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Exam 9: Understanding Sampling Distributions: Statistics As Random Variables

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Question 51

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A fourth grade class of 28 students is given a standardized math test. The mean score of the 12 boys is 25 with a standard deviation of 3. The mean score of the 16 girls is 24 with a standard deviation of 4. What is the standard error for the sampling distribution of $\overline{x}_{boys}$ - $\overline{x}_{girls}$ ?

A) 1
B) 1.87
C) 1.75
D) 1.32

E) All of the above
F) A) and C)

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Question 52

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For a random sample of 9 women, the average resting pulse rate is $\overline{x}$ = 76 beats per minute, and the sample standard deviation is s = 5. What is the standard error of the mean?

A) 0.557
B) 0.745
C) 1.667
D) 2.778

E) None of the above
F) All of the above

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Question 53

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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 standard deviation of the sampling distribution of the difference in sample means?

It is estimated that only about 10% of 3-year olds already know how to write their (first) name. Out of 85 random sampled 3-year olds, 11 are able to write their name. What is the z-score associated with the observed sample proportion of 3-year olds already know how to write their name?

Which of the following is an example of a difference in two proportions based on independent samples?

Use the following information for questions: The level of nitrogen oxides (NOX) in the exhaust of a particular car model can be modeled by the normal distribution with mean 0.9 grams per mile and standard deviation of 0.16 grams per mile. A random sample of 4 of these NOX levels is to be selected. -What is the probability that the NOX level of all 4 cars in the sample is greater than 1 gram per mile?

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 standard deviation of the sampling distribution of the sample mean difference in age between the husband and the wife?

Use the following information for questions: In the population of female students, motivation scores follow a normal distribution with an average of 29 and a standard deviation of 4.3. A random sample of female students is to be taken. They are all asked to answer the motivation questionnaire so that their motivation scores can be determined. -If the sample size is 20, what is the probability that the mean score of the female students in the sample will fall below 27?

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 mean of the sampling distribution of $\hat{p}$ ?

Use the following information for questions: Every student taking elementary statistics at a large university (1,100 students) participated in a class project by rolling a 6-sided die 100 times. Each individual student determined the proportion of his or her 100 rolls for which the result was a "1". The instructor plans to draw a histogram of the 1,100 sample proportions. -What will be the approximate standard deviation for the 1,100 sample proportions?

Use the following information for questions: People who go on a (leisure) vacation tend to gain weight. It is thought that the average difference between weight before the vacation and after the vacation is roughly 2 pounds with a standard deviation of 2.5 pounds. Ten vacationing people are to be randomly selected and their weights measured, both upon departure and return of their vacation. -If we can assume that the differences are normally distributed, what is probability that the ten vacationing people actually lost weight, on average?

Use the following information for questions: We wish to conduct a hypothesis test to determine if, on average, the mother in a family spends more time doing house work per week than the father, for families where both parents have a full time job. -Suppose that in reality, mothers and fathers spend an equal amount of time, on average, doing housework each week. What would be the mean of the sampling distribution of the sample mean difference in time, based on random samples taken from the population under study (mothers and fathers from families where both parents have a full time job) ?

Use the following information for questions: A group of 50 students had their blood pressures measured before and after watching a movie containing violence. The mean blood pressure before the movie is to be compared with the mean pressure after the movie. In general, blood pressure increases by 5 points after watching something violent happening with a standard deviation of 7 points. -If we can assume that the differences are normally distributed, what is probability that the difference (in either direction) in average blood pressure is no more than 3 points?

Use the following information for questions: Management of an airline uses a normal distribution to model the value claimed for a lost piece of luggage on domestic flights. The mean of the distribution is $600 and the standard deviation is $85. Suppose a random sample of 65 pieces of luggage is to be selected. -What is the standard deviation of the sampling distribution of the sample mean claimed value of the 65 pieces of lost luggage?

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 and the average amount these 100 households have set aside for retirement calculated. What is the expected value of the average amount set aside for retirement in the sample of 100 households?

Suppose that 20% of a random sample of n = 64 men are currently single. What is the standard error of the proportion of single men in the sample?

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 expected value of the average exam score of the 20 students in your lab section?

Use the following information for questions: In the population of female students, motivation scores follow a normal distribution with an average of 29 and a standard deviation of 4.3. A random sample of female students is to be taken. They are all asked to answer the motivation questionnaire so that their motivation scores can be determined. -If the sample size is 50, what is the probability that the mean score of the female students in the sample will fall below 27?

Which statement is true about p and $\hat{p}$ ?

Use the following information for questions: A candy factory makes 20% (p = 0.20) of all its candies with chocolate liquor. A random sample of n =100 candies is taken, and $\hat{p}$ = proportion of candies in the sample made with chocolate liquor is calculated. -What is the z-score for $\hat{p}$ = 0.10?