AP Statistics Chapter 9 Homework Name_______________________________________________________ Date_______________________ Period___________
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) A candy company claims that 6% of the jelly beans in its spring mix are pink. Suppose that the candies are packaged at random in small bags containing about 70 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?
2) A national study found that 13% of college seniors regret their choice of major. Suppose a group of 115 college seniors is selected at random. May the Normal model be used to describe the distribution of the proportion of seniors in the sample who regret their choice of major? 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? 4) A realtor has been told that 42% of homeowners in a city prefer to have a finished basement. She surveys a group of 400 homeowners randomly chosen from her client list. Find the mean of the proportion of homeowners in this sample who prefer a finished basement. Find the standard deviation of the sample proportion. 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? 6) A candy company claims that its jelly bean mix contains 20% blue jelly beans. Suppose that the candies are packaged at random in small bags containing about 330 jelly beans. Find the mean of the proportion of blue jelly beans in a bag. 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 68-95-99.7 Rule to provide the appropriate response. 7) If each student tosses the coin 200 times, about 99.7% of the sample proportions should be between what two numbers?
8) 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 Answer the question. 9) Information on a packet of seeds claims that the germination rate is 75%. Should you be surprised if 139 of the 150 seeds in the packet germinate?
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Solve the problem. 10) One hospital has found that 13.95% of its patients require specially equipped beds. If the hospital has 258 beds, what percentage of the beds should be specially equipped if the hospital wishes to be "pretty sure" of having enough of these beds? Assume that the hospital wants only a 5% chance that they could run short of these beds, even when the hospital is fully occupied. 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) 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 ?
12) The mean annual income for women in one city is $28,520 and the standard deviation of the incomes is $5600. The distribution of incomes is skewed to the right. Suppose a sample of 12 women is selected at random from the city and the mean income, x is determined for the women in the sample. May the Normal model be used to describe the sampling distribution of the mean, x ?
Describe the indicated sampling distribution. 13) A museum offers several levels of membership, as shown in the table. Member Amount Percent of Category of Donation ($) Members Individual $50 43 Family $80 28 Family Plus $120 16 Sponsor $200 9 Patron $500 4 Suppose that 20 museum members are selected at random. Describe the sampling distribution of the mean donation for these 20 members. In particular, state whether the distribution of the sample mean is normal or approximately normal, and give its mean and standard deviation.
At a large university, students have an average credit card debt of $2500, with a standard deviation of $1200. A random sample of students is selected and interviewed about their credit card debt. Use the 68-95-99.7 Rule to answer the question about the mean credit card debt for the students in this sample. 14) If we imagine all the possible random samples of 100 students at this university, 95% of the samples should have means between what two numbers?
Find the specified probability, from a table of Normal probabilities. Assume that the necessary conditions and assumptions are met. 15) The weight of crackers in a box is stated to be 16 ounces. The amount that the packaging machine puts in the boxes is believed to have a Normal model with mean 16.15 ounces and standard deviation 0.3 ounces. What is the probability that the mean weight of a 10-box case of crackers is below 16 ounces?
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Find the indicated probability. 16) A museum offers several levels of membership, as shown in the table. Member Amount Percent of Category of Donation ($) Members Individual $50 43 Family $80 28 Family Plus $120 16 Sponsor $200 9 Patron $500 4 During a membership drive, a volunteer enrolls 20 new members. If these 20 new members can be considered a random sample of all the museum's members, what is the probability that the mean donation from the new members is at least $150?
17) You pay $10 and roll a die. If you get a five or six, you win $30. If not, you get to roll again. If you get a 5 or 6 on the second roll, you get your $10 back. Suppose you play this game 30 times. What's the probability that your mean winnings are more than $5? Find the specified probability, from a table of Normal probabilities. Assume that the necessary conditions and assumptions are met. 18) 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 less than $5000 on dinner?
Solve the problem. 19) Packages received by a parcel service have a mean weight of 12.3 pounds with a standard deviation of 3.2 pounds. On one particular day, the parcel service receives 38 packages. There is a 2.5% chance that the mean weight of these 38 packages will be above what value? 20) Bill's hourly earnings have a skewed distribution with a mean of $84.20 and a standard deviation of $6.90. In a typical week he works 30 hours. In the best 2.5% of such weeks, his average hourly earnings are above what amount? (In other words, there is a 2.5% chance that his average hourly earning for the 30 hours will be greater than what value?) Provide an appropriate response. 21) The distribution of the lengths of time that employees work at one company is slightly skewed to the right and the mean and standard deviation of the times are known. Is it possible to determine the probability that a randomly selected employee works longer than 5 years at the company? Is it possible to determine the probability that the mean time for 20 randomly selected employees is more than 5 years? Explain your responses. 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 $25,000.
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23) A zoo offers several levels of membership, as shown in the table. Member Charge for Percent of Category Membership ($) Members Individual $55 35 Family $80 25 Family Plus $90 20 Sponsor $100 10 Zoo Keeper $200 10 If we select a random sample of 50 zoo members, would you expect their membership charges to follow a Normal model? Explain.
24) The President's job approval rating is always a hot topic. Your local paper conducts a poll of 100 randomly selected adults to determine the President's job approval rating. A CNN/USA Today/Gallup poll conducts a poll of 1010 randomly selected adults. Which poll is more likely to report that the President's approval rating is below 50%, assuming that his actual approval rating is 54%? Explain.
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