91Ó°ÊÓ

Observe the data and try to determine whether the data is symmetric or skewed. There is no straightforward method to do this, but look for similarity in values on either side of the middle value.

Mean is calculated by summing all the values and dividing by the total number of values present. To find the median, you'll need to sort the data in ascending order and find the middle value. In this case, the data is small, so we can sort it manually: $$ \begin{array}{rrrr} 8 & 11 & 52 & 57\\ 65 & 87 & 93 & 97\\ 109 & 120 & 127 & 133\\ 139 & 142 & 144 & 147\\ 148 & 157 & 162 & 165 \end{array} $$ Mean: $$ \frac{8+11+52+57+65+87+93+97+109+120+127+133+139+142+144+147+148+157+162+165}{20} = 107.8 $$ Median: The middle value will be the average of the values at positions 10 and 11 (97 and 109). $$ \frac{97+109}{2} = 103 $$ Since the mean and median are very similar, we can conclude that the data is roughly symmetric, with no significant skewness.

Construct a box plot of the data. The box plot consists of a box (containing the middle 50% of the data) and whiskers extending from the ends of the box to the minimum and maximum values. 1. Calculate the first quartile (Q1), which represents the 25th percentile of the data: The position of Q1 is at \(0.25 \cdot (20+1) = 5.25\). It's between the 5th value (65) and 6th value (87). $$ Q1 = \frac{65+87}{2} = 76 $$ 2. Calculate the third quartile (Q3), which represents the 75th percentile of the data: The position of Q3 is at \(0.75 \cdot (20+1) = 15.75\). It's between the 15th value (144) and the 16th (147). $$ Q3 = \frac{144+147}{2} = 145.5 $$ 3. Calculate the interquartile range (IQR): $$ IQR = Q3 - Q1 = 145.5 - 76 = 69.5 $$ 4. Draw the box plot: - Draw a horizontal scale from the minimum value (8) to the maximum value (165). - Draw a box from Q1 (76) to Q3 (145.5). - Draw a line inside the box at the median (103). - Draw whiskers from the ends of the box to the minimum value (8) and the maximum value (165). The box plot is symmetric, with the median line in the center of the box, and the whiskers have similar lengths. This confirms our conclusions from part b that the data is roughly symmetric and not significantly skewed.