MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Ch. 8 Sampling Distributions 8.1 Distribution of the Sample Mean 1 Describe the distribution of the sample mean: normal population. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine μ x and σ x from the given parameters of the population and the sample size. Round the answer to the nearest thousandth where appropriate. 1) μ = 58, σ = 25, n = 25 A) μ x = 58, σ x = 5 B) μ x = 58, σ x = 25 C) μ x = 58, σ x = 1 D) μ x = 11.6, σ x = 5 2) μ = 44, σ = 8, n = 21 A) μ x = 44, σ x = B) μ x = 44, σ x = C) μ x = 44, σ x = 8 D) μ x = , σ x = Provide an appropriate response. 3) What are the values of μ x and σ x for the sampling distribution of the sample mean shown? A) μ x = 340, σ x = 50 B) μ x = 50, σ x = 340 C) μ x = 340, σ x = 150 D) μ x = 340, σ x = 100 Page 1

2 4) What are the values of μ x and σ x for the sampling distribution of the sample mean shown? A) μ x = 12, σ x = 0.05 B) μ x = 0.05, σ x = 12 C) μ x = 12, σ x = 0.15 D) μ x = 12, σ x = 0.1 5) The sampling distribution of the sample mean is shown. If the sample size is n = 16, what is the standard deviation of the population from which the sample was drawn? Round to the nearest thousandth where appropriate A) σ = 200 B) σ = 800 C) σ = 12.5 D) σ = ) The sampling distribution of the sample mean is shown. If the sample size is n = 9, what is the standard deviation of the population from which the sample was drawn? Round to the nearest thousandth where appropriate A) σ = 0.09 B) σ = 0.27 C) σ = 0.01 D) σ = Page 2

3 7) Suppose a population has a mean of 7 for some characteristic of interest and a standard deviation of 9.6. A sample is drawn from this population of size 64. What is the standard error of the mean? A) 1.2 B) 0.7 C) 0.15 D) 3.3 8) Suppose a population has a mean weight of 180 pounds and a standard deviation of 25 pounds. A sample of 100 items is drawn from this population. What is the standard error of the mean? A) 2.5 B) 1.8 C) 18.0 D) ) The number of violent crimes committed in a day possesses a distribution with a mean of 2.3 crimes per day and a standard deviation of 2 crimes per day. A random sample of 100 days was observed, and the mean number of crimes for the sample was calculated. Describe the sampling distribution of the sample mean. A) approximately normal with mean = 2.3 and standard deviation = 0.2 B) approximately normal with mean = 2.3 and standard deviation = 2 C) shape unknown with mean = 2.3 and standard deviation = 2 D) shape unknown with mean = 2.3 and standard deviation = 0.2 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) A random sample of size n is to be drawn from a population with μ= 600 and σ = 100. What size sample would be necessary in order to ensure a standard error of 10? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) The amount of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of 0.1 ounce. Suppose 400 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 400 bags exceeded 10.6 ounces. A) approximately 0 B) C) D) ) The average score of all golfers for a particular course has a mean of 75 and a standard deviation of 3.5. Suppose 49 golfers played the course today. Find the probability that the average score of the 49 golfers exceeded 76. A) B) C) D) ) One year, professional sports players salaries averaged $1.6 million with a standard deviation of $0.9 million. Suppose a sample of 400 major league players was taken. Find the approximate probability that the average salary of the 400 players exceeded $1.1 million. A) approximately 1 B) approximately 0 C) D) ) Furnace repair bills are normally distributed with a mean of 273 dollars and a standard deviation of 25 dollars. If 100 of these repair bills are randomly selected, find the probability that they have a mean cost between 273 dollars and 275 dollars. A) B) C) D) ) Assume that the heights of men are normally distributed with a mean of 69.8 inches and a standard deviation of 2.8 inches. If 64 men are randomly selected, find the probability that they have a mean height greater than 70.8 inches. A) B) C) D) ) Assume that blood pressure readings are normally distributed with a mean of 117 and a standard deviation of 9.6. If 144 people are randomly selected, find the probability that their mean blood pressure will be less than 119. A) B) C) D) Page 3

