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Exam 9: Understanding Sampling Distributions: Statistics As Random Variables

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Question 111

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Use the following information for questions: Grade school students, especially the ones in higher grades, tend to spend a lot of time on the computer already. The average amount of time 4^th grade boys spend on the computer is 5 hours per week, with a standard deviation of 1 hour. For 5^th grade boys the average increases to 7 hours per week with a standard deviation of 1.5 hours. Random sample of 20 4^th grade students and 25 5^th grade students are to be selected. -What is the standard deviation of the sampling distribution of the difference in sample means?

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Question 112

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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 snapping turtles is higher than 0.25?

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Question 113

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Which of the following statements is true about the standard deviation of $\hat{p}$ ?

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. -What is the correct notation for the value 0.3 cm?

Use the following information for questions: When people get married for the first time, the husband is, on average, 2 years older than the wife. The standard deviation of the difference in age is roughly 2.5 years. Suppose it is reasonable to assume that the distribution of the difference in age between the husband and the wife is normal. Twelve recently married couples (all first marriages) are to be selected and the average difference in age is to be calculated. -What is the mean of the sampling distribution of the sample mean difference in age between the husband and the wife?

For a random sample of 40 students, the average statistics midterm score is 73 point (out of a possible 100 points) . The standard deviation in the sample was 7.3. What is the standard error of the mean?

Use the following information for questions: When people get married for the first time, the husband is, on average, 2 years older than the wife. The standard deviation of the difference in age is roughly 2.5 years. Suppose it is reasonable to assume that the distribution of the difference in age between the husband and the wife is normal. Twelve recently married couples (all first marriages) are to be selected and the average difference in age is to be calculated. -What is the probability that the average age of the wives in the sample is older than the average age of the husbands?

Use the following information for questions: Based on the 2000 Census, 31.8% of grandparents in California are the primary caregivers for their grandchildren. Suppose n = 1000 grandparents are to be sampled from this population and the sample proportion of grandparents as primary caregivers ( $\hat{p}$ ) is to be calculated. -What is the approximate probability that they find less than 30% primary caregivers in the sample?

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 are the degrees of freedom for the t-distribution?

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. -What would have been the null value in this scenario for the difference in school grades between the two groups?

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 standard error of the sample proportion?

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 probability that the difference between the two sample proportions (frequent use of painkillers - not frequent use of painkillers) is greater than 10 percentage points (0.10) ?

A store manager is trying to decide whether to price oranges by weight, with a fixed cost per pound, or by the piece, with a fixed cost per orange. He is concerned that customers will choose the largest ones if there is a fixed price per orange. For one week the oranges are priced by the piece rather than by weight, and during this time the mean weight of the oranges purchased is recorded for all customers who buy 4 of them. The manager knows the population of weights of individual oranges is bell-shaped with mean of 8 ounces and a standard deviation of 1.6 ounces. If the 4 oranges each customer chooses are equivalent to a random sample, what should be the approximate mean and standard deviation of the distribution of the mean weight of 4 oranges?

Use the following information for questions: High school students can be categorized into two groups by the amount of activities they are involved in. Let group 1 consist of all high school students who are very involved in sports and other activities and group 2 consist of all high school students who aren't. The distributions of GPAs in both groups are approximately normal. The mean and standard deviation for group 1 are 2.9 and 0.4, respectively. The mean and standard deviation for group 2 are 2.7 and 0.5, respectively. Independent random samples of 50 high school students are to be selected from both groups (for a total of 100 students) . -If the sample mean GPA is to be calculated for both groups and we calculate the difference as involved in activities - not so involved in activities, what is the expected value for the difference in sample means?

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 mean of the sampling distribution of the difference between the two sample proportions (English majors - engineering majors)?

Five hundred (500) random samples of size n = 900 are taken from a large population in which 10% of the subjects are left-handed. The proportion of subjects in the sample that is left-handed is found for each sample and a histogram of these 500 proportions is drawn. Which interval covers the range into which about 68% of the values in the histogram will fall?

Suppose we select a random sample of n = 100 students and find that the proportion of students who said they believe in love at first sight is 0.43. Which statement is not necessarily true?

Use the following information for questions: Suppose on a highway with a speed limit of 65 mph, the speed of cars are independent and normally distributed with an average speed = 65 mph and standard deviation = 5 mph. -What is the probability that the sample mean speed in a random sample of n = 10 cars exceeds 68 mph?

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 difference between the sample means is greater than $100?

Use the following information for questions: Based on the current rate of savings, the average American household will live on 59 percent of pre-retirement income once they retire, a recent report says. The report also says that Americans are behind in funding their retirement years. The report recommended saving enough to be able to live on 85 percent of income in retirement. However, the typical household has put aside only $18,750 for retirement. Suppose we can assume that the true average amount a typical household has put aside for retirement is $18,750. Also assume that these amounts vary with a standard deviation of $12,500. The distribution of these amounts is known to be right-skewed. -Suppose 100 households are to be randomly selected. Is the probability that the average amount these 100 households have set aside for retirement is more than $20,000 (roughly) equal to 0.1587?