I-1. AP Statistics Test A Data Analysis Part I

Transcription

1 AP Statistics Test A Data Analysis Part I Name 1. School administrators collect data on students attending the school. Which of the following variables is quantitative? A) class (freshman, soph., junior, senior) B) grade point average C) whether the student is in AP classes D) whether the student has taken the SAT E) none of these 2. Which of the following variables would most likely follow a Normal model? A) family income B) heights of singers in a co-ed choir C) weights of adult male elephants D) scores on an easy test E) all of these 3. A professor has kept records on grades that students have earned in his class. If he wants to examine the percentage of students earning the grades A, B, C, D, and F during the most recent term, which kind of plot could he make? A) boxplot B) timeplot C) dotplot D) pie chart E) histogram 4. Which is true of the data shown in the histogram? 6 Histogram of Data 5 Frequency Data 2 4 I. The distribution is approximately symmetric. II. The mean and median are approximately equal. III. The median and IQR summarize the data better than the mean and standard deviation. A) I only B) III only C) I and II D) I and III E) I, II, and III 5. Two sections of a class took the same quiz. Section A had 15 students who had a mean score of 80, and Section B had 20 students who had a mean score of 90. Overall, what was the approximate mean score for all of the students on the quiz? A) 84.3 B) 85.0 C) 85.7 D) none of these E) It cannot be determined. 6. Your Stats teacher tells you your test score was the 3 rd quartile for the class. Which is true? I. You got 75% on the test. II. You can t really tell what this means without knowing the standard deviation. III. You can t really tell what this means unless the class distribution is nearly Normal. A) none of these B) I only C) II only D) III only E) II and III I-1

2 7. Suppose that a Normal model described student scores in a history class. Parker has a standardized score (z-score) of This means that Parker A) is 2.5 points above average for the class. B) is 2.5 standard deviations above average for the class. C) has a standard deviation of 2.5. D) has a score that is 2.5 times the average for the class. E) None of the above. 8. The advantage of making a stem-and-leaf display instead of a dotplot is that a stem-andleaf display A) satisfies the area principle. B) shows the shape of the distribution better than a dotplot. C) preserves the individual data values. D) A stem-and-leaf display is for quantitative data, while a dotplot shows categorical data E) none of these 9. The five-number summary of credit hours for 24 students in a statistics class is: I-2 Min Q1 Median Q3 Max Which statement is true? A) There are no outliers in the data. B) There is at least one low outlier in the data. C) There is at least one high outlier in the data. D) There are both low and high outliers in the data. E) None of the above. 10.Which of the following summaries are changed by adding a constant to each data value? I. the mean II. the median III. the standard deviation A) I only B) III only C) I and II D) I and III E) I, II, and III 11. Cats and dogs The table shows whether students in an introductory statistics class like dogs and/or cats. Like Dogs Yes No Total Yes Like Cats No Total a. What is the marginal distribution (in %) of liking dogs? b. What is the conditional distribution (in %) of liking dogs for students who like cats? c. What kind of display(s) would you use to examine the association between liking dogs and liking cats? (Just name a graph.) d. Do liking dogs and liking cats appear to be independent? Give statistical evidence to support your conclusion.

3 12. House calls A local plumber makes house calls. She charges $30 to come out to the house and $40 per hour for her services. For example, a 4-hour service call costs $30 + $40(4) = $190. a. The table shows summary statistics for the past month. Fill in the table to find out the cost of the service calls. Statistic Hours of Service Call Cost of Service Call Mean 4.5 Median 3.5 SD 1.2 IQR 2.0 Minimum 0.5 b. This past month, the time the plumber spent on one service call corresponded to a z-score of What was the z-score for the cost of that service call? 13. Health insurance The World Almanac and Book of Facts 2004 reported the percent of people not covered by health insurance in the 50 states and Washington, D.C., for the year Computer output gives these summaries for the percent of people not covered by health insurance: Min Q1 Median Q3 Max Mean SD a. Were any of the states outliers? Explain how you made your decision. b. A histogram of the data is as follows: 14 Histogram of % not covered Frequency % not covered Is it more appropriate to use the mean and standard deviation or the median and IQR to describe these data? Explain. I-3

