DO NOT POST THESE ANSWERS ONLINE © BFW Publishers 2018 Chapter 2 Full Solutions Section 2.1 Check Your Understanding, page 95: 1. c 2. Her daughter weighs more than 87% of girls her age and she is taller than 67% of girls her age. 3. About 65% of calls lasted less than 30 minutes. This means that about 35% of calls lasted 30 minutes or longer. 4. The first quartile (25th percentile) is at about Q1 = 13 minutes. The third quartile (75th percentile) is at about Q3 = 32 minutes. This suggests that the IQR =−=32 13 19 minutes. Check Your Understanding, page 97: − 6762 1. z = −= 166.1 . Interpretation: Lynette’s height is 1.166 standard deviations below the mean height of 29.4 the class. 74− 76 2. Because Brent’s z-score is −0.85, we know that −=0.85 . Solving for σ we find that σ = 2.35 inches. σ Check Your Understanding, page 103: 1. Converting the cost of the rides from dollars to cents will not change the shape. However, it will multiply the mean and the standard deviation by 100. 2. Adding 25 cents to the cost of each ride will not change the shape of the distribution, nor will it change the variability. It will, however, add 25 cents to the measures of center (mean, median). 3. Converting the costs to z-scores will not change the shape of the distribution. It will change the mean to 0 and the standard deviation to 1. Chapter 2: Modeling Distributions of Data 1 DO NOT POST THESE ANSWERS ONLINE © BFW Publishers 2018 Section 2.1 Exercises 2.1 (a) Because 17 of the 20 students (85%) own fewer pairs of shoes than Jackson (who owns 22 pairs of shoes), Jackson is at the 85th percentile in the number of pairs of shoes distribution. (b) Interpretation: 45% of the boys had fewer pairs of shoes than Raul did. Raul was at the 45th percentile. This means that 45% of the 20 boys, or 9 boys, have fewer pairs of shoes. Therefore, Raul’s response is the 10th value in the ordered list. Putting the data in order we get: 4, 5, 5, 5, 6, 7, 7, 7, 7, 8, 10, 10, 10, 10, 11, 12, 14, 22, 35, and 38. Raul owns 8 pairs of shoes. 2.2 (a) Because 4 of the 50 states (8%) have smaller values for the percent of residents aged 65 and older than Colorado, Colorado is at the 8th percentile in the distribution. Interpretation: 8% of states have a smaller percent of residents aged 65 and older than the state of Colorado. (b) Interpretation: 80% of states have a smaller percent of residents aged 65 and older than the state of Rhode Island. Rhode Island is at the 80th percentile. This means that 80% of the 50 states, or 40 states, have a smaller percent of residents aged 65 and older. Therefore, Rhode Island is the 41st value in the ordered list. Counting the observations in the stemplot allows us to determine that 13.9% of Rhode Island’s residents are aged 65 and older. 2.3 (a) The head circumferences (sorted in ascending order) are 20.8, 20.8, 21, 21.5, 21.6, 21.7, 21.9, 22, 22.2, 22.3, 22.4, 22.5, 22.5, 22.6, 22.6, 22.7, 22.7, 22.7, 23, 23, 23.1, 23.3, 23.4, 23.5, 23.5, 23.9, 23.9, 24, 24.2, 25.6. Because 10 of the 30 observations (33.3%) are below Antawn’s head circumference (22.4 inches), Antawn is at the 33.3rd percentile in the head circumference distribution. (b) The head circumferences (sorted in ascending order) are 20.8, 20.8, 21, 21.5, 21.6, 21.7, 21.9, 22, 22.2, 22.3, 22.4, 22.5, 22.5, 22.6, 22.6, 22.7, 22.7, 22.7, 23, 23, 23.1, 23.3, 23.4, 23.5, 23.5, 23.9, 23.9, 24, 24.2, 25.6. In this case, the 90th percentile means that 90% of the 30 players, or 27 players, will have a smaller head circumference than this player. Therefore, the player at the 90th percentile will have a head circumference that is the 28th value in the ordered list, or equivalently, the third largest head circumference. The player with a head circumference of 24 inches is at the 90th percentile of the distribution. 