Chapter 9

STT 201

STATISTICAL METHODS

TAKE HOME MIDTERM EXAMINATION IV

DUE DATE: TUESDAY NOVEMBER 25, 2014

SCANTRONS WILL BE PROVIDED DURING THE CLASS MEETING BEFORE THE DUE DATE

EARLY SUBMISSION: IF YOU SUBMIT YOUR WORK ON OR BEFORE THURSDAY NOVEMBER 13, 2014, YOU EARN AN EXTRA 4 POINTS.

  1. Which of the following statements is correct about a parameter and a statistic associated with repeated random samples of the same size from the same population?
  2. Values of a parameter will vary from sample to sample but values of a statistic will not.
  3. Values of both a parameter and a statistic may vary from sample to sample.
  4. Values of a parameter will vary according to the sampling distribution for that parameter.
  5. Values of a statistic will vary according to the sampling distribution for that statistic.
  6. None of the above
  1. Which of the following statements best describes the relationship between a parameter and a statistic?
  2. A parameter has a sampling distribution with the statistic as its mean.
  3. A parameter has a sampling distribution that can be used to determine what values the statistic is likely to have in repeated samples.
  4. A parameter is used to estimate a statistic.
  5. A statistic is used to estimate a parameter.
  6. None of the above
  1. Which one of the following statements is false?
  2. The standard error measures the variability of a population parameter.
  3. The standard error of a sample statistic measures, roughly, the average difference between the values of the statistic and the population parameter.
  4. Assuming a fixed value of s = sample standard deviation, the standard error of the mean decreases as the sample size increases.
  5. The standard error of a sample proportion decreases as the sample size increases.
  6. None of the above
  1. Which one of the following statements is false?
  2. A sampling distribution is the probability distribution of a sample statistic. It describes how values of a sample statistic vary across all possible random samples of a specific size that can be taken from a population.
  3. For all five scenarios considered, the sampling distribution is approximately normal as long as the sample size(s) are large enough.
  4. The mean value of a sampling distribution is the mean value of a sample statistic over all possible random samples. For the five scenarios, this mean equals the value of the statistic.
  5. The standard deviation of a sampling distribution measures the variation between all possible values of the sample statistic and their mean over all possible random samples. For the five scenarios, this mean equals the value of the parameter.
  6. None of the above
  1. 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.

What is the correct notation for the difference ?

  1. 12
  2. p1p2
  3. None of the above
  1. If the size of a sample randomly selected sample from a population is increased from n = 100 to n = 400, then the standard deviation of will
  2. remain the same.
  3. increase by a factor of 4.
  4. decrease by a factor of 4.
  5. decrease by a factor of 2.
  6. None of the above
  1. The mean of the sampling distribution for a sample proportion depends on the value(s) of
  1. the true population proportion but not the sample size.
  2. the sample size but not the true population proportion.
  3. the sample size and the true population proportion.
  4. neither the sample size nor the true population proportion.
  5. None of the above

Questions 8 – 10

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.

  1. What is the mean of the sampling distribution for the sample proportion of city residents who can name the news anchor on the evening news at their station?
  2. 400
  3. 0.10
  4. 0.12
  5. 48
  6. None of the above
  1. 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?
  2. 0.015
  3. 0.0162
  4. 0.1056
  5. 0.12
  6. None of the above
  1. What is the approximate probability that the anchor will be fired?
  2. 0.02
  3. 1.23
  4. 0.11
  5. 0.89
  6. None of the above
  1. Suppose that the mean of the sampling distribution for the difference in two sample proportions is 0. This tells us that
  1. The two sample proportions are both 0.
  2. The two sample proportions are equal to each other.
  3. The two population proportions are both 0.
  4. The two population proportions are equal to each other.
  5. None of the above

Questions 12 – 14

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.

  1. Which of the following choices correctly denotes the difference between the proportions of men with hypertension (frequent use of painkillers – not frequent use of painkillers)?
  2. None of the above.
  3. None of the above
  1. 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?
  2. 0
  3. 0.0333
  4. 0.0471
  5. 0.5773
  6. None of the above
  1. 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)?
  2. 0.0013
  3. 0.0169
  4. 0.0337
  5. 0.9999
  6. None of the above
  1. Which of the following statements is true about the standard deviation of ?
  1. It decreases as the sample size n increases.
  2. It increases as the sample size n increases.
  3. It does not change as the sample size n increases.
  4. It changes each time a new sample is drawn.
  5. None of the above
  1. The standard deviation of the sampling distribution for a sample mean depends on the value(s) of
  1. the sample size and the population standard deviation.
  2. the sample size but not the population standard deviation.
  3. the population standard deviation but not the sample size
  4. neither the sample size nor the population standard deviation.
  5. None of the above
  1. Consider a random sample with sample mean . If the sample size is increased from n = 40 to n = 360, then the standard deviation of will
  2. remain the same.
  3. increase by a factor of 9 (will be multiplied by 9).
  4. decrease by a factor of 9 (will be multiplied by 1/9).
  5. decrease by a factor of 3 (will be multiplied by 1/3).
  6. None of the above
  1. 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?
  2. mean = 32 ounces, standard deviation = 6.2 ounces
  3. mean = 8 ounces, standard deviation = 1.6 ounces
  4. mean = 8 ounces, standard deviation = 0.8 ounces
  5. mean = 2 ounces, standard deviation = 0.4 ounces
  6. None of the above
  1. 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?
  1. < 0.0001
  2. 0.0681
  3. 0.1977
  4. 0.3557
  5. None of the above

Questions 20 – 22

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.

