91Ó°ÊÓ

Since the data set is already sorted in ascending order, we need to find the positions of the first and third quartiles using the positions given by \( Q_1 = \frac{n+1}{4} \) and \( Q_3 = \frac{3(n+1)}{4} \), where \( n \) is the number of observations.The number of observations \( n \) is 30.To find the first quartile (\( Q_1 \)), calculate:\[ Q_1 = \frac{30+1}{4} = 7.75 \]The position is 7.75, indicating \( Q_1 \) is between the 7th and 8th values in the list (31 and 34 respectively). The value of \( Q_1 \) is calculated as:\[ Q_1 = 31 + 0.75(34 - 31) = 33.25 \]To find the third quartile (\( Q_3 \)), calculate:\[ Q_3 = \frac{3(30+1)}{4} = 23.25 \]The position is 23.25, indicating \( Q_3 \) is between the 23rd and 24th values in the list (50 and 51 respectively). The value of \( Q_3 \) is calculated as:\[ Q_3 = 50 + 0.25(51 - 50) = 50.25 \]

Deciles divide the data set into ten equal parts. The positions for the second and eighth decile are given by:For the second decile (\(D_2\)), \[ D_2 = \frac{2(n+1)}{10} = \frac{2(30+1)}{10} = 6.2 \]The position is 6.2. The 6th and 7th values in the list are 27 and 31 respectively. So:\[ D_2 = 27 + 0.2(31 - 27) = 27.8 \]For the eighth decile (\(D_8\)), \[ D_8 = \frac{8(n+1)}{10} = \frac{8(30+1)}{10} = 24.8 \]The position is 24.8. The 24th and 25th values in the list are 51 and 53 respectively. So:\[ D_8 = 51 + 0.8(53 - 51) = 52.6 \]

Percentiles divide the data into 100 equal parts. For the 67th percentile, the position is given by:\[ P_{67} = \frac{67(n+1)}{100} = \frac{67(30+1)}{100} = 20.77 \]This position suggests the 20th and 21st items, which are both 47. Thus, the 67th percentile value is simply 47.