2.4 (a) The number of text messages sent (sorted in ascending order) are: 0, 0, 0, 1, 1, 3, 3, 5, 5, 7, 8, 8, 9, 14, 25, 25, 26, 29, 42, 44, 52, 72, 92, 98, 118. Because 18 of the 25 students (72%) sent fewer text messages than Sunny (who sent 42 texts), Sunny is at the 72nd percentile in the distribution of number of text messages sent in the past 24 hours. (b) The number of text messages sent (sorted in ascending order) are: 0, 0, 0, 1, 1, 3, 3, 5, 5, 7, 8, 8, 9, 14, 25, 25, 26, 29, 42, 44, 52, 72, 92, 98, 118. Joelle is at the 12th percentile. This means that 12% of the 25 students in the class, or 3 students sent fewer text messages than she did. Therefore, the number of texts Joelle sent will be the 4th value in the ordered list. Joelle sent 1 text message in the past 24 hours. 2.5 This means that the speed limit is set at such a speed that 85% of the vehicle speeds are slower than the posted speed. 2 Starnes/Tabor, The Practice of Statistics, 6e DO NOT POST THESE ANSWERS ONLINE © BFW Publishers 2018 2.6 Larry’s wife should tell him that being at the 90th percentile for blood pressures is not a good thing. It means that 90% of men like him have a lower blood pressure than him! When it comes to blood pressure, a high number is not desirable. Larry may need treatment for his high blood pressure. 2.7 The girl in question weighs more than 48% of girls her age, but is taller than 78% of the girls her age. Because she is taller than 78% of girls, but only weighs more than 48% of girls, she is probably fairly thin. 2.8 Peter’s time was slower than 80% of his previous race times that season, but it was slower than only 50% of the racers at the league championship meet. Because this time was relatively slow for Peter but at the median for the runners in the league championship, Peter must be a good runner. 2.9 (a) No. A sprint time of 8 seconds is not unusually slow. A student with an 8 second sprint is at the 75th percentile, so 25% of the students took that long or longer. (b) The 20th percentile of the distribution is approximately 6.7 seconds. Interpretation: 20% of the students completed the 50-yard sprint in less than 6.7 seconds. 2.10 (a) No. North Dakota, with a median household income of $55,766, is not an unusually wealthy state. North Dakota is at the 65th percentile of median household income. That means that 35% of states have a median household income as large as or larger than North Dakota. (b) The 90th percentile of the distribution of median household income is approximately $65,000. Interpretation: 90% of the states have a median household income of less than $65,000. 2.11 (a) The IQR is calculated by subtracting Q1 from Q3. The first quartile is the 25th percentile. Find 25 on the y- axis, read over to the line and then down to the x-axis to get about Q1 = 4%. The 3rd quartile is the 75th percentile. Find 75 on the y-axis, read over to the line and then down to the x-axis to get about Q3 = 14%. Therefore, the IQR is approximately 14 – 4 = 10%. (b) Arizona, which had 15.1% foreign-born residents that year, is at the 90th percentile. (c) The graph is fairly flat between 20% and 27.5% foreign-born residents because there were very few states that had 20% to 27.5% foreign-born residents that year. (d) Chapter 2: Modeling Distributions of Data 3 DO NOT POST THESE ANSWERS ONLINE © BFW Publishers 2018 2.12 (a) The IQR is calculated by subtracting Q1 from Q3. The first quartile is the 25th percentile. Find 25 on the y- axis, read over to the line and then down to the x-axis to get about Q1 = $19. The 3rd quartile is the 75th percentile. Find 75 on the y-axis, read over to the line and then down to the x-axis to get about Q3 = $46. Therefore, the IQR is approximately $46−= $19 $27. (b) The person who spent $19.50 is just above what we have called the 25th percentile. It appears that $19.50 is at about the 26th percentile. (c) The graph is steepest between $10 and $30 because more shoppers spent amounts in this interval than any other interval. (d) The graph is below: 2.13 1.9− 8.73 (a) The z-score for Montana is z = = −1.12 . Interpretation: Montana’s percent of foreign-born 6.12 residents is 1.12 standard deviations below the mean percent of foreign-born residents for all states. (b) If we let x denote the percent of foreign-born residents in New York at that time, then we can solve for x in the x − 8.73 equation 2.10 = .