Instructor: Doug Ensley Course: MAT Applied Statistics - Ensley

Transcription

1 Student: Date: Instructor: Doug Ensley Course: MAT Applied Statistics - Ensley Assignment: Online 04 - Sections 2.5 and A travel magazine recently presented data on the annual number of vacation days averaged by residents of eight different countries. They reported 41 days for Italy, 38 for France, 35 for Germany, 32 for Brazil, 27 for Britain, 26 for Canada, 24 for Japan, and 10 for the United States. Complete parts (a) through (d). a. Report the median. days b. By finding the median of the four values below the median, report the first quartile. days c. Find the third quartile. days d. Interpret the values found in parts (a)-(c) in the context of these data. % of the countries have residents who take fewer than 25.0 vacation days, half of the countries have residents who take fewer than vacation days, and 75% of the countries have residents who take (1) 36.5 vacation days per year. The middle 50% of the countries have residents who take an average of between and 36.5 vacation days annually. (1) fewer than more than exactly ID: of 12 9/2/15, 1:12 PM

2 2. The given data table is divided by categories (Cat) and number of observations (Obs) of general data. Use the data table to answer the following questions. a. Find and interpret the median. b. Find the first quartile (Q1) and the third quartile (Q3). c. Find and interpret the mean. Full data set Cat Obs Cat Obs Cat Obs Cat Obs A 3.8 F 17.1 L 30.1 R 43.5 B 6.4 G 20.6 M 31.9 S 45.2 C 9.9 H 23.9 N 33.1 T 48.4 D 13.2 J 25.3 P 35.7 U 51.3 E 15.3 K 26.7 Q 39.5 V 52.9 a. The median is. (Round to the nearest tenth as needed.) What information is given by the median? A. Approximately 50% of the data lie below this value. B. Approximately 50% of the data lie within 1 standard deviation of this value. C. This value is the average of the data set. D. Approximately 50% of the data are more than 1 standard deviation away from this value. b. The first quartile, Q1, is. (Round to the nearest tenth as needed.) The third quartile, Q3, is. (Round to the nearest tenth as needed.) c. The mean is. (Round to the nearest tenth as needed.) What information is given by the mean? A. This value is the average of the data set. B. Approximately 50% of the data are more than 1 standard deviation away from this value. C. Approximately 50% of the data lie within 1 standard deviation of this value. D. Approximately 50% of the data lie below this value. ID: of 12 9/2/15, 1:12 PM

3 3. The "high school female athletes" data file has data for 57 high school female athletes on the maximum number of pounds they were able to bench press, which is a measure of strength. For these data x = 86.2, Q1 = 76, median = 86, Q3 = 96. Complete parts (a) and (b). a. Interpret the quartiles. One (1) had a maximum bench press less than pounds, and one fourth had a maximum bench press (2) than 96 pounds. b. Would you guess that the distribution is skewed, or roughly symmetric? A. There's not enough evidence to decide. B. Skewed C. Roughly symmetric (1) fourth fifth third (2) greater less ID: of 12 9/2/15, 1:12 PM

4 4. Here is the five-number summary for the distribution of a cigarette tax (in cents) for all the states in a certain country. Use this information to answer parts a through d. Minimum = 9, Q1 = 30, Median = 47, Q3 = 99, Maximum = 150 a. About what proportion of the states have cigarette taxes (i) greater than 30 cents and (ii) greater than 99 cents? (i) About % of the states have cigarette taxes greater than 30 cents. (ii) About % of the states have cigarette taxes greater than 99 cents. b. Between what two values are the middle 50% of the observations found? The lower bound of the middle 50% is. (Type a whole number.) and the upper bound of the middle 50% is c. Find and interpret the interquartile range. The interquartile range (IQR) is. What is the relevance of the IQR? A. The IQR summarizes the range for the lower half of the data. B. The IQR summarizes the range for the upper half of the data. C. The IQR summarizes the range within one standard deviation of the mean. D. The IQR summarizes the range for the middle half of the data. d. Based on the summary, do you think this distribution was bell-shaped? If so, why? If not, why not, and what shape would you expect? A. The distribution is skewed right because the median is closer to Q1. Further proof is given by the values of the minim relative to Q1 and Q3, respectively. B. The distribution is bell-shaped because the median is exactly between Q1 and Q3. Further proof is given by the value maximum relative to Q1 and Q3, respectively. C. The distribution is skewed left because the median is closer to Q3. Further proof is given by the values of the minimu to Q1 and Q3, respectively. ID: of 12 9/2/15, 1:12 PM

