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AP Statistics Sem 1Study Guide Name: Part 2: Multiple Choice. Circle the letter corresponding to the best answer. 1. You measure the age, marital status and earned income of an SRS of 1463 women. The number and type of variables you have measured is (a) 1463; all quantitative. (b) four; two categorical and two quantitative. (c) four; one categorical and three quantitative. (d) three; two categorical and one quantitative. (e) three; one categorical and two quantitative. 2. Consumers’ Union measured the gas mileage in miles per gallon of 38 1978–1979 model automobiles on a special test track. The pie chart below provides information about the country of manufacture of the model cars used by Consumers Union. Based on the pie chart, we may conclude that: (a) Japanese cars get significantly lower gas mileage than cars of other countries. This is because their slice of the pie is at the bottom of the chart. (b) U.S cars get significantly higher gas mileage than cars from other countries. (c) Swedish cars get gas mileages that are between those of Japanese and U.S. cars. (d) Mercedes, Audi, Porsche, and BMW represent approximately a quarter of the cars tested. (e) More than half of the cars in the study were from the United States. 3. A researcher reports that, on average, the participants in his study lost 10.4 pounds after two months on his new diet. A friend of yours comments that she tried the diet for two months and lost no weight, so clearly the report was a fraud. Which of the following statements is correct? (a) Your friend must not have followed the diet correctly, since she did not lose weight. (b) Since your friend did not lose weight, the report must not be correct. (c) The report only gives the average. This does not imply that all participants in the study lost 10.4 pounds or even that all lost weight. Your friend’s experience does not necessarily contradict the study results. (d) In order for the study to be correct, we must now add your friend’s results to those of the study and recompute the new average. (e) Your friend is an outlier. 1 4. The following is an ogive on the number of ounces of alcohol (one ounce is about 30 mL) consumed per week in a sample of 150 students. A study wished to classify the students as “light”, “moderate”, “heavy” and “problem” drinkers by the amount consumed per week. About what percentage of students are moderate drinkers, that is consume between 4 and 8 ounces per week? (a) 60% (b) 20% (c) 40% (d) 80% (e) 50% 5. “Normal” body temperature varies by time of day. A series of readings was taken of the body temperature of a subject. The mean reading was found to be 36.5° C with a standard deviation of 0.3° C. When converted to °F, the mean and standard deviation are (°F = °C(1.8) + 32). (a) 97.7, 32 (b) 97.7, 0.30 (c) 97.7, 0.54 (d) 97.7, 0.97 (e) 97.7, 1.80 6. The following is a histogram showing the actual frequency of the closing prices on the New York exchange of a particular stock. Based on the frequency histogram for New York Stock exchange, the class that contains the 80th percentile is: (a) 20-30 (b) 10-20 (c) 40-50 (d) 50-60 (e) 30-40 2 7. Which of the following is likely to have a mean that is smaller than the median? (a) The salaries of all National Football League players. (b) The scores of students (out of 100 points) on a very easy exam in which most get nearly perfect scores but a few do very poorly. (c) The prices of homes in a large city. (d) The scores of students (out of 100 points) on a very difficult exam in which most get poor scores but a few do very well. (e) Amounts awarded by civil court juries. 8. There are three children in a room, ages three, four, and five. If a four-year-old child enters the room the (a) mean age will stay the same but the variance will increase. (b) mean age will stay the same but the variance will decrease. (c) mean age and variance will stay the same. (d) mean age and variance will increase. (e) mean age and variance will decrease. 9. The weights of the male and female students in a class are summarized in the following boxplots: (a) (b) (c) (d) (e) Which of the following is NOT correct? About 50% of the male students have weights between 150 and 185 pounds. About 25% of female students have weights more than 130 pounds. The median weight of male students is about 162 pounds. The mean weight of female students is about 120 pounds because of symmetry. The male students have less variability than the female students. 