Question

the highway mileage (mpg) for a sample of 8 different models of a car company can be found below. find the mean, median, mode, and sample standard deviation. round to one decimal place as needed. 20, 23, 26, 28, 30, 32, 35, 35 a. mean = 38.88 b. median = 32 c. mode = 26,33 d. sample standard deviation = question help: message instructor post to forum

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 8\) and \(\sum_{i=1}^{8}x_{i}=20 + 23+26+28+30+32+35+35=229\). So \(\bar{x}=\frac{229}{8}=28.625\approx28.6\) (the original value of mean in the problem is wrong).

Step2: Calculate the median

For \(n = 8\) (even - numbered data set), the median \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\). \(x_{4}=28\), \(x_{5}=30\). So \(M=\frac{28 + 30}{2}=29\) (the original value of median in the problem is wrong).

Step3: Calculate the mode

The mode is the most - frequently occurring value. In the data set \(20,23,26,28,30,32,35,35\), the mode is \(35\) (the original value of mode in the problem is wrong).

Step4: Calculate the sample standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
First, \((x_{1}-\bar{x})^{2}=(20 - 28.6)^{2}=(- 8.6)^{2}=73.96\)
\((x_{2}-\bar{x})^{2}=(23 - 28.6)^{2}=(-5.6)^{2}=31.36\)
\((x_{3}-\bar{x})^{2}=(26 - 28.6)^{2}=(-2.6)^{2}=6.76\)
\((x_{4}-\bar{x})^{2}=(28 - 28.6)^{2}=(-0.6)^{2}=0.36\)
\((x_{5}-\bar{x})^{2}=(30 - 28.6)^{2}=(1.4)^{2}=1.96\)
\((x_{6}-\bar{x})^{2}=(32 - 28.6)^{2}=(3.4)^{2}=11.56\)
\((x_{7}-\bar{x})^{2}=(35 - 28.6)^{2}=(6.4)^{2}=40.96\)
\((x_{8}-\bar{x})^{2}=(35 - 28.6)^{2}=(6.4)^{2}=40.96\)

\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=73.96+31.36 + 6.76+0.36+1.96+11.56+40.96+40.96=207.8\)

\(s=\sqrt{\frac{207.8}{8 - 1}}=\sqrt{\frac{207.8}{7}}\approx\sqrt{29.69}\approx5.5\)

Answer:

a. Mean \(=28.6\)
b. Median \(=29\)
c. Mode \(=35\)
d. Sample Standard Deviation \(=5.5\)