4 14. Veterinary costs Costs for standard veterinary services at a local animal hospital follow a Normal model with a mean of $80 and a standard deviation of $20. a. Draw and clearly label this model. b. Is it unusual to have a veterinary bill for $125? Explain. c. What is the IQR for the costs of standard veterinary services? Show your work. 15. Soda cans A machine that fills cans with soda fills according to a Normal model with mean 12.1 ounces and standard deviation 0.05 ounces. a. If the cans claim to have 12 ounces of soda each, what percent of cans are under-filled? b. Management wants to ensure that only 1% of cans are under-filled. i. Scenario 1: If the mean fill of the cans remains at 12.1 ounces, what standard deviation does the filling machine need to have to achieve this goal? ii. Scenario 2: If the standard deviation is to remain at 0.05 ounces, what mean does the filling machine need to have to achieve this goal? I-4

5 AP Statistics Test A Data Analysis Part I Key 1. B 2. C 3. D 4. C 5. C 6. A 7. B 8. C 9. A 10. C 11. Cats and dogs a. Yes: 90.7% No: 9.3% b. Yes: 90.2% No: 9.8% c. segmented bar graph or pie charts d. Perhaps. There is little difference between the percents of those who like dogs, depending on whether they like cats. Of those who like cats, only 90.2% like dogs, compared to 90.7% overall. 12. House calls a. The table shows summary statistics for the past month. Fill in the table to find out the cost of the service calls. b Statistic Hours of Service Call Cost of Service Call Mean 4.5 $210 Median 3.5 $170 SD 1.2 $48 IQR 2.0 $80 Minimum 0.5 $ Health insurance a. Were any of the states outliers? Explain how you made your decision. IQR = Q3 Q1 = = (IQR) = 1.5(5.9) = 8.85 Q (IQR) = = < Max, so there is at least one high outlier Q1 1.5(IQR) = = 1.95 < Min, so there are no low outliers b. It is more appropriate to use the median and IQR to describe these data, since the distribution is skewed right. I-5

6 14. Veterinary costs a b. z = = 2.25, more than 2 standard deviations above the mean bill. A veterinary bill 20 of $125 is unusual. c. Q1 has z = and Q3 has z = +0.67, so 0.67 = y = y y = ( 20)= 66.6 and y = ( 20)= The IQR = Q3 Q1 = = $ Soda cans A machine that fills cans with soda fills according to a Normal model with mean 12.1 ounces and standard deviation 0.05 ounces a. z = = 2.0, which suggests that 2.28% of cans are under-filled b. A z-score of has 1% to its left, meaning that 1% of the cans would be under-filled i = 2.33σ = 0.1 σ = 0.1 σ 2.33 need to be ounces. ii = 12 µ 0.05 would need to be ounces. = The standard deviation would 2.33( 0.05)= 12 µ µ = ( 0.05)=12.12 The mean I-6