10. When testing water for chemical impurities, results are often reported as bdl, that is, below detection limit. The following are the measurements of the amount of lead in a series of water samples taken from inner-city households (ppm). (a) (b) (c) (d) (e) 5, 7, 12, bdl, 10, 8, bdl, 20, 6 Which of the following is correct? The mean lead level in the water is about 10 ppm. The mean lead level in the water is about 8 ppm. The median lead level in the water is 7 ppm. The median lead level in the water is 8 ppm. Neither the mean nor the median can be computed because some values are unknown. 11. The heights of American men aged 18 to 24 are approximately normally distributed with mean 68 inches and standard deviation 2.5 inches. Half of all young men are shorter than (a) 65.5 inches (b) 68 inches (c) 70.5 inches (d) can't tell, because the median height is not given (e) none of the above 3 12. Use the information in the previous problem. Only about 5% of young men have heights outside the range (a) 65.5 inches to 70.5 inches (b) 63 inches to 73 inches (c) 60.5 inches to 75.5 inches (d) 58 inches to 78 inches (e) none of the above 13. For the density curve shown to the right, which statement is true? (a) The density curve is symmetric. (b) The density curve is skewed right. (c) The area under the curve between 0 and 1 is 1. (d) The density curve is normal. (e) None of the above is correct. 14. For the density curve shown in question 3, which statement is true? (a) The mean and median are equal. (b) The mean is greater than the median. (c) The mean is less than the median. (d) The mean could be either greater than or less than the median. (e) None is the above is correct. 15. Increasing the frequencies in the tails of a distribution will: (a) not affect the standard deviation as long as the increases are balanced on each side of the mean (b) not affect the standard deviation (c) increase the standard deviation (d) reduce the standard deviation (e) none of the above 16. For the density curve shown, what is the mean? (a) 0 (b) 0.25 (c) 0.50 (d) 0.75 (e) None of the above 17. A smooth curve which approximates the shape of a histogram and describes the overall pattern of a distribution is called (a) a stemplot (b) a normal plot (c) a normal probability plot (d) a density curve (e) none of the above 18. A normal density curve has which of the following properties? (a) It is symmetric. (b) It has a peak centered above its mean. (c) The spread of the curve is proportional to it standard deviation. (d) All of the properties, (a) to (c), are correct. (e) None of the properties, (a) to (c), is correct. 4 19. The area under the standard normal curve corresponding to –0.3<Z<1.6 is (a) 0.3273 (b) 0.4713 (c) 0.5631 (d) 0.9542 (e) None of the above 20. Which of the following histograms would be best approximated by a normal distribution: (a) (b) (c) (d) 21. In a statistics course, a linear regression equation was computed to predict the final exam score from the score on the first test. The equation was y = 10 + .9x where y is the final exam score and x is the score on the first test. Carla scored 95 on the first test. What is the predicted value of her score on the final exam? (a) 95 (b) 85.5 (c) 90 (d) 95.5 (e) None of the above 22. Refer to the previous problem. On the final exam Carla scored 98. What is the value of her residual? (a) 98 (b) 2.5 (c) –2.5 (d) 0 (e) None of the above 23. A study of the fuel economy for various automobiles plotted the fuel consumption (in liters of gasoline used per 100 kilometers traveled) vs. speed (in kilometers per hour). A least squares line was fit to the data. Here is the residual plot from this least squares fit. What does the pattern of the residuals tell you about the linear model? (a) The evidence is inconclusive. (b) The residual plot confirms the linearity of the fuel economy data. (c) The residual plot does not confirm the linearity of the data. (d) The residual plot clearly contradicts the linearity of the data. (e) None of the above 24. All but one of the following statements contains a blunder. Which statement is correct? (a) There is a correlation of 0.54 between the position a football player plays and their weight. (b) The correlation between planting rate and yield of corn was found to be r=0.23. (c) The correlation between the gas mileage of a car and its weight is r=0.71 MPG. (d) We found a high correlation (r=1.09) between the height and age of children. (e) We found a correlation of r=–.63 between gender and political party preference. 