5 5. During a recent semester at a large national university, students having accounts on a mainframe computer had hard drive use (in kilobytes) described by the five-number summary, minimum = 430, Q1 = 444, median = 596, Q3 = 1053, and maximum = 430,000. Complete parts a and b below. a. Would you expect this distribution to be symmetric, skewed to the right, or skewed to the left? Explain. Fill in the blanks to complete the statement below. The distribution is (1), because the median is (2) b. Use the 1.5 IQR criterion to determine all potential outliers that are present. Choose the correct answer below. A. Since the minimum value is not within 1.5 IQR of Q1, there is at least one outlier. B. Since the maximum value is not within 1.5 IQR of Q3, there is at least one outlier. C. Since the minimum value is within 1.5 IQR of Q1 and the maximum value is within 1.5 IQR of Q3, there are no po D. Since the minimum value is not within 1.5 IQR of Q1 and the maximum value is not within 1.5 IQR of Q3, there ar (1) symmetric skewed to the right skewed to the left (2) closer to the minimum. closer to the maximum. exactly halfway between the minimum and the maximum. ID: The scores on an exam have mean = 87, standard deviation = 13, minimum = 66, Q1 = 77, median = 81, Q3 = 103, and maximum = 120. State which of these values are used in a box plot and then sketch the box plot. Which of these values are used in the box plot? Select all that apply. A. Q3 B. maximum C. mean D. minimum E. standard deviation F. Q1 G. median Choose the correct box plot below. A. B. C. D ID: of 12 9/2/15, 1:12 PM

6 7. A survey was conducted to determine how many miles per day employees of a company used public transportation. The sample values are below. Identify the five-number summary, and draw a box plot Identify the five-number summary. minimum = Q1 = median = Q3 = maximum = Choose the correct box plot below. A. B. C. D ID: The unemployment rates for various countries range from 4.8 to 11.4, with Q1 = 5.9, median = 6.2, Q3 = 9.8, a mean of 6.9, and standard deviation of 3.3. Use this information to answer parts a through c. a. In a box plot, what would be the values at the outer edges of the box, and what would be the values to which the whiskers extend? The lower edge of the box is and the upper edge of the box is. The lower whisker extends down to and the upper whisker extends up to. b. The highest unemployment rate was Is it an outlier according to the three standard deviation criterion? Explain. A. No, because is it less than three standard deviations from the mean. B. Yes, because is it more than three standard deviations from the mean. C. Yes, because is it less than three standard deviations from the mean. D. No, because is it more than three standard deviations from the mean. c. What unemployment value for a country would have a z-score equal to 0? A country with a z-score of 0 would have an unemployment rate of. ID: of 12 9/2/15, 1:12 PM

7 9. The carbon dioxide emissions of a group of nations had a mean of 8.5 and standard deviation of 2.9. a. One country's observation was Find and interpret its z-score relative to the distribution of values for the group of nations. b. Another country's observation was 2.2. Find and interpret its z-score. a. Find the z-score for the observation of z = (Round to two decimal places as needed.) What does this z-score imply? A. The observation 14.4 is not an outlier because it is less than 3 standard deviations from the mean. B. The observation 14.4 is not an outlier because its z-score is positive. C. The observation 14.4 is an outlier because it is greater than 3 standard deviations from the mean. D. The observation 14.4 is an outlier because its z-score is negative. b. Find the z-score for the observation of 2.2. z = (Round to two decimal places as needed.) What does this z-score imply? A. The observation 2.2 is an outlier because its z-score is negative. B. The observation 2.2 is not an outlier because its z-score is positive. C. The observation 2.2 is an outlier because it is greater than 3 standard deviations from the mean. D. The observation 2.2 is not an outlier because it is less than 3 standard deviations from the mean. ID: For a sample of 297 female heights, the mean was 64.3 inches and the standard deviation was 2.2 inches. The shortest person in this sample had a height of 55 inches. a. Find the z-score for the height of 55 inches. b. What does the negative sign for the z-score represent? c. Is this observation a potential outlier according to the three standard deviation distance criterion? Explain. a. Find the z-score. z = (Round to one decimal place as needed.) b. What does the negative sign for the z-score represent? A. The observation is not a potential outlier. B. The observation is a potential outlier. C. The observation is above the mean. D. The observation is below the mean. c. Is this observation a potential outlier according to the three standard deviation distance criterion? Explain. A. Yes, because the z-score is negative. B. No, because the z-score is negative. C. Yes, because it is greater than three standard deviations from the mean. D. No, because it is less than three standard deviations from the mean. ID: of 12 9/2/15, 1:12 PM