4 17) According to the law of large numbers, as more observations are added to the sample, the difference between the sample mean and the population mean A) Tends to become smaller B) Tends to become larger C) Remains about the same D) Is inversely affected by the data added 18) The standard error of the mean is given by σ A) B) μ - x C) μ - x D) μ ± σ n 2 Describe the distribution of the sample mean: nonnormal population. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 19) The amount of money collected by a snack bar at a large university has been recorded daily for the past five years. Records indicate that the mean daily amount collected is $3900 and the standard deviation is $600. The distribution is skewed to the right due to several high volume days (including football game days). Suppose that 100 days were randomly selected from the five years and the average amount collected from those days was recorded. Which of the following describes the sampling distribution of the sample mean? A) normally distributed with a mean of $3900 and a standard deviation of $60 B) normally distributed with a mean of $3900 and a standard deviation of $600 C) normally distributed with a mean of $390 and a standard deviation of $60 D) skewed to the right with a mean of $3900 and a standard deviation of $600 20) A farmer was interested in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 60 or more flights on average in the next 40 rows down which he drives his tractor? A) B) C) D) ) A farmer was interest in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor? A) B) C) D) ) A farmer was interested in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 52 or fewer flights or 60 or more flights on average in the next 40 rows down which he drives his tractor? A) B) C) D) ) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will exceed $7500? A) B) C) D) Page 4

5 24) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will be less than $7000? A) B) C) D) ) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will be between $7000 and $7500? A) B) C) D) ) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, how many groups can be formed? A) 10 B) 5 C) 30 D) 60 27) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, what is the probability of the population mean falling between 5.5 and 6.5 years? A) 0.5 B) 0.4 C) 0.55 D) ) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, what is the probability of the population mean being more than 6 years? A) 0.4 B) 0.5 C) 0.1 D) ) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time how many, how many groups can be formed? A) 20 B) 10 C) 30 D) 15 30) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of at least $26? A) 0.55 B) 0.50 C) 0.60 D) ) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of between $40 and $45? A) 0.15 B) 0.05 C) 0.25 D) ) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of equal to or less than $25? A) 0.45 B) 0.86 C) 0.50 D) 0.35 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 33) The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 14 minutes and a standard deviation of 1 minutes. If data were collected from 40 randomly selected walks, describe the sampling distribution of x, the sample mean time. Page 5

6 8.2 Distribution of the Sample Proportion 1 Describe the sampling distribution of a sample proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Describe the sampling distribution of p^. 1) N = 20,000, n = 600, p = 0.3 A) Approximately normal; μp = 0.3, σp = B) Exactly normal; μp = 0.3, σp = C) Approximately normal; μp = 0.3, σp = D) Binomial; μp = 180, σp = Provide an appropriate response. 2) Professor Whata Guy surveyed a random sample of 420 statistics students. One of the questions was ʺWill you take another mathematics class?ʺ The results showed that 252 of the students said yes. What is the sample proportion, p^ of students who say they will take another math class? A) 0.6 B) 0.42 C) D) Page 6 3) To assess attitudes towards issues that affect the residents of a village, the village randomly chose 800 families to participate in a survey of life attitudes. The village received 628 completed surveys. What is the sample proportion of completed surveys? A) B) C) D) ) A national caterer determined that 87% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of The 144 people are asked to sample the catererʹs food. Will the distribution of p^, the sample proportion saying that the food is delicious, be approximately normal? Answer Yes or No. A) Yes B) No 5) A national caterer determined that 87% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of The 144 people are asked to sample the catererʹs food. If p^ is the sample proportion saying that the food is delicious, what is the mean of the sampling distribution of p^? A) 0.87 B) 1.25 C) 0.19 D) ) A national caterer determined that 37% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of The 144 people are asked to sample the catererʹs food. If p^ is the sample proportion saying that the food is delicious, what is the standard deviation of the sampling distribution of p^? A) 0.04 B) C) 0.48 D) ) A greenhouse in a tri-county area has kept track of its customers for the last several years and has determined that it has about 10,000 regular customers. Of those customers, 28% of them plant a vegetable garden in the spring. The greenhouse obtains a random sample of 800 of its customers. Is it safe to assume that the sampling distribution of p^, the sample proportion of customers that plant a vegetable garden, is approximately normal? Answer Yes or No. A) No B) Yes 8) A greenhouse in a tri-county area has kept track of its customers for the last several years and has determined that 28% of its customers plant a vegetable garden in the spring. The greenhouse obtains a random sample of 1000 of its customers. What is the mean of the sampling distribution of p^, the sample proportion of customers that plant a vegetable garden in the spring? A) 0.28 B) 0.72 C) 2800 D) 0.002

