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1 Chapter 18 Review Name Date Period Write complete answers, using complete sentences where necessary.show your work to get full credit. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the Normal model may be used to describe the distribution of the sample proportions. If the Normal model may be used, list the conditions and explain why each is satisfied. If Normal model may not be used, explain which condition is not satisfied. 1) In a large statistics class, the professor has each student toss a coin 12 times and calculate the 1) proportion of tosses that come up tails. The students then report their results, and the professor plots a histogram of these several proportions. May a Normal model be used here? A) A Normal model may not be used because the population distribution is not Normal. B) A Normal model may be used: Coin flips are independent of each other - no need to check the 10% condition Success/Failure condition is satisfied: np = nq = 6 which are both less than 10 C) A Normal model may be used: Coin flips are independent of each other - no need to check the 10% condition Success/Failure condition is satisfied: np = nq = 12 which are both greater than 10 D) A Normal model should not be used because the sample size is not large enough to satisfy the success/failure condition. For this sample size, np = 6 = nq = 6 which are both less than 10. E) A Normal model should not be used because the 10% condition is not satisfied: the sample size, 12, is larger than 10% of the population of all coins flips. 2) A candy company claims that 25% of the jelly beans in its spring mix are pink. Suppose that the candies are packaged at random in small bags containing about 300 jelly beans. A class of students opens several bags, counts the various colors of jelly beans, and calculates the proportion that are pink in each bag. Is it appropriate to use a Normal model to describe the distribution of the proportion of pink jelly beans? A) A Normal model is not appropriate because the 10% condition is not satisfied: the sample size, 300, is larger than 10% of the population of all jelly beans. B) A Normal model is not appropriate because the success/failure condition is not satisfied: np = 75 and nq = 225 neither of which is less than 10 C) A Normal model is appropriate: Randomization condition is satisfied: the 300 jelly beans in the bag are selected at random and can be considered representative of all jelly beans 10% condition is satisfied: the sample size, 300, is less than 10% of the population of all jelly beans. success/failure condition is satisfied: np = 75 and nq = 225 are both greater than 10 D) A Normal model is not appropriate because the population distribution is not Normal. E) A Normal model is not appropriate because the randomization condition is not satisfied: the 300 jelly beans in the bag are not a simple random sample and cannot be considered representative of all jelly beans. 2) 1

2 Find the mean of the sample proportion. 3) Assume that 25% of students at a university wear contact lenses. We randomly pick 200 students. What is the mean of the proportion of students in this group who may wear contact lenses? A) µ = 7.07% B) µ = 12.5% C) µ = 50% D) µ = 3.06% E) µ = 25% 3) Find the standard deviation of the sample proportion. 4) Based on past experience, a bank believes that 5% of the people who receive loans will not make payments on time. The bank has recently approved 300 loans. What is the standard deviation of the proportion of clients in this group who may not make timely payments? A) = 3.87% B) = 1.26% C) = 3.77% D) = 1.58% E) = 1.29% 4) 5) Assume that 15% of students at a university wear contact lenses. We randomly pick 200 students. What is the standard deviation of the proportion of students in this group who may wear contact lenses? A) = 2.74% B) = 5.05% C) = 2.52% D) = 5.48% E) = 6.38% 5) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. In a large class, the professor has each person toss a coin several times and calculate the proportion of his or her tosses that come up heads. The students then report their results, and the professor plots a histogram of these proportions. Use the Rule to provide the appropriate response. 6) If each student tosses the coin 50 times, about 68% of the sample proportions should be 6) between what two numbers? 7) The Atilla Barbell Company makes bars for weight lifting. The weights of the bars are independent and are normally distributed with a mean of 720 ounces (45 pounds) and a standard deviation of 4 ounces. The bars are shipped 10 in a box to the retailers. The weights of the empty boxes are normally distributed with a mean of 320 ounces and a standard deviation of 8 ounces. The weights of the boxes filled with 10 bars are expected to be normally distributed with a mean of 7,520 ounces and a standard deviation of 7) Find the specified probability, from a table of Normal probabilities. Assume that the necessary conditions and assumptions are met. 8) Based on past experience, a bank believes that 7% of the people who receive loans will not 8) make payments on time. The bank has recently approved 300 loans. What is the probability that over 8% of these clients will not make timely payments? 2

