class follow an approximate normal distribution with an average score of 56 and a standard deviation of 5. The average exam score in your lab was 59.5. The 20 students in your lab sections will be considered a random sample of all students who take this class. -What is the standard deviation of the distribution of the average exam score of the 20 students in your lab section?

The household income in a certain community is known to have a right skewed distribution with a mean of $42,000 and a standard deviation of $5,000. Without knowing the exact distribution we cannot find the probability that a randomly selected household has a yearly income less than $41,000. Suppose we randomly sample 80 incomes from this population, what is the (approximate) probability that the average income in the sample is less than $41,000?

Use the following information for questions: A television station plans to ask a random sample of 400 city residents if they can name the news anchor on the evening news at their station. They plan to fire the news anchor if fewer than 10% of the residents in the sample can do so. Suppose that in fact 12% of city residents could name the anchor if asked. -What is the standard deviation of the sampling distribution for the sample proportion of city residents who can name the news anchor on the evening news at their station?

Use the following information for questions: Data is to be gathered to estimate the proportion of residents who will vote for Mr. Treehouse in the coming city elections. A random sample of 650 voters is interviewed and 306 of them say that they will support Mr. Treehouse. -Use the Empirical Rule to find values that fill in the blanks in the following sentence: In about 95% of all randomly selected samples of n = 650 voters, we would estimate the population proportion of all residents who will vote for Mr. Treehouse in the coming city elections to be between _____ and _____.

Use the following information for questions: Many undergraduate students are thinking about getting a graduate degree. Do male and female students have equal opinions about this? A survey of undergraduate students at public universities in the state of California revealed the following results: Out of 265 female students, 131 were seriously thinking about applying to graduate school. Out of 233 male students, 130 were seriously thinking about applying to graduate school. We wish to compare the proportions of male and female students who are thinking about getting a graduate degree by calculating a confidence interval for the difference between males and females. -What is the correct notation for the parameter that is estimated by the confidence interval?

Use the following information for questions: In each situation, indicate whether it would make more sense to find a confidence interval for the population parameter or to conduct a hypothesis test for a specific null value. If your answer is "a hypothesis test," specify the null value that you would test. -A high school teacher is interested to know if students who take a foreign language will get higher grades if their final exam is an oral exam instead of a written exam. The following year he asks all of the Spanish teachers to give their students an oral exam at the end of the course. The average exam score in the previous years was 68 (out of a possible 100 points).

Use the following information for questions: Data collected over several years shows that of all patients that come to the doctor's office with a sore throat, 6% of them have a contagious virus. The next 188 patients with a sore throat are tested for this virus. -What is the standard deviation of the sampling distribution of $\hat{p}$ ?

Use the following information for questions: In each situation, indicate whether it would make more sense to find a confidence interval for the population parameter or to conduct a hypothesis test for a specific null value. -A consumer agency wishes to find out how much more private hospitals and medical clinics charge for certain procedures than the large (research) hospitals.

Use the following information for questions: For each study, decide if the two samples are : -In a random sample of 100 students (60 women and 40 men), the average hours of sleep during finals week were compared.

Use the following information for questions: A large company is making all of its employees take a stress test. Test scores on this stress test follow a normal distribution with a mean score of 3 and a standard deviation of 0.9. A random sample of 35 employees is to be taken. -What is the probability that the sample mean stress score exceeds 3.5?

Use the following information for questions: A new study finds that frequent use of painkillers does not substantially increase a healthy man's risk of developing hypertension (high blood pressure). In other words, the proportion of healthy men with hypertension is the same for those who use painkillers frequently and those who don't. According to the American Heart Association, 1 in 3 American adults have hypertension. -Suppose random samples of 200 men who use painkillers frequently and 200 men who don't are to be selected. What is the standard deviation for the difference between the two sample proportions?

Use the following information for questions: A large company is making all of its employees take a stress test. Test scores on this stress test follow a normal distribution with a mean score of 3 and a standard deviation of 0.9. A random sample of 35 employees is to be taken. -What is the expected value of the sample mean stress score in the sample of 35 employees?

In which of the following scenarios is a paired design not appropriate?

For a random sample of 10 men, the mean head circumference is $\overline{x}$ = 57.3 cm and the sample standard deviation is s = 2 cm. What is the standard error of the mean?