8 11. The MINITAB vertical side-by-side box plots shown below compare the values reported by the UN of per capita carbon dioxide emissions for for two nations in a certain year. Complete parts (a) through (c) CO2 Nation 1 x CO2 Nation 2 x a. Give the approximate value of carbon dioxide emissions for the outlier shown. (Round to the nearest integer as needed.) b. What shape would you predict for the distribution in Nation 2? Why? A. Skewed to the left since the distance between Q2 and large and the lower whisker is much shorter than the B. Skewed to the right since the distance between Q2 a much larger than between Q1 and Q2 and the upper much longer than the lower one C. There can be many values represented by the shorte and just a few represented by the longer one. So, the enough evidence for prediction. c. Summarize how the carbon dioxide emissions compare in the two nations. A. The emissions levels in Nation 1 and in Nation 2 are because their whiskers overlap. B. The emissions are much higher in Nation 1 than in Na Roughly 75% of the levels reported in Nation 2 are le smallest level reported in Nation 1. C. The emissions are much lower in Nation 2 than in Na of the levels reported in Nation 2 are less than those Nation 1. ID: The six full-time employees of a tanning salon near campus had annual incomes last year of $ 8100, $ 8300, $ 8300, $ 8800, $ 9200, $ The owner made $ 450,000. a. For the seven annual incomes at the salon, report the mean and the median. b. Why is it misleading for the owner to boast to her friends that the average salary at the salon is more than $ 70,000? a. The mean salary is $. The median salary is $. b. Why is the boast misleading? A. It is misleading because no one actually makes $ 70,000. B. It is misleading because the mean is so influenced by the owner's salary that it is not a typical value. C. It is misleading because the mean is actually less than $ 70,000. D. The boast is not misleading. ID: of 12 9/2/15, 1:12 PM

9 13. A college newspaper reported the results of a survey of students taken on campus. One question asked was "Do you think the current war has made us safer?" The figure shows the way the magazine reported the results. Use this information to complete parts a and b Has War Made Us Safer? 36% Yes 61% No a. Explain what is wrong with the way this bar chart was constructed. A. The graph uses figures, so the relative percentages a misleading. B. The graph does not have a heading, so it is unclear w graph is about. C. The graph is portraying more than one group when th differ greatly, so the size of the bars is misleading. D. The vertical axis does not start at zero, so the relative percentages are visually misleading. b. Explain why you would not see this error made with a pie chart. A. A pie chart can always display more than one group. B. A pie chart never uses figures, so the relative percen always be clear. C. A pie chart always has a heading, so it will always be what the pie chart is depicting. D. A pie chart always uses percentages, so the relative the slices will represent the relative percentages in ea catagory. ID: of 12 9/2/15, 1:12 PM

10 (1) fewer than A. Approximately 50% of the data lie below this value A. This value is the average of the data set. 3. (1) fourth 76 (2) greater C. Roughly symmetric D. The IQR summarizes the range for the middle half of the data. A. The distribution is skewed right because the median is closer to Q1. Further proof is given by the values of the minimum and maximum relative to Q1 and Q3, respectively. 5. (1) skewed to the right (2) closer to the minimum. B. Since the maximum value is not within 1.5 IQR of Q3, there is at least one outlier. 6. A. Q3, B. maximum, D. minimum, F. Q1, G. median 10 of 12 9/2/15, 1:12 PM

11 B D A. No, because is it less than three standard deviations from the mean A. The observation 14.4 is not an outlier because it is less than 3 standard deviations from the mean D. The observation 2.2 is not an outlier because it is less than 3 standard deviations from the mean D. The observation is below the mean. C. Yes, because it is greater than three standard deviations from the mean. 11 of 12 9/2/15, 1:12 PM

12 B. Skewed to the right since the distance between Q2 and Q3 is much larger than between Q1 and Q2 and the upper whisker is much longer than the lower one B. The emissions are much higher in Nation 1 than in Nation 2. Roughly 75% of the levels reported in Nation 2 are less than the smallest level reported in Nation , B. It is misleading because the mean is so influenced by the owner's salary that it is not a typical value. 13. D. The vertical axis does not start at zero, so the relative percentages are visually misleading. D. A pie chart always uses percentages, so the relative sizes of the slices will represent the relative percentages in each catagory. 12 of 12 9/2/15, 1:12 PM