3 9) Researchers believe that 7% of children have a gene that may be linked to a certain childhood disease. In an effort to track 50 of these children, researchers test 950 newborns for the presence of this gene. What is the probability that they find enough subjects for their study? 9) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 10) In a large class, the professor has each person toss a coin 200 times and calculate the proportion of his or her tosses that were tails. The students then report their results, and the professor records the proportions. One student claims to have tossed her coin 200 times and found 58% tails. What do you think of this claim? Explain your response. A) This is a typical result. Her proportion is only 1.60 standard deviations above the mean. B) This is an extremely unlikely result. Her proportion is about 5.2 standard deviations above the mean. C) This is an unusual result. Her proportion is about 2.26 standard deviations above the mean. D) This is a typical result. Her proportion is only 2.26 standard deviations above the mean. E) This is a fairly unusual result. Her proportion is about 1.60 standard deviations above the mean. 10) Determine whether the Normal model may be used to describe the distribution of the sample means. If the Normal model may be used, list the conditions and explain why each is satisfied. If Normal model may not be used, explain which condition is not satisfied. 11) A researcher believes that scores on an IQ test for students at a certain college are skewed to the 11) left with a mean of 72 and a standard deviation of 15. The college has a total of 850 students. The researcher selects 200 students at random and determines the mean test score, x, for the students in the sample. May the Normal model be used to describe the sampling distribution of the mean, x? A) No, Normal model may not be used. Large enough sample condition is not satisfied: since the original population is skewed to the left, 200 is not a large enough sample B) No, Normal model may not be used. 10% condition is not satisfied : the 200 students in the sample represent more than 10% of students at the college. This means that the independence assumption will not be satisfied. C) No, Normal model may not be used since scores for students at the college are not normally distributed D) No, Normal model may not be used. Randomization condition is not satisfied since the 200 students in the sample may not be representative of all students at the college. E) Yes, Normal model may be used. Randomization condition: The students were selected at random Independence assumption: It is reasonable to think that scores of randomly selected students are mutually independent. Large enough sample condition: a sample of 200 is certainly large enough, whatever the distribution of the scores in the original population 10% condition is satisfied since the 200 students represent more than 10% of students at the college. 3

4 12) The weights of men in a certain city are normally distributed with a mean of 153 lb and a standard deviation of 22 lb. Suppose a sample of 3 men is selected at random from the city and the mean weight, x is determined for the men in the sample. May the Normal model be used to describe the sampling distribution of the mean, x? A) Yes, Normal model may be used. Randomization condition: The men were selected at random Independence assumption: It is reasonable to think that weights of randomly selected men are mutually independent. Large enough sample condition: Since the original population is normally distributed, a sample of 3 is large enough, in fact any sample would be large enough. 10% condition: the 3 men in the sample certainly represent less than 10% of men in the city. B) No, Normal model may not be used: Large enough sample condition is not satisfied: since the sample size is only 3 C) Randomization condition is not satisfied: The men in the sample do not represent a simple random sample and may not be representative of all men in the city D) No, Normal model may not be used: 10% condition: is not satisfied since the 3 men in the sample represent less than 10% of men in the city. E) Independence assumption is not satisfied: Since the men may be related, the chance of selecting a heavy man depends on who has already been selected. 12) Describe the indicated sampling distribution. 13) The weights of people in a certain population are normally distributed with a mean of 158 lb and a standard deviation of 25 lb. Describe the sampling distribution of the mean for samples of size 9. In particular, state whether the distribution of the sample mean is normal or approximately normal and give its mean and standard deviation. A) Normal, mean = 158 lb, standard deviation = 8.33 lb B) Approximately normal, mean = 158 lb, standard deviation = 2.78 lb C) Normal, mean = 158 lb, standard deviation = 2.78 lb D) Approximately normal, mean = 158 lb, standard deviation = 8.33 lb E) Normal, mean = 158 lb, standard deviation = 25 lb 13) 14) Let x represent the number which shows up when a balanced die is rolled. Then x is a random variable with a mean of 3.5 and a standard deviation of Let x denote the mean of the numbers obtained when the die is rolled 36 times. Determine the sampling distribution of x. In particular, state whether the distribution of the sample mean is normal or approximately normal and give its mean and standard deviation. A) Normal, mean = 3.5, standard deviation = 0.29 B) Normal, mean = 3.5, standard deviation = 0.05 C) Approximately normal, mean = 3.5, standard deviation = 0.05 D) Approximately normal, mean = 3.5, standard deviation = 0.29 E) Approximately normal, mean = 3.5, standard deviation = ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. At a large university, students have an average credit card debt of $2500, with a standard deviation of $1200. A random sample of students is selected and interviewed about their credit card debt. Use the Rule to answer the question about the mean credit card debt for the students in this sample. 15) If we imagine all the possible random samples of 250 students at this university, 68% of 15) the samples should have means between what two numbers? 4