5 25. What do we call a sample that consists of the entire population? (a) A stratum (b) A multistage sample (c) A mistake. A sample can never be the entire population. (d) A census (e) None of the above. The answer is _________________________. 26. A member of Congress wants to know what his constituents think of proposed legislation on health insurance. His staff reports that 228 letters have been received on the subject, of which 193 oppose the legislation. What is the population in this situation? (a) The constituents (b) The 228 letters received (c) The 193 opposing the legislation (d) Congress (e) None of the above. The answer is _____________________________. 27. Which of the following is a method for improving the accuracy of a sample? (a) Use no more than 3 or 4 words in any question (b) When possible, avoid the use of human interviewers, relying on computerized dialing instead (c) Use large sample sizes (d) Use smaller sample sizes (e) None of the above. The answer is _____________________________. 28. We say that the design of a study is biased if which of the following is true? (a) A racial or sexual preference is suspected (b) Random placebos have been used (c) Certain outcomes are systematically favored (d) The correlation is greater than 1 or less than –1 (e) None of the above. The answer is _____________________________. 29. Control groups are used in experiments in order to . . . (a) Control the effects of lurking variables such as the placebo effect (b) Control the subjects of a study so as to insure all participate equally (c) Guarantee that someone other than the investigators, who have a vested interest in the outcome, control how the experiment is conducted (d) Achieve a proper and uniform level of randomization (e) None of the above. The answer is ______________________________. 30. The probability of any outcome of a random phenomenon is (a) The precise degree of randomness present in the phenomenon (b) Any number as long as it is between 0 and 1 (c) Either 0 or 1, depending on whether or not the phenomenon can actually occur or not (d) The proportion of a very long series of repetitions on which the outcome occurs (e) None of the above 31. A randomly selected student is asked to respond to yes, no, or maybe to the question, “Do you intend to vote in the next presidential election?” The sample space is { yes, no, maybe }. Which of the following represent a legitimate assignment of probabilities for this sample space? (a) 0.4, 0.4, 0.2 (b) 0.4, 0.6, 0.4 (c) 0.3, 0.3, 0.3 (d) 0.5, 0.3, –0.2 (e) None of the above 6 32. If you choose a card at random from a well-shuffled deck of 52 cards, what is the probability that the card chosen is not a heart? (a) 0.25 (b) 0.50 (c) 0.75 (d) 1 (e) None of the above 33. You play tennis regularly with a friend, and from past experience, you believe that the outcome of each match is independent. For any given match you have a probability of .6 of winning. The probability that you win the next two matches is (a) 0.16 (b) 0.36 (c) 0.4 (d) 0.6 (e) 1.2 34. If P(A) = 0.24 and P(B) = 0.52 and A and B are independent, what is P(A or B)? (a) 0.1248 (b) 0.28 (c) 0.6352 (d) 0.76 35. Suppose X is a random variable with mean µ. Suppose we observe X many times and keep track of the average of the observed values. The law of large numbers says that (a) The value of µ will get larger and larger as we observe X. (b) As we observe X more and more, this average and the value of µ will get larger and larger. (c) This average will get closer and closer to µ as we observe X more and more often. (d) As we observe X more and more, this average will get to be a larger and larger multiple of µ. (e) None of the above 36. In a population of students, the number of calculators owned is a random variable X with P(X = 0) = 0.2, P(X = 1) = 0.6, and P(X = 2) = 0.2. The mean of this probability distribution is (a) 0 (b) 2 (c) 1 (d) 0.5 (e) The answer cannot be computed from the information given. 37. Refer to the previous problem. The variance of this probability distribution is (a) 1 (b) 0.63 (c) 0.5 (d) 0.4 (e) The answer cannot be computed from the information given. 