Name: Class: _ Date: _ AP Statistics - Chapter 6,7 Quiz Short Answer Scenario 6-1 The probability distribution below is for the random variable X = number of mice caught in traps during a single night in small apartment building. X 0 1 2 3 4 5 P(X) 0.12 0.20 0.31 0.14 0.16 0.07 1. Use Scenario 6-1. Describe in words and find its value. 2. Express the event trapping at least one mouse in terms of X and find its probability. Scenario 6-2 Joe the barber charges $32 for a shave and haircut and $20 for just a haircut. Based on experience, he determines that the probability that a randomly selected customer comes in for a shave and haircut is 0.85, the rest of his customers come in for just a haircut. Let J = what Joe charges a randomly-selected customer. 3. Use Scenario 6-2. Give the probability distribution for J. 4. Use Scenario 6-2. Find and interpret the mean of J,. 5. Use Scenario 6-2. Find and interpret the standard deviation of J,. Scenario 6-7 The manager of a children s puppet theatre has determined that the number of adult tickets he sells for a Saturday afternoon show is a random variable with a mean of 28.3 tickets and a standard deviation of 5.3 tickets. The mean number of children s tickets he sells is 42.5, with a standard deviation of 8.1. 6. Use Scenario 6-7. The children s tickets sell for $6. Let T = the money he collects from all ticket sales (adults and children) on a random Saturday. Assume (unrealistically, perhaps) that the number of tickets sold to adults is independent of the number sold to children. What are the mean and standard deviation of T? 7. Mr. Voss and Mr. Cull bowl every Tuesday night. Over the past few years, Mr. Voss s scores have been approximately Normally distributed with a mean of 212 and a standard deviation of 31. During the same period, Mr. Cull s scores have also been approximately Normally distributed with a mean of 230 and a standard deviation of 40. Assuming their scores are independent, what is the probability that Mr. Voss scores higher than Mr. Cull on a randomly-selected Tuesday night? 8. Identify each underlined number as a parameter or statistic. Use appropriate notation to describe each number. A 1993 survey conducted by the Richmond Times-Dispatch one week before election day asked voters which candidate for the state s attorney general they would vote for. 37% of the respondents said they would vote for the Democratic candidate. On election day, 41% actually voted for the Democratic candidate. 1
Name: 9. A large pet store that specializes in tropical fish has several thousand guppies. The store claims that the guppies have a mean length of 5 cm and a standard deviation of 0.5 cm. You come to the store and buy 10 randomly-selected guppies and find that the mean length of your 10 guppies is 4.8 cm. This makes you suspect that the mean fish length is not what the store says it is. To explore this further, you assume that the length of guppies is Normally distributed and use a computer to simulate 200 samples of 10 guppies from the store s claimed population. Below is a dotplot of the means from these 200 samples. (a) What is the population in this situation, and what population parameters have we been given? (b) The distribution of one sample is described in the opening paragraph. What information have we been given about this sample? (c) Is the dotplot above a sampling distribution? Explain. (d) Do you think the store is being honest about the length of its guppies? Justify your answer. 10. Suppose that in a certain community, 40% of the residents would answer Yes to the question, Do you know the names of at least five other people who live on your block? Suppose you plan to take a random sample of 100 people from this community and calculate the proportion of people in your sample whose response to this question is Yes. (a) What are the parameter and the statistic in this situation? (b) What does the sampling distribution of this statistic describe? (c) What does it mean to say that the statistic in this case is a unbiased estimator of the parameter? (d) Suppose that in a much larger community, 40% of the residents would also answer Yes to the question. If you took a sample of 100 individuals from this much larger community, would the sampling distribution of the statistic be different? In what way? (e) If you took a sample of 50 individuals instead of 100 from the original community, would the sampling distribution of the statistic change? In what way? 2
Name: 11. Below are histograms of the values taken by three sample statistics in several hundred samples from the same population. The true value of the population parameter is marked on each histogram. (a) Which statistic has the largest bias among these three? Justify your answer. (b) Which statistic has the lowest variability among these three? Justify your answer. (c) Based on the performance of the three statistics in many samples, which is preferred as an estimate of the parameter? Why? 12. According to the 2000 U.S. Census, 80% of Americans over the age of 25 have earned a high school diploma. Suppose we take a random sample of 120 Americans and record the proportion, of individuals in our sample that have a high school diploma. (a) What are the mean and standard deviation of the sampling distribution of (b) What is the approximate shape of the sampling distribution? Justify your answer. (c) Suppose our sample size was 30 instead of 120. Compare the shape, center, and spread of this sampling distribution to the one in parts (a) and (b). (d) You live in a small town with only 500 residents over the age of 25. What is the largest possible sample you can take from your town and still be able to calculate the standard deviation of sampling distribution of using the method presented in the textbook? Explain. 13. The customer care manager at a cell phone company keeps track of how long each help-line caller spends on hold before speaking to a customer service representative. He finds that the distribution of wait times for all callers has a mean of 12 minutes with a standard deviation of 5 minutes. The distribution is moderately skewed to the right. Suppose the manager takes a random sample of 10 callers and calculates their mean wait time, (a) What is the mean of the sampling distribution of (b) Is it possible to calculate the standard deviation of If it is, do the calculation. If it isn t, explain why. (c) Do you know the approximate shape of the sampling distribution of If so, describe the shape and justify your answer. If not, explain why not. 3
