Problem Set 08 Sampling Distribution of Sample Mean

Transcription

1 Problem Set 08 Sampling Distribution of Sample Mean MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the requested probability. 1) The table reports the distribution of pocket money, in bills, of the 6 students in a statistics seminar. 1) Student Hannah Ming Keshaun Tameeka Jose Vaishali Amount, in dollars For a random sample of size two, find the probability, expressed as a percent rounded to the nearest tenth, that the sample mean will be within $1 of the population mean. A) 66.7% B) 80.0% C) 78.6% D) 73.3% 2) The table reports the GPA for each of five students in a statistics class. 2) Student Maria Alvin Elvis Ingrid Rashad GPA For a random sample of size two, find the probability, expressed as a percent, that the sample mean will be within 0.1 of the population mean. A) 80% B) 30% C) 70% D) 20% 3) The test scores of 5 students are under consideration. The following is the dotplot for the sampling distribution of the sample mean for samples of size 2. 3) Find the probability, expressed as a percent, that the sample mean will be equal to the population mean. A) 5% B) 10% C) 20% D) 30% 4) The test scores of 5 students are under consideration. The following is the dotplot for the sampling distribution of the sample mean for samples of size 2. 4) Find the probability, expressed as a percent, that the sample mean will be within 1 point of the population mean. A) 30% B) 20% C) 25% D) 10% 1

2 Provide an appropriate response. 5) What is the sampling distribution of a statistic? A) The distribution of all possible sizes of samples from a population that can be used to make observations of the statistic B) The distribution of observations of the statistic for all possible sizes of samples from a population C) The distribution of all possible observations of the statistic for samples of a given size from a population D) The distribution of observations of a variable in a sample for a given value of the statistic 5) 6) As a general rule, you cannot expect to exactly determine the sampling distribution of a statistic. Why? A) Many populations are too large. B) Many populations are not uniform. C) Many populations are not normal. D) Many populations are too small. 6) 7) Which of the following is not synonymous with the sampling distribution of the sample mean? A) Distribution of a variable in a sample of a given size for a given x B) Distribution of all possible sample means from samples of a given size C) Distribution of the variable x D) Distribution of x 7) 8) What generally happens to the sampling distribution of the sample mean as the sample size is decreased? A) It becomes more tightly concentrated around the population mean. B) It becomes less tightly concentrated around the population mean. C) It is unaffected. D) None of the above 8) Use the given table to determine the mean, μ x, of the variable x for the given sample size. 9) Sample x 4, 5, , 5, 9 6 4, 8, 9 7 5, 8, A) 7 B) 6 C) 8.7 D) 6.5 9) For samples of the specified size from the population described, find the mean and standard deviation of the sample mean x. 10) The mean and the standard deviation of the sampled population are, respectively, and ) n = 64 A) μ = 21.9; σ = 2.7 B) μ = 2.7; σ = = 211.7; σ x = 2.7 D) μ x = 309.1; σ x = 0.8 2

3 11) The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain range, the annual snowfall has a mean of 106 inches and a standard deviation of 10 inches. Suppose the snowfalls are sampled during randomly picked years. For samples of size 25, determine the mean and standard deviation of x. A) μ = 10; σ = 106 B) μ = 106; σ = 2 = 106; σ x = 10 D) μ x = 2; σ x = ) 12) One barge from Inland Waterways, Inc. can carry a load of lb. Records of past trips show that the weights of the cans that it carries have a mean of 109 lb and a standard deviation of 14 lb. For samples of size 49, find the mean and standard deviation of x. A) μ = 109; σ = 2 B) μ = 2; σ = 109 = 14; σ x = 109 D) μ x = 109; σ x = 14 12) 13) One truck from Lakeland Trucking, Inc. can carry a load of lb. Records show that the weights of boxes that it carries have a mean of 75 lb and a standard deviation of 16 lb. For samples of size 64, find the mean and standard deviation of x. A) μ = 75; σ = 16 B) μ = 2; σ = 75 = 16; σ x = 75 D) μ x = 75; σ x = 2 13) Provide an appropriate response. 14) The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B denote the events described below. 14) Event A: The height of a randomly selected person is 5 inches or more from the population mean. Event B: The mean height in a random sample of 16 people is 5 inches or more from the population mean. True or false, the probability of event A is greater than the probability of event B? A) True B) False 15) The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B denote the events described below. 15) Event A: The height of a randomly selected person is within 3 inches of the population mean. Event B: The mean height in a random sample of 16 people is within 3 inches of the population mean. True or false, the probability of event A is greater than the probability of event B? A) True B) False Identify the distribution of the sample mean. In particular, state whether the distribution of x is normal or approximately normal and give its mean and standard deviation. 16) The weights of people in a certain population are normally distributed with a mean of 152 lb and a standard deviation of 22 lb. Determine the sampling distribution of the mean for samples of size 2. A) Approximately normal, mean = 152 lb, standard deviation = lb B) Normal, mean = 152 lb, standard deviation = lb C) Normal, mean = 152 lb, standard deviation = 22 lb D) Approximately normal, mean = 152 lb, standard deviation = 11 lb 16) 3

