What percentage of the general U.S. population are high-school dropouts? The Statistical Abstract of the United States, 120 th Edition, gives the percentage of high-school dropouts by state. For convenience, the data are sorted in increasing order. \[ \begin{array}{c} 5,6,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11 \\ 11,11,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,15 \end{array} \] (a) Which percentile is the value 10 ? (Round to the nearest whole value above.) 10 is the \( \square \) th percentile. (b) Find the 95th percentile on the data. Locator \( = \) \( \square \) \[ 95 \text { th percentile }=14 \] (c) Find the \( Q_{1} \) value on the data. \[ \text { Locator }=13 \] \[ Q_{1}=9 \] (d) Find the Q3 value on the data. \[ \text { Locator }=\square \] \[ Q_{3}=12 \] (e) Find the \( Q_{2} \) value on the data. \[ Q_{2}=10 \] (f) Find the IQR of the data. \[ I Q R=3 \] (g) What are the Upper and Lower limits of the boxplot? \[ \text { Upper Limit }=16.5 \] Lower Limit \( =4.5 \) (b) With five-number-summary (in the order Min = Q1 - Q2 - Q3 - Max) being 5-9-10 = 12 15, are there any outliers? The only outlier is 5 The only outlier is 15 Outliers are 15 , and 14 Outliers are 5, and 15. o No outliers

Let's solve the problem step by step, addressing each part clearly. ### Given Data The data set of high-school dropout percentages is: \[ 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15 \] ### (a) Which percentile is the value 10? To find the percentile of a value in a data set, we use the formula: \[ P = \left( \frac{N_L}{N} \right) \times 100 \] where \( N_L \) is the number of values less than the given value, and \( N \) is the total number of values. 1. Count the number of values less than 10: - Values less than 10: \( 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9 \) (total of 18 values) 2. Total number of values \( N = 45 \). Now, calculate the percentile: \[ P = \left( \frac{18}{45} \right) \times 100 = 40 \] Thus, 10 is the \( 40 \)th percentile. ### (b) Find the 95th percentile on the data. To find the 95th percentile, we calculate the locator: \[ L = \frac{95}{100} \times (N + 1) = 0.95 \times 46 = 43.7 \] Since we round up, we take the 44th value in the ordered list. The 44th value is \( 14 \). ### (c) Find the \( Q_1 \) value on the data. To find \( Q_1 \): \[ L = \frac{25}{100} \times (N + 1) = 0.25 \times 46 = 11.5 \] Rounding up gives us the 12th value, which is \( 9 \). ### (d) Find the \( Q_3 \) value on the data. To find \( Q_3 \): \[ L = \frac{75}{100} \times (N + 1) = 0.75 \times 46 = 34.5 \] Rounding up gives us the 35th value, which is \( 12 \). ### (e) Find the \( Q_2 \) value on the data. The median \( Q_2 \) is the 23rd value: \[ Q_2 = 10 \] ### (f) Find the IQR of the data. The Interquartile Range (IQR) is calculated as: \[ IQR = Q_3 - Q_1 = 12 - 9 = 3 \] ### (g) What are the Upper and Lower limits of the boxplot? The limits are calculated as follows: - Upper Limit: \( Q_3 + 1.5 \times IQR = 12 + 1.5 \times 3 = 12 + 4.5 = 16.5 \) - Lower Limit: \( Q_1 - 1.5 \times IQR = 9 - 1.5 \times 3 = 9 - 4.5 = 4.5 \) ### Outliers To determine outliers, we check values outside the limits: - Outliers are values below \( 4.5 \) or above \( 16.5 \). - The only outlier is \( 5 \) (since all other values are within the limits). ### Summary of Answers - (a) 10 is the \( 40 \)th percentile. - (b) 95th percentile = \( 14 \). - (c) \( Q_1 = 9 \). - (d) \( Q_3 = 12 \). - (e) \( Q_2 = 10 \). - (f) \( IQR = 3 \). - (g) Upper Limit = \( 16.5 \), Lower Limit = \( 4.5 \). - Outliers: The only outlier is \( 5 \).