38. 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 probability that exactly 2 students have experienced math anxiety is (a) 0.3020 (b) 0.2634 (c) 0.2013 (d) 0.5 (e) 1 (f) None of the above 7 39. Refer to the previous problem. The standard deviation of the number of students in the sample who have experienced math anxiety is (a) 0.0160 (b) 1.265 (c) 0.2530 (d) 1 (e) .2070 40. In a certain large population, 40% of households have a total annual income of $70,000. A simple random sample of 4 of these households is selected. What is the probability that 2 or more of the households in the survey have an annual income of over $70,000? (a) 0.3456 (b) 0.4000 (c) 0.5000 (d) 0.5248 (e) The answer cannot be computed from the information given. 41. A factory makes silicon chips for use in computers. It is known that about 90% of the chips meets specifications. Every hour a sample of 18 chips is selected at random for testing. Assume a binomial distribution is valid. Suppose we collect a large number of these samples of 18 chips and determine the number meeting specifications in each sample. What is the approximate mean of the number of chips meeting specifications? (a) 16.20 (b) 1.62 (c) 4.02 (d) 16.00 (e) The answer cannot be computed from the information given. 42. Which of the following are true statements? I. The expected value of a geometric random variable is determined by the formula (1 – p)n–1p. II. If X is a geometric random variable and the probability of success is .85, then the probability distribution of X will be skewed left, since 85 is closer to 1 than to 0. III. An important difference between binomial and geometric random variables is that there is a fixed number of trials in a binomial setting, and the number of trials varies in a geometric setting. (a) I only (b) II only (c) III only (d) I, II, and III (e) None of the above gives the complete set of true responses. 43. Following a dramatic drop of 500 points in the Dow Jones Industrial Average in September 1998, a poll conducted for the Associated Press found that 92% of those polled said that a year from now their family financial situation will be as good as it is today or better. The number 92% is a (a) Statistic (b) Sample (c) Parameter (d) Population (e) None of the above. The answer is . 44. In a large population, 46% of the households own VCR’s. A simple random sample of 100 households is to be contacted and the sample proportion computed. The mean of the sampling distribution of the sample proportion is (a) 46 (b) 0.46 (c) About 0.46, but not exactly 0.46 (d) 0.00248 (e) The answer cannot be computed from the information given 8 45. If a population has a standard deviation , then the standard deviation of the mean of 100 randomly selected items from this population is (a) (b) 100 (c) /10 (d) /100 (e) 0.1 46. The distribution of values taken by a statistic in all possible samples of the same size from the same population is (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) None of the above. The answer is . 47. 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, the statistic is said to be (a) Random (b) Biased (c) A proportion (d) Unbiased (e) None of the above. The answer is . 48. A simple random sample of 1000 Americans found that 61% were satisfied with the service provided by the dealer from which they bought their car. A simple random sample of 1000 Canadians found that 58% were satisfied with the service provided by the dealer from which they bought their car. The sampling variability associated with these statistics is (a) Exactly the same. (b) Smaller for the sample of Canadians because the population of Canada is smaller than that of the United States, hence the sample is a larger proportion of the population. (c) Smaller for the sample of Canadians because the percentage satisfied was smaller than that for the Americans. (d) Larger for the Canadians because Canadian citizens are more widely dispersed throughout the country than in the United States, hence they have more variable views. (e) About the same. 49. You want to compute a 96% confidence interval for a population mean. Assume that the population standard deviation is known to be 10 and the sample size is 50. The value of z* to be used in this calculation is (a) 1.960 (b) 1.645 (c) 1.7507 (d) 2.0537 (e) None of the above. The answer is . 