7 AP Statistics Test B Data Analysis Part I Name 1. The SPCA collects the following data about the dogs they house. Which is categorical? A) breed B) age C) weight D) number of days housed E) veterinary costs 2. Which of those variables about German Shepherds is most likely to be described by a Normal model? 3. The SPCA has kept these data records for the past 20 years. If they want to show the trend in the number of dogs they have housed, what kind of plot should they make? A) boxplot B) timeplot C) bar graph D) pie chart E) histogram 4. The veterinary bills for the dogs are summarized in the ogive shown. Estimate the IQR of these expenses. A) $50 B) $75 C) $100 D) $150 E) $ Last weekend police ticketed 18 men whose mean speed was 72 miles per hour, and 30 women going an average of 64 mph. Overall, what was the mean speed of all the people ticketed? A) 67 mph B) 68 mph C) 69 mph D) none of those E) It cannot be determined. 6. Which is true of the data shown in the histogram? I. The distribution is skewed to the right. II. The mean is probably smaller than the median. III. We should use median and IQR to summarize these data. A) I only B) II only C) III only D) II and III only E) I, II, and III 7. The best estimate of the standard deviation of the mens weights displayed in this dotplot is A) 10 B) 15 C) 25 D) 35 E) If we want to discuss any gaps and clusters in a data set, which of the following should not be chosen to display the data set? A) histogram B) stem-and-leaf plot C) boxplot D) dotplot E) any of these would work 9. Suppose a Normal model describes the number of pages printer ink cartridges last. If we keep track of printed pages for the 47 printers at a company s office, which must be true? I. The page counts for those ink cartridges will be normally distributed. II. The histogram for those page counts will be symmetric. III. 95% of those page counts will be within 2 standard deviations of the mean. A) none B) I only C) II only D) II and III E) I, II, and III $50 $100 $150 $200 $ I-7

8 10. Suppose that a Normal model describes fuel economy (miles per gallon) for automobiles and that a Saturn has a standardized score (z-score) of This means that Saturns... A) get 2.2 miles per gallon. B) get 2.2 times the gas mileage of the average car. C) get 2.2 mpg more than the average car. D) have a standard deviation of 2.2 mpg. E) achieve fuel economy that is 2.2 standard deviations better than the average car. 11. Commuting to work The table shows how a company s employees commute to work. a. What is the marginal distribution (in %) of mode of transportation? Car Bus Train b. What is the conditional distribution (in %) of mode of transportation for management? Car Bus Train Transportation Job Class Car Bus Train Total Management Labor Total c. What kind of display would you use to show the association between job class and mode of transportation? (Just name a graph.) d. Do job classification and mode of transportation appear to be independent? Give statistical evidence to support your conclusion. 12. Book sales A publishing company pays its sales staff $600 a week plus a commission of $0.50 per book sold. For example, a salesman who sold 440 books earned (440) = $820. a. The table shows summary statistics for the number of books the large sales staff sold last week. Fill in the table to show the statistics for the pay these people earned. b. The newest employee had a pretty good week. Among all the salespeople her pay corresponded to a z-score of What was the z-score of the number of books she sold? Statistic Books Sold $ Earned Mean 640 Standard deviation 360 IQR 450 Maximum 1420 I-8

9 13. Cordless phones In their October 2003 issue, Consumer Reports evaluated the price and performance of 23 models of cordless phones. Computer output gives these summaries for the prices: Min Q1 Median Q3 Max MidRange Mean TrMean SD a. Were any of the prices outliers? Explain how you made your decision. b. One of the manufacturers advertises a cordless phone economy-priced at only $ Would you consider that to be a very low price? Explain. 14. Exercising Owners of an exercise gym believe that a Normal model is useful in projecting the number of clients who will exercise in their gym each week. They use a mean of 800 clients and a standard deviation of 90 clients. a. Draw and clearly label this model. b. What is the first quartile of the weekly number of clients? [Show work] c. An owner of another gym reports that 5% of the time their gym has fewer than 450 clients, and 40% of the time the gym has more than 1085 clients. What parameters should that owner use for his Normal model? N(, ) I-9

10 15. Concrete thickness A roadway construction process uses a machine that pours concrete onto the roadway and measures the thickness of the concrete so the roadway will measure up to the required depth in inches. The concrete thickness needs to be consistent across the road, but the machine isn t perfect and it is costly to operate. Since there s a safety hazard if the roadway is thinner than the minimum 23 inch thickness, the company sets the machine to average 26 inches for the batches of concrete. They believe the thickness level of the machine s concrete output can be described by a Normal model with standard deviation 1.75 inches. [Show work] a. What percent of the concrete roadway is under the minimum depth? b. The company s lawyers insist that no more than 3% of the output be under the limit. Because of the expense of operating the machine, they cannot afford to reset the mean to a higher value. Instead they will try to reduce the standard deviation to achieve the only 3% under goal. What standard deviation must they attain? c. Explain what achieving a smaller standard deviation means in this context. I-10

