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 \( = \) \( \square \) \[ 95 \mathrm{th} \text { percentile }=14 \] (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 \( I Q R \) 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 \( =4.5 \) (h) With five-number-summary (in the order Min = \( Q_{1}=Q_{2}-Q_{3}-\operatorname{Max} \) ) 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. ### 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 value of interest, 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 \( N = 46 \). Now, calculate the percentile: \[ P = \left( \frac{18}{46} \right) \times 100 \approx 39.13 \] Rounding up gives us the 40th percentile. **Answer:** 10 is the \( 40 \)th percentile. ### (b) Find the 95th percentile on the data. To find the 95th percentile, we calculate the locator: \[ L = \frac{95}{100} \times (N + 1) = 0.95 \times 47 = 44.65 \] Rounding up gives us the 45th position. The value at the 45th position is 14. **Answer:** \( 95 \text{th percentile } = 14 \) ### (c) Find the \( Q_{1} \) value on the data. To find \( Q_1 \): \[ L = \frac{25}{100} \times (N + 1) = 0.25 \times 47 = 11.75 \] Rounding up gives us the 12th position. The value at the 12th position is 9. **Answer:** \( Q_{1} = 9 \) ### (d) Find the \( Q_{3} \) value on the data. To find \( Q_3 \): \[ L = \frac{75}{100} \times (N + 1) = 0.75 \times 47 = 35.25 \] Rounding up gives us the 36th position. The value at the 36th position is 12. **Answer:** \( Q_{3} = 12 \) ### (e) Find the \( Q_{2} \) value on the data. The median \( Q_2 \) is found at: \[ L = \frac{50}{100} \times (N + 1) = 0.5 \times 47 = 23.5 \] Rounding up gives us the 24th position. The value at the 24th position is 10. **Answer:** \( Q_{2} = 10 \) ### (f) Find the \( IQR \) of the data. The interquartile range \( IQR \) is calculated as: \[ IQR = Q_3 - Q_1 = 12 - 9 = 3 \] **Answer:** \( IQR = 3 \) ### (g) What are the Upper and Lower limits of the boxplot? - **Upper Limit**: \( Q_3 + 1.5 \times IQR = 12 + 1.5 \times 3 = 12 + 4.5 = 16.5 \) - **Lower Limit**: \( Q_1 - 1.5 \times IQR = 9 - 1.5 \times 3 = 9 - 4.5 = 4.5 \) **Answer:** Upper Limit = 16.5, Lower Limit = 4.5 ### (h) Are there any outliers? To find outliers, we check for values outside the limits: - Outliers are values less than 4.5 or greater than 16.5. - The data set has no values below 5 or above 15. **Answer:** No outliers. ### Summary of Answers (a) 40 (b) 14 (c) 9 (d) 12 (e) 10 (f) 3 (g) Upper Limit = 16.5, Lower Limit = 4.5 (h) No outliers.