5 Find the specified probability, from a table of Normal probabilities. Assume that the necessary conditions and assumptions are met. 16) The number of hours per week that high school seniors spend on computers is normally 16) distributed, with a mean of 6 hours and a standard deviation of 2 hours. 80 students are chosen at random. Let y represent the mean number of hours spent on the computer for this group. Find the probability that y is between 6.2 and ) The weight of crackers in a box is stated to be 16 ounces. The amount that the packaging machine puts in the boxes is believed to have a Normal model with mean ounces and standard deviation 0.3 ounces. What is the probability that the mean weight of a 10-box case of crackers is below 16 ounces? 17) 18) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 91 inches, and a standard deviation of 12 inches. What is the probability that the mean annual snowfall during 36 randomly picked years will exceed 93.8 inches? 18) 19) A restaurant's receipts show that the cost of customers' dinners has a skewed distribution with a mean of $54 and a standard deviation of $18. What is the probability that the next 100 customers will spend a total of at least $5800 on dinner? 19) 5

6 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 20) At a shoe factory, the time taken to polish a finished shoe has a mean of 3.7 minutes and a standard deviation of 0.48 minutes. If 44 shoes are polished, there is a 5% chance that the mean time to polish the shoes is below what value? A) 3.51 minutes B) 3.89 minutes C) 3.58 minutes D) 3.82 minutes E) 3.53 minutes 20) Provide an appropriate response. 21) A researcher selects samples of size n from a population and determines the mean test score for each sample. The mean and standard deviation of the sampling distribution are 60 and 12 respectively. If the sample size is multiplied by a factor of 4, what will the mean and standard deviation of the new sampling distribution be? A) mean will be 60, standard deviation will be 3 B) mean will be 60, standard deviation will be 6 C) mean will be 30, standard deviation will be 3 D) mean will be 30, standard deviation will be 6 E) mean will be 60, standard deviation will be 24 21) 22) In which of the following situations does the Central Limit Theorem allow use of a Normal model for the sampling distribution model: A: Weights of students are normally distributed. We wish to determine the probability that the mean weight for a random sample of 4 students is greater than 150 pounds. B: The distribution of test scores of students is slightly skewed to the right. We wish to determine the probability that the mean score for a random sample of 8 students is greater than 80. C: The distribution of incomes of students is strongly skewed to the right. We wish to determine the probability that the mean income for a random sample of 100 students is greater than $25,000. A) A and C B) A only C) C only D) A, B, and C E) A and B 22) 23) A sample is chosen randomly from a population that was strongly skewed to the right. Describe the sampling distribution model for the sample mean if the sample size is small. A) Skewed right, center at µ, standard deviation /n B) Normal, center at µ, standard deviation / n C) There is not enough information to describe the sampling distribution model. D) Normal, center at µ, standard deviation /n E) Skewed right, center at µ, standard deviation / n 23) 6

7 24) A certain population is bimodal. We want to estimate its mean, so we will collect a sample. Which should be true if we use a large sample rather than a small one? I. The distribution of our sample data will be more clearly bimodal. II. The sampling distribution of the sample means will be approximately normal. III. The variability of the sample means will be smaller. A) I, II, and III B) III only C) I only D) II and III E) II only 24) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 25) A can of pumpkin pie mix contains a mean of 30 ounces and a standard deviation of 2 ounces. The contents of the cans are normally distributed. What is the probability that four randomly selected cans of pumpkin pie mix contain a total of more than 126 ounces? 25) 7