50. You want to estimate the mean SAT score for a population of students with a 90% confidence interval. Assume that a sample size of (a) 16 (b) 271 (c) 38 (d) 1476 (e) None of the above. The answer is . 9 51. A significance test gives a P-value of 0.04. From this we can (a) Reject H0 at the 1% significance level (b) Reject H0 at the 5% significance level (c) Say that the probability that H0 is false is 0.04 (d) Say that the probability that H0 is true is 0.04 (e) None of the above. The answer is . 52. A significance test was performed to test the null hypothesis H0: µ = 2 versus the alternative Ha: µ 2. The test statistic is z = 1.40. The P-value for this test is approximately (a) 0.16 (b) 0.08 (c) 0.003 (d) 0.92 (e) 0.70 (f) None of the above. The answer is . 53. You have measured the systolic blood pressure of a random sample of 25 employees of a company located near you. A 95% confidence interval for the mean systolic blood pressure for the employees of this company is (122, 138). Which of the following statements gives a valid interpretation of this interval? (a) Ninety-five percent of the sample of employees has a systolic blood pressure between 122 and 138. (b) Ninety-five percent of the population of employees has a systolic blood pressure between 122 and 138. (c) If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure. (d) The probability that the population mean blood pressure is between 122 and 138 is .95. (e) If the procedure were repeated many times, 95% of the sample means would be between 122 and 138. (f) None of the above. The answer is . 54. An analyst, using a random sample of n = 500 families, obtained a 90% confidence interval for mean monthly family income for a large population: ($600, $800). If the analyst had used a 99% confidence coefficient instead, the confidence interval would be: (a) Narrower and would involve a larger risk of being incorrect (b) Wider and would involve a smaller risk of being incorrect (c) Narrower and would involve a smaller risk of being incorrect (d) Wider and would involve a larger risk of being incorrect (e) Wider but it cannot be determined whether the risk of being incorrect would be larger or smaller 55. In preparing to use a t procedure, suppose we were not sure if the population was normal. In which of the following circumstances would we not be safe using a t procedure? (a) A stemplot of the data is roughly bell shaped. (b) A histogram of the data shows moderate skewness. (c) A stemplot of the data has a large outlier. (d) The sample standard deviation is large. (e) The t procedures are robust, so it is always safe. 56. The weights of 9 men have mean x = 175 pounds and standard deviation s = 15 pounds. What is the standard error of the mean? (a) 58.3 (b) 19.4 (c) 5 (d) 1.7 (e) None of the above. The answer is . 10 MULTIPLE CHOICE ANSWERS 1.e 2.e 3.c 4.c 5.c 6.e 7.b 8.b 9.e 10.c 11.b 12.b 13.c 14.c 15.c 16.c 17.d 18.d 19.c 20.c 21.d 22.b 23.d 24.b 25.d 26.a 27.c 28.c 29.a 30.b 31.a 32.c 33.b 34.c 35.c 36.c 37.d 38.a 39.b 40.d 41.a 42.c 43.a 44.b 45.c 46.d 47.d 48.e 49.d 50.b 51.omit 52.omit 53.c 54.b 55.c 56.c 11 Part 2: Free Response Communicate your thinking clearly and completely. 1. The test grades for a certain class were entered into a Minitab worksheet, and then “Descriptive Statistics” were requested. The results were: MTB > Describe 'Grades'. N Grades Grades MEAN MEDIAN TRMEAN STDEV SEMEAN 28 74.71 76.00 75.50 12.61 2.38 MIN MAX Q1 Q3 35.00 94.00 68.00 84.00 You happened to see, on a scrap of paper, that the lowest grades were 35, 57, 59, 60, . . . but you don’t know what the other individual grades are. Nevertheless, a knowledgeable user of statistics can tell a lot about the dataset simply by studying the set of descriptive statistics above. (a) Write a brief description of what the results tell you about the distribution of grades. Be sure to address: the general shape of the distribution unusual features, including possible outliers the middle 50% of the data any significance in the difference between the mean and the median (b) Construct a modified boxplot for these data. 12 2. The following table gives the Nielsen ratings for the top 50 prime-time television shows for the first half of the 199495 viewing season. Each ratings point represents 954,000 households. Show Home Improvement Grace Under Fire Seinfeld E.R. NYPD Blue 60 