11 AP Statistics Test B Data Analysis Part I Key 1. A 2. C 3. B 4. C 5. A 6. D 7. C 8. C 9. A 10. E 11. Commuting to work a. Car: 19.5% Bus: 30% Train: 50.5% b. Car: 28.9% Bus: 22.2% Train: 48.9% c. segmented bar graph, or pie charts d. No, there is a difference between the percents in two types of transportation - Car and Bus categories, depending on the Job Classification. Car Bus Train Management 28.9% 22.2% 48.9% Labor 17.0% 32.1% 50.9% Although about half of each group take the train, management are more likely than labor to come by car and less likely to take a bus. 12. Book sales. a. The table shows summary statistics for the number of books the large sales staff sold last week. Fill in the table to show the statistics for the pay these people earned. b Statistic Books Sold $ Earned Mean 640 $920 Standard deviation 360 $180 IQR 450 $225 Maximum 1420 $ Cordless phones a. IQR = Q 3 Q 1 = = (IQR) = 1.5(80) = 120 Q (IQR) = = 230; Q 1-1.5(IQR) = = -90; Max (200) < Q (IQR), so no high outliers. Min (15) > Q 1-1.5(IQR), so no low outliers. b. A price of $31.95 is slightly above the first quartile, so over 25% of phones cost less. No, the advertised price would not be a very low price. (Or: The advertised price is only 0.76 standard deviations below the mean. This is not an unusually low price.) I-11

12 14. Exercising a. Draw and clearly label this model. b. Q 1 => P = 0.25 and z = 0.674, = x = x 800 x = So the first quartile is at 740 clients. c. 450 (5 th percentile) has z = (60 th percentile) has z = = 0.253σ σ σ µ (334.5) = 1085 µ N(1000.4, 334.5) 15. Concrete thickness a. The concrete roadway is under minimum depth when less then 23 inches in thickness z = 1.75 depth = 1.7 P = 0.043, so the model suggests about 4.3% is under the minimum b. P = 0.03 z = 1.88, so 1.88 = σ ; then σ = 1.6 inches c. A smaller standard deviation means that the thickness of the concrete will be more consistent. I-12

13 AP Statistics Test C Data Analysis Part I Name 1. We collect these data from 50 male students. Which variable is categorical? A) eye color B) head circumference C) hours of homework last week D) number of cigarettes smoked daily E) number of TV sets at home 2. Which of those variables is most likely to be bimodal? 3. Which of those variables is most likely to follow a Normal model? 4. The mean number of hours worked for the 30 males was 6, and for the 20 females was 9. The overall mean number of hours worked A) is 6.5 B) is 7.2 C) is 7.5 D) is none of these. E) cannot be determined. 5. We might choose to display data with a stemplot rather than a boxplot because a stemplot I. reveals the shape of the distribution. II. is better for large data sets. III. displays the actual data. A) I only B) II only C) III only D) I and III E) I, II, and III 6. Which is true of the data whose distribution is shown? I. The distribution is skewed to the right. II. The mean is probably smaller than the median. III. We should summarize with mean and standard deviation. A) I only B) II only C) I and II D) II and III E) I, II, and III 7. The standard deviation of the data displayed in this dotplot is most likely to be A) 5. B) 8. C) 12. D) 18. E) Suppose that a Normal model describes the acidity (ph) of rainwater, and that water tested after last week s storm had a z score of 1.8. This means that the acidity of that rain A) had a ph of 1.8. B) varied with a standard deviation of 1.8 C) had a ph 1.8 higher than average rainfall. D) had a ph 1.8 times that of average rainwater. E) had a ph 1.8 standard deviations higher than that of average rainwater. 9. The ages of people attending the opening show of a new movie are summarized in the ogive shown. Estimate the IQR of the ages. A) 5 B) 13 C) 21 D) 30 E) Environmental researchers have collected rain acidity data for several decades. They want to see if there is any evidence that attempts to reduce industrial pollution have produced a trend toward less acidic rainfall. They should display their data in a(n) A) contingency table B) bar graph C) boxplot D) histogram E) timeplot Age I-13