  1. What is the expected value of the average exam score of the 20 students in your lab section?
  2. 5
  3. 20
  4. 56
  5. 59.5
  6. None of the above
  1. What is the standard deviation of the distribution of the average exam score of the 20 students in your lab section?
  2. 0.25
  3. 1.12
  4. 1.25
  5. 5
  6. None of the above
  1. What is the probability that the average score of a random sample of 20 students exceeds 59.5?
  2. < 0.0001
  3. 0.0009
  4. 0.0026
  5. 0.2420
  6. None of the above

Questions 23 – 26

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).

  1. 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?
  2. 0
  3. 0.2
  4. 0.4
  5. 0.6
  6. None of the above
  1. 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 standard deviation of the sampling distribution of the difference in sample means?
  2. 0.0082
  3. 0.0905
  4. 0.18
  5. 0.45
  6. None of the above
  1. What is the probability that the average GPA in the sample of students who are not so involved is higher than the average GPA in the sample of students who are very involved?
  2. < 0.0001
  3. 0.0136
  4. 0.0582
  5. 0.9864
  6. None of the above
  1. What is the probability that the average GPAs in the two samples differ by no more than 0.1?
  2. 0.1341
  3. 0.2266
  4. 0.2415
  5. 0.8659
  6. None of the above
  1. Which statement is not true about confidence intervals?
  2. A confidence interval is an interval of values computed from sample data that is likely to include the true population value.
  3. An approximate formula for a 95% confidence interval is sample estimate  margin of error.
  4. A confidence interval between 20% and 40% means that the population proportion lies between 20% and 40%.
  5. A 99% confidence interval procedure has a higher probability of producing intervals that will include the population parameter than a 95% confidence interval procedure.
  6. All of the above
  1. Which statement is not true about the 95% confidence level?
  2. Confidence intervals computed by using the same procedure will include the true population value for 95% of all possible random samples taken from the population.
  3. The procedure that is used to determine the confidence interval will provide an interval that includes the population parameter with probability of 0.95.
  4. The probability that the true value of the population parameter falls between the bounds of an already computed confidence interval is roughly 95%.
  5. If we consider all possible randomly selected samples of the same size from a population, the 95% is the percentage of those samples for which the confidence interval includes the population parameter.
  6. All of the above

Questions 29 – 31

A randomly selected sample of 400 students at a university with 15-week semesters was asked whether or not they think the semester should be shortened to 14 weeks (with longer classes). Forty-six percent of the 400 students surveyed answered yes.

  1. Which one of the following statements about the number 46% is correct?
  2. It is a sample statistic.
  3. It is a population parameter.
  4. It is a margin of error.
  5. It is a standard error.
  6. None of the above
  1. Which one of the following description of the number 46% is not correct?
  2. point estimate
  3. population parameter
  4. sample estimate
  5. sample statistic
  6. None of the above
  1. The multiplier for a confidence interval is determined by
  2. the desired level of confidence and the sample size.
  3. the desired level of confidence but not the sample size.
  4. the sample size but not the desired level of confidence.
  5. neither the sample size nor the level of confidence.
  6. None of the above

Questions 32 – 35

In a survey of n = 950 randomly selected individuals, 17% answered yes to the question “Do you think the use of marijuana should be made legal or not?”

  1. A 95% confidence interval for the proportion of all Americans in favor of legalizing marijuana is
  2. 0.150 to 0.190
  3. 0.146 to 0.194
  4. 0.142 to 0.198
  5. 0.139 to 0.201
  6. None of the above
  1. A 90% confidence interval for the proportion of all Americans in favor of legalizing marijuana is
  2. 0.150 to 0.190
  3. 0.146 to 0.194
  4. 0.142 to 0.198
  5. 0.139 to 0.201
  6. None of the above
  1. A 98% confidence interval for the proportion of all Americans in favor of legalizing marijuana is
  2. 0.150 to 0.190
  3. 0.146 to 0.194
  4. 0.142 to 0.198
  5. 0.139 to 0.201
  6. None of the above
  1. A 99% confidence interval for the proportion of all Americans in favor of legalizing marijuana is
  2. 0.150 to 0.190
  3. 0.146 to 0.194
  4. 0.142 to 0.198
  5. 0.139 to 0.201
  6. None of the above

Questions 36 – 38

A 95% confidence interval for the proportion of young adults who skip breakfast is found to be 0.20 to 0.27.