Minutes Monday Night Football Roseanne Murder, She Wrote Ellen Frasier Mad About You Murphy Brown CBS Sunday Movie Madman of the People NBC Monday Movie Friends Dave’s World 20/20 Me and the Boys Full House Law and Order Love and War Northern Exposure The Nanny Network ABC ABC NBC NBC ABC CBS ABC ABC CBS ABC NBC NBC CBS CBS NBC NBC NBC CBS ABC ABC ABC NBC CBS CBS CBS Rating 20.8 20.0 19.7 18.5 18.1 17.9 17.4 17.3 16.2 15.5 15.0 14.8 14.8 14.7 14.6 14.1 13.9 13.8 13.7 13.5 13.0 12.9 12.8 12.8 12.8 Show ABC Sunday Movie Wings NBC Sunday Movie Step by Step Beverly Hills, 90210 Walker, Texas Ranger Dr. Quinn, Med. Woman Family Matters Dateline NBC Tuesday CBS Tuesday Movie Boy Meets World John Larroquette Show Hangin’ with Mr. Cooper Turning Point Earth 2 Due South Rescue 911 Chicago Hope Fresh Prince of Bel-Air Blossom All-American Girl Diagnosis Murder The Cosby Mysteries Picket Fences SeaQuest DSV Network ABC NBC NBC ABC FOX CBS CBS ABC NBC CBS ABC NBC ABC ABC NBC CBS CBS CBS NBC NBC ABC CBS NBC CBS NBC Rating 12.8 12.6 12.0 11.7 11.6 11.6 11.5 11.5 11.5 11.4 11.4 11.4 11.3 11.2 10.9 10.9 10.9 10.7 10.6 10.6 10.5 10.5 10.5 10.3 10.2 (a) Make a stemplot of the ratings for the 16 programs that were broadcast by ABC. (b) How does ABC compare with NBC in the race for high ratings? Give appropriate graphical and numerical evidence to support your answer. 13 3. The best male long jumpers for State College since 1973 have averaged a jump of 263.0 inches with a standard deviation of 14.0 inches. The best female long jumpers have averaged 201.2 inches with a standard deviation of 7.7 inches. This year Joey jumped 275 inches and his sister, Carla, jumped 207 inches. Both are State College students. Assume that male and female jumps are normally distributed. Within their groups, which athlete had the more impressive performance? Explain briefly. 4. The length of pregnancies from conception to natural birth among a certain female population is a normally distributed random variable with mean 270 and standard deviation 10 days. (a) What is the percent of pregnancies that last more than 300 days? (b) How short must a pregnancy be in order to fall in the shortest 10% of all pregnancies? 14 Exercises 5–8 relate to the following. Joey read in his biology book that fish activity increases with water temperature, and he decided to investigate this issue by conducting an experiment. On nine successive days, he measures fish activity and water temperature in his aquarium. Larger values of his measure of fish activity denote more activity. The figure below presents the scatterplot of his data. 5. What does the scatterplot reveal? o o 450 o o o Activity o o 300 o o 69.0 72.0 75.0 78.0 81.0 Water Temp H2Otemp 6. One of the following numbers is the correlation coefficient between fish activity and water temperature; circle the correct number. –0.20 0.03 0.52 0.86 7. Write your best guess for the correlation coefficient for water temperature versus fish activity. Then briefly explain your reasoning. 8. Suppose a new point at (66, 500), i.e., water temperature = 66∞F and fish activity = 500, is added to the plot. Describe the effect, if any, that this new point will have on the correlation coefficient of fish activity versus water temperature? 15 Exercises 9-–10 relate to the following. At summer camp, one of Carla’s counselors told her that air temperature can be determined from the number of cricket chirps. 9. What is the explanatory variable, and what is the response variable? (Note: this is in the context of this problem, not in the biological sense.) EXPLANATORY: RESPONSE: To determine a formula, Carla collected data on temperature and number of chirps per minute on 12 occasions. She entered the data into lists L1 and L2 of her TI-83 and then did STATS ‰ CALC ‰ 2-Var Stats. Here are some of the results: x = 166.8, sx = 31.0 y = 78.83 sy = 9.11 r = 0.461 10. Use this information to determine the equation of the LSRL. Bias is present in each of the following sampling designs. In each case, identify the type of bias involved and state whether you think the sampling frequency obtained is lower or higher than the actual population parameter. 11. A political pollster seeks information about the proportion of American adults that oppose gun controls. He asks a SRS of 1000 American adults: "Do you agree or disagree with the following statement: Americans should preserve their constitutional right to keep and bear arms." A total of 910, or 91%, said "agree" (that is, 910 out of the 1000 oppose gun controls). 