Question

A researcher collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed limit was 25mph. The recorded speed of 20 randomly selected vehicles is given below: 2038243927402837293530313219331834213623 Calculate the sample mean, standard deviation, and the median of the above data. (Round to one decimal place

Answer 1

`\bar x =29.7`

`\sigma = 6.81`

`Median = 30.5`

Explanation:

Given

`Data:20,38,24,39,27,40,28,37,29,35,30,31,32,19,33,18,34,21,36,23`

`n = 20`

Solving (a): The sample mean

This is calculated as:

`\bar x =(\sum x)/(n)`

`\bar x =(20+38+24+39+27+40+28+37+29+35+30+31+32+19+33+18+34+21+36+23)/(20)`

`\bar x =(594)/(20)`

`\bar x =29.7`

Solving (b): The standard deviation

This is calculated as;

`\sigma = \sqrt{(\sum(x-\bar x)^2)/(n)}`

So, we have:

`\sigma = \sqrt{(928.2)/(20)}`

`\sigma = √(46.41)`

`\sigma = 6.81`

Solving (c): The median

First, sort the data in ascending order

`Sorted:18,19,20,21,23,24,27,28,29,30,31,32,33,34,35,36,37,38,39,40`

The position of the median is calculated as:

`Median = (n+1)/(2)`

`Median = (20+1)/(2)`

`Median = (21)/(2)`

`Median = 10.5th`

The 10.5th item represents the mean of the 10th and 11th item.

So, median is:

`Median = (30+31)/(2)`

`Median = (61)/(2)`

`Median = 30.5`