Q 99. - The Practice of Statistics for AP

Question

The figure displays computer output from Data Desk for data on the amount spent by $50$ grocery shoppers. 

(a) What would you guess is the shape of the distribution based only on the computer output? Explain.

 (b) Interpret the value of the standard deviation. 

(c) Are there any outliers? Justify your answer. 

Step-by-Step Solution

Answer

Part (a) The shape of the distribution is right-skewed.

Part (b) The standard deviation is worth $\$ 21.6974$

Part (c) There could be more outliers.

Part (a) Step 1: Given information

Given: 

Part (a) Step 2: Concept

Qualitative variables are indicated individual variables whereas quantitative categories are indicated numerical variables. 

Part (a) Step 3: Explanation

We can see that the mean is much larger than the median in the output. If the mean is significantly bigger than the median, the distribution is right-skewed (because the mean is influenced by large outliers, while the median remains unaffected).

Part (b) Step 1: Calculation

The standard deviation (Std Dev) value is reported as follows in the output: $s_x = 21.6974$

This suggests that the difference between a typical shopper's spending and the mean spending is $\$21.6974$

Part (c) Step 1: Calculation

Given in the output: $$\begin{gathered} Q_1 = 19.27 \\ {Q }_3= 45.4 \\ Max = 93.34 \end{gathered}$$

The third quartile's interquartile range is reduced by the first quartile: $IQR = {Q }_3- Q_1 = 45.4 - 19.27 = 26.13$

Outliers are observations that are more than 1.5 times the IQR above or below Q3, or more than 1.5 times the IQR below Ql. $$\begin{gathered} {Q }_3+ 1.5IQR = 45.4 + 1.5 (26.13) = 84.595 \\ {Q }_3- 1.5IQR = 19.27 + 1.5 (26.13) =-19.925 \end{gathered}$$

The maximum of $93.34$is thus bigger than $84.595$ indicating that the maximum is an anomaly (there might be more outliers). Therefore, there might be more outliers.