12. A flour company in Minneapolis wants to know what percentage of local households bake at least twice a week. A company representative calls 500 households during the daytime and finds that 50% of them bake at least twice a week. 16 Suppose you are given a standard 6-sided die and told that the die is “loaded” in such a way that while the numbers 1, 3, 4, and 6 are equally likely to turn up, the numbers 2 and 5 are three times as likely to turn up as any of the other numbers. 13. The die is rolled once and the number turning up is observed. Use the information given above to fill in the following table: Outcome Probability 1 2 3 4 5 6 14. Let A be the event: the number rolled is a prime number (a number is prime if its only factors are 1 and the number itself; note that 1 is not prime). List the outcomes in A and find P(A). 15. Let B be the event: the number rolled is an even number. List the outcomes in B, and find P(B). 16. Are events A and B disjoint? Explain briefly. 17. Determine if events A and B are independent. 17 A box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. A person is charged $20 to select one bill. 18. Identify the random variable. X = 19. Construct a probability distribution for this data. 20. Find the expected value. 21. Is the game fair? Explain briefly. According to government data, 20% of employed women have never been married. 22. What is the random variable X of interest here? Define X. Is X normal, binomial, or geometric? 23. If 10 employed women are selected at random, what is the probability that 2 or fewer have never been married? 24. What are the mean and standard deviation of X? 25. Find the probability that the number of employed women who have never been married is within 1 standard deviation of its mean. 26. Describe the four conditions that describe a binomial setting. 18 A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let p denote the proportion in the sample that says they support the increase. Suppose that 40% of all adults in Ohio support the increase. 27. If p̂ is the proportion of the sample who support the increase, what is the mean of p̂ ? 28.What is the standard deviation of p̂ ? 29. Explain why you can use the formula for the standard deviation of p̂ in this setting. 30. Check that you can use the normal approximation for the distribution of p̂ . 31. How large a sample would be needed to guarantee that the standard deviation of p̂ is no more than 0.01? Explain. A certain beverage company is suspected of underfilling its cans of soft drink. The company advertises that its cans contain, on the average, 12 ounces of soda with standard deviation 0.4 ounce. 32. Compute the probability that a random sample of 50 cans produces a sample mean fill of 11.9 ounces or less. (A sketch of the distribution is required.) 33. Suppose that each of the 25 students in a statistics class collects a random sample of 50 cans and calculates the mean number of ounces of soda. Describe the approximate shape of the distribution for these 25 values of x . 34. What important principle that we studied is used to answer the previous question? 19 35. There are many ways to measure the reading ability of children. Research designed to improve reading performance is dependent on good measures of the outcome. One frequently used test is the DRP or Degree of Reading Power. A researcher suspects that the mean score µ of all third graders in Henrico County Schools is different from the national mean, which is 32. To test her suspicion, she administers the DRP to an SRS of 44 Henrico County third-grade students. Their scores were: 40 47 52 47 26 19 25 35 39 26 35 48 14 35 35 22 42 34 33 33 18 15 29 41 25 44 34 51 43 40 41 27 46 38 49 14 27 31 28 54 19 46 52 45 She then asked Minitab to calculate some descriptive statistics from this data set: MTB > Describe 'DRPscore'. N DRPscore MEAN MEDIAN TRMEAN STDEV SEMEAN 44 35.09 35.00 35.25 11.19 1.69 MIN MAX Q1 Q3 DRPscore 14.00 54.00 26.25 44.75 You may assume that DRP scores are approximately normal, and that the standard deviation of scores in Henrico County Schools is known to be (a) Construct a 90% confidence interval for the mean DRP score in Henrico County Schools. Follow the Inference Toolbox. 20