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Exam 9: Understanding Sampling Distributions: Statistics As Random Variables

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Question 71

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Use the following information for questions: A Statistics instructor asked a random sample of female and male students how many hours they exercise each week. Minitab output for a comparison of females and males is shown below. -What is the correct notation for the parameter that is estimated by the confidence interval given in the output?

A) p
B) p₁ -p₂
C) $\mu$
D) $\mu$ ₁ - $\mu$ ₂

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Question 72

Multiple Choice

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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. -The school board wishes to find out if the newly adapted teaching method for K-5 math has resulted in higher average grades.

A) Confidence interval
B) Hypothesis test

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Question 73

Short Answer

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Use the following information for questions: The distribution of quiz, exam, and course grades is known to be left-skewed. The course grades (measured as percentage scores) in upper level statistics classes have a mean of 68 with a standard deviation of 15. -Suppose 50 course grades are to be randomly selected and the average score in the sample calculated. What is the (approximate) probability that the average score in the sample is more than 5 points removed from the true average of 68?

Use the following information for questions: A comparison is to be made between the proportion of second graders that cannot read at second grade level and the proportion of third graders that cannot read at second grade level. School records from schools across the state are collected and records for 123 second graders and 146 third graders are randomly selected. Of the sampled second graders, 25 seem to be not reading at second grade level. Of the sample third graders, 26 do not read at second grade level. -We know that if the sampling procedure were repeated, we would probably not observe 25/123 and 26/146 students who cannot read at second grade level again. What is the mean of the sampling distribution of the difference between the two proportions based on 123 second graders and 146 third graders?

Use the following information for questions: Suppose that 80% of all English majors and 85% of all engineering majors at a Minnesota college wear winter boots when there is snow on the ground. Two independent random samples of 40 English majors and 60 engineering majors are to be selected during a day with snow on the ground and the proportions of students with winter boots are to be calculated. -What is the standard deviation of the sampling distribution of the difference between the two sample proportions (English majors - engineering majors)?

Based on the 2000 Census, 31.8% of grandparents in California are the primary caregivers for their grandchildren. In a random sample of 1000 grandparents, 296 are the primary caregivers for their grandchildren. What is the z-score associated with the observed sample proportion of grandparents who are the primary caregivers for their grandchildren?

For results based on a small random sample from a bell-shaped distribution, the distribution of the sample mean is

Use the following information for questions: According to a recent survey, teenagers who eat with their families at least five times a week are more likely to get better grades in school and much less likely to have substance abuse problems. Two groups of teenagers were studied: those who eat with their families at least five times a week and those who don't. The difference in school grades and the difference in likelihood to have substance abuse problems were both found to be statistically significant. -When comparing the likelihood to have substance abuse problems between the two groups, what does statistical significance mean?

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. -If the sample mean number of accidents is to be calculated for both the male and female drivers and we calculate the difference as male - female, what is the standard deviation of the sampling distribution of the difference in sample means?

In a random sample of 1000 students, 80% were in favor of longer hours at the school library. What is the standard error of the sample proportion?

Use the following information for questions: Ms. Jackson of the Employment Agency believes that the agency receives an average of 16 complaints per week. Mr. Vermeer, the boss, thinks it is less. A sample of 10 weeks yields an average of 14 complaints per week with a standard deviation of 3 complaints. -What is the value of the observed t-statistic when comparing the sample mean to the average perceived by Ms. Jackson?

Use the following information for questions: Items produced by a certain process are supposed to weigh 90 grams. However, the process is such that there is variability in the items produced and they do not all weigh exactly 90 grams. The distribution of weights is normal with a mean of 89.8 grams and a standard deviation of 1.1 gram. -Suppose we have a random sample of 2 of these items. What is the probability that both these items weigh more than 91 grams?

Use the following information for questions: Let p denote the proportion of adults in major minority groups in the US who read an ethnic newspaper at least several times per week. According to the nonprofit organization New California Media the true value of p is 45 percent. We decide to take SRS of 200 adults in major minority groups and calculate $\hat{p}$ , the sample proportion of adults who read an ethnic newspaper at least several times per week. -What is the mean of the sampling distribution of $\hat{p}$ ?

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. -What is the sample proportion of residents who will vote for Mr. Treehouse in the coming city elections?

Which of the following describes the relationship between the Central Limit Theorem and the Rule for Sample Means?

Use the following information for questions: A group of 50 students each measured the length of their right arm and the length of their left arm. The right arm lengths were compared to the left arm lengths. The average difference between right and left arm lengths was 0.3 cm with a standard deviation of 1 cm. -(Optional) Why do you think is the average difference between right and left arm lengths positive?

Use the following information for questions: In a pond with a large population of turtles, the percent of three different types of turtles are presented in the table below. A random sample of 100 turtles is taken from the pond. -What is the approximate probability that the sample proportion of chicken turtles is lower than 0.25?

Use the following information for questions: College students spend a lot of their money, not on things they would like to buy, but on text books. College text books have increased in price significantly over the past few years. The amount students spend on text books is approximately normally distributed with a standard deviation of $50. The average amount spent on books each semester is $340 for undergraduate students and $250 for graduate students. Independent random samples of 100 undergraduate students and 80 graduate students are to be selected and the average amount they spent on text books last semester is to be compared (undergraduate - graduate). -What is the probability that the undergraduate students in the sample spent more on books, on average, than the graduate students in the sample?

Use the following information for questions: The distribution of quiz, exam, and course grades is known to be left-skewed. The course grades (measured as percentage scores) in upper level statistics classes have a mean of 68 with a standard deviation of 15. -Suppose 50 course grades are to be randomly selected and the average score in the sample calculated. What is the (approximate) probability that the average score in the sample is higher than 75?

The sample mean number of accidents is to be calculated for a sample of husbands and wives and we wish to determine the difference in mean number of accidents for these husband-wife couples. Which statistic is being studied?