7 9) If a population proportion is believed to be 0.6, how many items must be sampled to ensure that the sampling distribution of p^ will be approximately normal? Assume that the size of the population is N = 10,000. A) 42 B) 30 C) 13 D) 60 10) A simple random sample of size n = 1040 is obtained from a population whose size is N = 1,200,000 and whose population proportion with a specified characteristic is p = Describe the sampling distribution of p^. A) Approximately normal; μp = 0.32, σp = B) Exactly normal; μp = 0.32, σp = C) Approximately normal; μp = 0.32, σp = D) Exactly normal; μp = 0.32, σp = ) According to a study conducted in one city, 38% of adults in the city have credit card debts of more than $2000. A simple random sample of n = 100 adults is obtained from the city. Describe the sampling distribution of p^, the sample proportion of adults who have credit card debts of more than $2000. A) Approximately normal; μp = 0.38, σp = B) Exactly normal; μp = 0.38, σp = C) Approximately normal; μp = 0.38, σp = D) Binomial; μp = 38, σp = Compute probabilities of a sample proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 12) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find less than 20 substandard welds? A) B) C) D) ) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find more than 25 substandard welds? A) B) C) D) ) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find between 10 and 20 substandard welds? A) B) C) D) ) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, would it be unusual for Smith to find 30 or more substandard welds? A) Yes B) No 16) The National Association of Realtors estimates that 23% of all homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at most 200 homes are going to be used as investment property? A) B) C) D) ) The National Association of Realtors estimates that 23% of all homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at least 175 homes are going to be used as investment property? A) B) C) D) Page 7

8 18) The National Association of Realtors estimates that 23% of all homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained and it was noted that 248 homes were to be used as investment property, would this be unusual? Answer Yes or No. A) Yes B) No 19) The National Association of Realtors estimates that 23% of all homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that between 175 and 200 homes are going to be used as investment property? A) B) C) D) Page 8

9 Ch. 8 Sampling Distributions Answer Key 8.1 Distribution of the Sample Mean 1 Describe the distribution of the sample mean: normal population. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) The standard error is σ x = σ. If the standard error is desired to be 10, we get: n 10 = σn = 100 n n 10 = n = = 10 n = ) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 2 Describe the distribution of the sample mean: nonnormal population. 19) A 20) A 21) A 22) A 23) A 24) A 25) A 26) A 27) A 28) A 29) A 30) A 31) A 32) A 33) By the Central Limit Theorem, the sampling distribution of x is approximately normal with μ x = μ = 14 minutes and σ x = σn = 1 = minutes Distribution of the Sample Proportion 1 Describe the sampling distribution of a sample proportion. 1) A 2) A 3) A 4) A 5) A 6) A Page 9

10 7) A 8) A 9) A 10) A 11) A 2 Compute probabilities of a sample proportion. 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A Page 10