14 11. Paying for purchases One day a store tracked the way shoppers paid for their purchases. Their data are summarized in the table. a. What percent of the men paid cash? b. What is the conditional relative frequency distribution of payment method for women? Cash Check Charge Total Male Female Total c. If you wanted to show the association between gender and method of payment visually, what kind of graph would you make? (Just name it.) d. Is there evidence of an association between gender and method of payment? Explain briefly. 12. Repair bills An automobile service shop reported the summary statistics shown for repair bills (in $) for their customers last month. a. Were any of the bills outliers? Show how you made your decision. Min Q1 Median Q3 Max Mean SD b. After checking out a problem with your car the service manager gives you an estimate of only $90. Is he right to imply that your bill will be unusually low? Explain briefly. 13. Salary conversions You learn that your company is sending you and several other employees to staff a new office in China. While there everyone will earn the equivalent of their current salary, converted to Chinese currency at the rate of 8 yuans per dollar. In addition, everyone will earn a weekly foreign living allowance of 200 yuans. For example, since you are earning $1000 per week, your weekly salary in China will be 1000 x = 8200 yuans. a. Shown are some summary statistics Statistic In the US In China describing the current salaries of this group being sent overseas. Fill in the table to show Minimum salary $400 what these statistics will be for the salaries Standard deviation $250 you all will earn while in China. Median $750 b. Among this group of employees going to China, your US salary has a z score of IQR $300 What will your new z-score be, based on everyone s China salary? I-14

15 14. Copy machines A manufacturer claims that lifespans for their copy machines (in months) can be described by a Normal model N(42,7). Show your work. a. Draw and clearly label the model. b. A company with a several large office buildings buys 200 of these copiers. The salesman tells the boss 190 (95%) of your new copiers will last between and months. Comment on this claim. c. What is the 3 rd quartile of copier lifespans? d. What percent of the copiers are expected to fail before 36 months? e. The manufacturer wants to reduce the 36-month failure rate to only 10%. Assuming the mean lifespan will stay the same, what standard deviation must they achieve? f. Briefly explain what that change in standard deviation means in this context. g. A competing manufacturer says that not only will 90% of their copiers last at least 36 months, 65% will last at least 42 months. What Normal model parameters is that manufacturer claiming? Show your work. N(, ) I-15

16 AP Statistics Test C Data Analysis Part I Key 1. A 2. D 3. B 4. B 5. D 6. A 7. C 8. E 9. B 10. E 11. Paying for purchases a. 45% b. 30% cash, 20% check, 50% charge c. segmented bar graphs, or pie charts d. Yes. Women are more likely to charge their purchases than men (50% to 30%) and less likely to pay cash (30% to 45%). 12. Repair bills a. Yes. IQR = = 220. The upper fence for outliers is one and a half IQR s above the third quartile, or (220) = 638. The maximum repair bill was $1442, well above $638, so it is certainly an outlier. b. No. $90 is higher than over 25% of the bills, so it is not unusually low. 13. Salary conversions a yuans, 2000, 6200, 2400 b. z = Copy machines a. b. 28, 56. The claim is probably false. This model should provide a useful estimate of what might happen, but is not certain to predict what actually will happen. c months d. 19.6% e. 4.7 months (should all include sketches of labeled curves) f. A smaller standard deviation means that the copiers would be more consistent in their lifespans g. For 36 months z = and for 42 months z = Thus the difference of 6 months is = standard deviations. The model is N(44.6, 6.7) I-16