  1. Which of the following is a correct interpretation of the 95% confidence level?
  2. There is a 95% probability that the true proportion of young adults who skip breakfast is between 0.20 and 0.27.
  3. In about 95% of all studies for which this procedure is used, the confidence interval will cover the true population proportion, but there is no way to know if this interval covers the true proportion or not.
  4. If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.
  5. The proportion of young adults who skip breakfast 95% of the time is between 0.20 and 0.27.
  6. None of the above
  1. Which of the following is the correct interpretation of the 95% confidence interval?
  2. There is a 95% probability that the proportion of young adults who skip breakfast is between 0.20 and 0.27.
  3. If this study were to be repeated with a sample of the same size, there is a 95% probability that the sample proportion would be between 0.20 and 0.27.
  4. We can be 95% confident that the sample proportion of young adults who skip breakfast is between 0.20 and 0.27.
  5. We can be 95% confident that the population proportion of young adults who skip breakfast is between 0.20 and 0.27.
  6. None of the above
  1. From the information provided, we can determine thatANS = A
  2. = 0.235 and margin of error = 0.035.
  3. = 0.235 and margin of error = 0.07.
  4. p = 0.235 and margin of error = 0.035.
  5. p = 0.235 and margin of error = 0.07.
  6. None of the above
  1. Suppose that 90%, 95%, 98%, and 99% confidence intervals are computed (using the same sample) for a population proportion. Which confidence level will give the narrowest interval?
  2. 99%
  3. 98%
  4. 95%
  5. 90%
  6. None of the above
  1. You plan to use a conservative margin of error of 2%. How large a sample size do you need?
  2. 100
  3. 400
  4. 2500
  5. None of the above
  6. None of the above
  1. Which of the following is the minimum sample size that could be used to guarantee that the margin of error for a confidence interval for a proportion is no more than 0.025?
  2. 100
  3. 400
  4. 1600
  5. 2500
  6. None of the above

Questions 42 and 44

Random samples from two age groups of brides (200 brides under 18 years and 100 brides at least twenty years old) showed that 50% of brides in the under 18 group were divorced after 15 years, while 40% of brides in the 20 or older age group were divorced after 15 years. The difference between the two proportions is 0.10, with a standard error of 0.0604.

  1. What is a 95% confidence interval for the difference between the population proportions of brides who are divorced within 15 years (brides under 18  brides at least 20)?
  2. (0.018, 0.218)
  3. (0.123, 0.023)
  4. (0.040, 0.160)
  5. None of the above
  6. None of the above
  1. What is a 99% confidence interval for the difference between the population proportions of brides who are divorced within 15 years (brides under 18  brides at least 20)?
  2. (0.123, 0.023)
  3. (0.056, 0.256)
  4. (0.040, 0.160)
  5. None of the above
  6. None of the above
  1. In a recent poll of 500 13-year-olds, many indicated to enjoy their relationships with their parents. Suppose that 200 of the 13-year olds were boys and 300 of them were girls. We wish to estimate the difference in proportions of 13-year old boys and girls who say that their parents are very involved in their lives. In the sample, 93 boys and 172 girls said that their parents are very involved in their lives. What is a 96% confidence interval for the difference in proportions (proportion of boys minus proportion of girls)?
  1. (0.2015, 0.0151)
  2. (0.1973, 0.0194)
  3. (0.1978, 0.0289)
  4. None of the above
  5. None of the above
  1. The following ranges are possible 95% confidence intervals for the percentage of Americans who think they work too many hours. For which one of the confidence intervals could you conclude that a majority of the population thinks they work too many hours?
  1. 41% to 49%
  2. 49% to 54%
  3. 42% to 51%
  4. 52% to 58%
  5. None of the above
  1. In a past General Social Survey, a random sample of men and women answered the question “Are you a member of any sports groups?” Based on the sample data, 95% confidence intervals for the population proportion who would answer yes are 0.13 to 0.19 for women and 0.25 to 0.33 for men. Based on these results, you can reasonably conclude that
  2. at least 25% of American men and American women belong to sports clubs.
  3. there is no conclusive evidence of a gender difference in the proportions of men and women who belong to sports clubs.
  4. there is conclusive evidence of a gender difference in proportions of American men and American women who belong to sports clubs.
  5. None of the above
  6. None of the above
  1. A CBS News poll taken in January 2010 asked a random sample of 1,090 adults in the US "Do you think the federal government is adequately prepared to deal with a major earthquake in the United States, or not?" A 95% confidence interval for the proportion of the population that thinks the government is prepared is .31 to .37. Based on this result, which one of the following statements is false?
  1. It is reasonable to say that a majority of adults in the US think the federal government is prepared to deal with a major earthquake.
  2. It is reasonable to say that fewer than half of adults in the US think the federal government is prepared to deal with a major earthquake.
  3. It is possible that approximately 35% of adults in the US think the federal government is prepared to deal with a major earthquake.
  4. A 99% confidence interval for the proportion of the population that thinks the government is prepared would be wider than the 95% confidence interval given above.
  5. None of the above

FOR QUESTIONS 48 – 80, READ CHAPTER 4, PAGES 124 – 134, AND CHAPTER 15, PAGES 599 – 610. YOU HAVE TO STUDY THESE PAGES INDEPENDENTLY.