Question

a group of 10 students was asked how many hours they slept the night before. below are their responses: 0, 7.5, 7.5, 8, 8.5, 8.5, 9, 9, 9, 9. what is the mean? what is the median? there is a low outlier in the list. if you remove the outlier, what are the revised mean and median? mean (without the outlier) (round to 2 decimal places.) median (without the outlier) which changed more by removing the outlier? mean median

Explanation:

Step1: Calculate the original mean

The data set is \(0, 7.5, 7.5, 8, 8.5, 8.5, 9, 9, 9, 9\). The sum of the data is \(0 + 7.5+7.5 + 8+8.5+8.5+9+9+9+9=76\). The number of data - points \(n = 10\). The mean \(\bar{x}=\frac{76}{10}=7.6\).

Step2: Calculate the original median

Since \(n = 10\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th ordered values. The \(\frac{10}{2}=5\)th and \((\frac{10}{2}+1) = 6\)th values are \(8.5\) and \(8.5\). So the median \(M=\frac{8.5 + 8.5}{2}=8.5\).

Step3: Identify the outlier and recalculate mean and median

The outlier is \(0\). After removing the outlier, the new data set has \(n=9\) data - points: \(7.5, 7.5, 8, 8.5, 8.5, 9, 9, 9, 9\). The sum of the new data set is \(7.5+7.5 + 8+8.5+8.5+9+9+9+9=76 - 0=76\). The new mean \(\bar{x}_{new}=\frac{76}{9}\approx8.44\). Since \(n = 9\) (an odd - numbered data set), the median is the \(\frac{n + 1}{2}=\frac{9+1}{2}=5\)th ordered value, which is \(8.5\).

Step4: Determine which is changed more

The change in the mean is \(|7.6-8.44| = 0.84\). The change in the median is \(|8.5 - 8.5|=0\). So the mean is changed more by removing the outlier.

Answer:

What is the mean? 7.6
What is the median? 8.5
What are the revised mean and median (without the outlier)? Mean: 8.44, Median: 8.5
Which changed more by removing the outlier? mean