Question

the fuel efficiency, in miles per gallon, of 12 small utility trucks was measured. the results are recorded in the table below. fuel efficiency (mpg) 22 26 22 26 16 23 23 32 24 23 25 21 find the mean and sample standard deviation of these data. round to the nearest hundredth. mean 23.92 sample standard deviation 3.68 suggested tutorials: learn it: find the mean, median, and mode of a data set. learn it: calculate the variance and standard deviation of a sample. resources read it

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$.
Here, $n = 12$ and $x=\{22,26,22,26,16,23,23,32,24,23,25,21\}$.
$\sum_{i=1}^{12}x_i=22 + 26+22+26+16+23+23+32+24+23+25+21=285$.
$\bar{x}=\frac{285}{12}=23.75$.

Step2: Calculate the sample variance

The formula for the sample variance $s^{2}$ is $s^{2}=\frac{1}{n - 1}\sum_{i = 1}^{n}(x_i-\bar{x})^{2}$.
$x_1-\bar{x}=22 - 23.75=-1.75$, $(x_1-\bar{x})^{2}=(-1.75)^{2}=3.0625$.
$x_2-\bar{x}=26 - 23.75 = 2.25$, $(x_2-\bar{x})^{2}=(2.25)^{2}=5.0625$.
$x_3-\bar{x}=22 - 23.75=-1.75$, $(x_3-\bar{x})^{2}=(-1.75)^{2}=3.0625$.
$x_4-\bar{x}=26 - 23.75 = 2.25$, $(x_4-\bar{x})^{2}=(2.25)^{2}=5.0625$.
$x_5-\bar{x}=16 - 23.75=-7.75$, $(x_5-\bar{x})^{2}=(-7.75)^{2}=60.0625$.
$x_6-\bar{x}=23 - 23.75=-0.75$, $(x_6-\bar{x})^{2}=(-0.75)^{2}=0.5625$.
$x_7-\bar{x}=23 - 23.75=-0.75$, $(x_7-\bar{x})^{2}=(-0.75)^{2}=0.5625$.
$x_8-\bar{x}=32 - 23.75 = 8.25$, $(x_8-\bar{x})^{2}=(8.25)^{2}=68.0625$.
$x_9-\bar{x}=24 - 23.75 = 0.25$, $(x_9-\bar{x})^{2}=(0.25)^{2}=0.0625$.
$x_{10}-\bar{x}=23 - 23.75=-0.75$, $(x_{10}-\bar{x})^{2}=(-0.75)^{2}=0.5625$.
$x_{11}-\bar{x}=25 - 23.75 = 1.25$, $(x_{11}-\bar{x})^{2}=(1.25)^{2}=1.5625$.
$x_{12}-\bar{x}=21 - 23.75=-2.75$, $(x_{12}-\bar{x})^{2}=(-2.75)^{2}=7.5625$.
$\sum_{i = 1}^{12}(x_i-\bar{x})^{2}=3.0625+5.0625+3.0625+5.0625+60.0625+0.5625+0.5625+68.0625+0.0625+0.5625+1.5625+7.5625 = 155.25$.
Since $n = 12$, $s^{2}=\frac{155.25}{12-1}=\frac{155.25}{11}\approx14.1136$.

Step3: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^{2}}$.
$s=\sqrt{14.1136}\approx3.76$.

Answer:

mean: $23.75$; sample standard deviation: $3.76$