4 17) The mean annual income for adult women in one city is $28,520 and the standard deviation of the incomes is $5700. The distribution of incomes is skewed to the right. Determine the sampling distribution of the mean for samples of size 132. A) Normal, mean = $28,520, standard deviation = $496 B) Approximately normal, mean = $28,520, standard deviation = $496 C) Normal, mean = $28,520, standard deviation = $43 D) Approximately normal, mean = $28,520, standard deviation = $ ) 18) For the population of one town, the number of siblings, x, is a random variable whose relative frequency histogram has a reverse J-shape. The mean number of siblings is 1.1 and the standard deviation is 1.3. Let x denote the mean number of siblings for a random sample of size 39. Determine the sampling distribution of the mean for samples of size 39. A) Normal, mean = 1.1, standard deviation = 1.3 B) Approximately normal, mean = 1.1, standard deviation = 1.3 C) Approximately normal, mean = 1.1, standard deviation = 0.21 D) Normal, mean = 1.1, standard deviation = ) Find the indicated probability or percentage for the sampling error. 19) The distribution of weekly salaries at a large company is right skewed with a mean of $1000 and a standard deviation of $350. What is the probability that the sampling error made in estimating the mean weekly salary for all employees of the company by the mean of a random sample of weekly salaries of 50 employees will be at most $50? A) B) C) D) Cannot be determined, because the distribution of the population is not normal. 19) 20) Scores on an aptitude test are distributed with a mean of 220 and a standard deviation of 30. The shape of the distribution is unspecified. What is the probability that the sampling error made in estimating the population mean by the mean of a random sample of 50 test scores will be at most 5 points? A) B) C) D) Cannot be determined, because the distribution of the population is not known to be normal. 20) 21) The amount of coffee that a filling machine puts into an 8-ounce jar is normally distributed with a mean of 8.2 ounces and a standard deviation of 0.18 ounce. What is the probability that the sampling error made in estimating the mean amount of coffee for all 8-ounce jars by the mean of a random sample of 100 jars will be at most 0.02 ounce? A) B) C) D) ) 22) The monthly expenditures on food by single adults in one city are normally distributed with a mean of $410 and a standard deviation of $70. What is the probability that the sampling error made in estimating the mean monthly expenditure of all single adults in that city by the mean of a random sample of 90 such adults will be at most $10? A) B) C) D) ) 4

5 23) Scores on a chemistry final exam are normally distributed with a mean of 280 and a standard deviation of 50. Determine the percentage of samples of size 4 that will have mean scores within 35 points of the population mean score of 280. A) 83.84% B) 91.92% C) 99.48% D) 51.60% 23) Provide an appropriate response. 24) The mean annual income for adult women in one city is $28,520 and the standard deviation of the incomes is $5,190. The distribution of incomes is skewed to the right. For samples of size 30, which of the following statements best describes the sampling distribution of the mean? A) The distribution of x is skewed to the right. B) x is approximately normally distributed. C) x is normally distributed. D) Nothing can be said about the distribution of x. 24) 25) Let x represent the number which shows up when a balanced die is rolled. Then x is a random variable with a uniform distribution. Let x denote the mean of the numbers obtained when the die is rolled 32 times. For samples of size 32, which of the following statements concerning the sampling distribution of the mean is true? A) The distribution of x is uniform. B) x is normally distributed. C) x is approximately normally distributed. D) None of the above statements is true. 25) 5

6 Answer Key Testname: PS 08 1) D 2) C 3) C 4) B 5) C 6) A 7) A 8) B 9) D 10) C 11) B 12) A 13) D 14) A 15) B 16) B 17) B 18) C 19) B 20) A 21) C 22) D 23) A 24) B 25) C 6