Name: 14. The weights of Granny Smith apples from a large orchard are Normally distributed with a mean of 380 gm and a standard deviation of 28 gm. (a) A single apple is selected at random from this orchard. What is the probability that it weighs more 400 gm? (b) Three apples are selected at random from this orchard. What is the probability that their mean weight is greater than 400 gm.? (c) Explain why the probabilities in (a) and (b) are not equal. 15. A certain beverage company is suspected of underfilling its cans of soft drink. The company advertises that its cans contain, on average, 12 ounces of soda with standard deviation 0.4 ounce. For the questions that follow, suppose that the company is telling the truth. (a) Can you calculate the probability that a single randomly selected can contains 11.9 ounces or less? If so, do it. If not, explain why you cannot. (b) A quality control inspector measures the contents of an SRS of 50 cans of the company s soda and calculates the sample mean. What are the mean and standard deviation of the sampling distribution of for samples of size n = 50? (c) The inspector in part (b) obtains a sample mean of ounces. Calculate the probability that a random sample of 50 cans produces a sample mean amount of 11.9 ounces or less. Be sure to explain why you can use a Normal calculation. (d) What would you conclude about whether the company is underfilling its cans of soda? Justify your answer. Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume that X is Normal with mean $360 and standard deviation $50. What is P(X > $400)? A. 0.2119 B. 0.2881 C. 0.5319 D. 0.7881 E. 0.8450 2. In a large population of college students, 20% of the students have experienced feelings of math anxiety. If you take a random sample of 10 students from this population, the mean and standard deviation of the number of students in the sample who have experienced math anxiety is: A. µ = 1.6; σ = 1.414 B. µ = 1.6; σ = 1.265 C. µ = 2; σ = 1.6 D. µ = 2; σ = 1.414 E. µ = 2; σ = 1.265 4
Name: Scenario 6-11 It has been estimated that about 30% of frozen chickens are contaminated with enough salmonella bacteria to cause illness if improperly cooked. Chickens are delivered to grocery stores in crates of 24. Assume the chickens are independently selected for inclusion in the crate. 3. Use Scenario 6-11. The probability that a certain crate has more than 4 contaminated chickens is A. 0.0424 B. 0.0686 C. 0.8889 D. 0.9313 E. 0.9576 4. What is distribution of values taken by a statistic in all possible samples of the same size from the same population called? A. the probability that the statistic is obtained. B. the population parameter. C. the variance of the values. D. the sampling distribution of the statistic. E. the distribution of sample data. 5. If a statistic used to estimate a parameter is such that the mean of its sampling distribution is equal to the true value of the parameter being estimated, what is the statistic said to be? A. random B. biased C. a proportion D. unbiased E. non-varying. 6. Suppose you take a random sample of size 25 from a population with mean of 120 and a standard deviation of 15. Your sample has a mean of 115 and a standard deviation of 13.8. Which of the following has a mean of 120 and a standard deviation of 3? A. the distribution of the population B. the distribution of the sample data. C. the sampling distribution of the sample mean. D. the sampling distribution of the population mean. E. No important distribution related to this situation has the given mean and standard deviation. 7. In order to use the formula to calculate the standard deviation of the sampling distribution of the sample mean, which of the following conditions must be met? I. II. The population s distribution is approximately Normal. III. The sample size is less than 10% of the population size. A. I only B. II only C. III only D. III and either I or II E. All three conditions must be met. 5
Name: 8. A student investigating study habits asks a simple random sample of 16 students at her school how many minutes they spent on their English homework the previous night. Suppose the actual parameter values for this variable are minutes and minutes. Which of the following best describes what we know about the sampling distribution of means for the student s sample? A. unknown; shape of distribution unknown B. distribution approximately Normal C. shape of distribution unknown D. distribution approximately Normal E. shape of distribution unknown 9. Olive weights are classified according to a unique set of adjectives implying great size. For example, the mean weight of olives classified as Colossal is 7.7 gm. Suppose a particular company s crop of Colossal olives is approximately Normally distributed with a mean of 7.7 gm and a standard deviation of 0.2 gm. Which of the following represents the probability that the mean weight of a random sample of 3 olives from this population is greater than 8 gm? A. D. B. E. C. 10. In a study of the effects of acid rain, a random sample of 100 trees from a particular forest is examined. Forty percent of the trees show some signs of damage. Which of the following statements is correct? A. 40% is a parameter B. 40% is a statistic C. 40% of all trees in the forest show some signs of damage D. More than 40% of the trees in the forest show some signs of damage E. Less than 40% of the trees in the forest show some signs of damage 6
Name: 11. A statistic is said to be unbiased if A. the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice. B. the mean of its sampling distribution is equal to the true value of the parameter being estimated. C. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. D. the value from any sample is equal to the parameter being estimated. E. it is used for honest purposes only. 12. The chipmunk population in a certain area is known to have a mean weight of 84 gm and a standard deviation of 18 gm. A wildlife biologist weighs 9 chipmunks that have been caught in live traps before releasing them. Which of the following best describes what we know about the sampling distribution of means for the biologist s sample? A. distribution approximately Normal B. shape of distribution unknown C. distribution approximately Normal D. unknown; distribution approximately Normal E. unknown; shape of distribution unknown 13. Suppose you are sampling from a distribution that is strongly skewed left. Which of the following statements about the sampling distribution of the sample mean is true? A. As the sample size increases, the shape of the sampling distribution gets closer and closer to a Normal distribution. B. As the sample size increases, the shape of the sampling distribution gets closer and closer to the shape of the population distribution. C. As the sample size increases, the mean of the sampling distribution gets closer to the population mean. D. Regardless of the sample size, the shape of the sampling distribution is similar to the shape of the population distribution. E. Regardless of the sample size, the standard deviation of the sampling distribution is approximately equal to the standard deviation of the population. 14. In order to use the formula to calculate the standard deviation of the sampling distribution of the sample proportion, which of the following conditions must be met? I. II. The population s distribution is approximately Normal. III. The sample size is less than 10% of the population size. A. I only B. II only C. III only D. I and III E. All three conditions must be met. 7