Question

you work for a consumer watchdog publication and are testing the advertising claims of a tire manufacturer. the manufacturer claims that the life spans of the tires are normally distributed, with a mean of 40,000 miles and a standard deviation of 3500 miles. you test 16 tires and get the following life spans. complete parts (a) through (c) below. 47,341 40,426 27,908 34,457 32,241 39,052 42,528 36,365 25,392 30,739 34,330 38,029 33,043 37,079 37,699 43,495 is it reasonable to assume that the life spans are normally distributed? why? choose the correct answer below. a. yes, because the histogram is neither symmetric nor bell - shaped b. yes, because the histogram is symmetric and bell - shaped. c. no, because the histogram is symmetric and bell - shaped. d. no, because the histogram is neither symmetric nor bell - shaped. (b) find the mean of your sample. the mean is 36257.8. (round to one decimal place as needed.) find the standard deviation of your sample. the standard deviation is . (round to one decimal place as needed.)

Explanation:

Step1: Recall sample - standard - deviation formula

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $n$ is the sample size, $x_{i}$ are the individual data - points, and $\bar{x}$ is the sample mean. Here, $n = 16$ and $\bar{x}=36257.8$.

Step2: Calculate the sum of squared differences

Let the data - points be $x_1 = 47341,x_2 = 40426,\cdots,x_{16}=43495$.
First, calculate $(x_1-\bar{x})^2=(47341 - 36257.8)^2=(11083.2)^2 = 122847030.24$, $(x_2-\bar{x})^2=(40426 - 36257.8)^2=(4168.2)^2 = 17373891.24$, and so on for all 16 data - points. Then sum them up: $\sum_{i = 1}^{16}(x_{i}-\bar{x})^{2}$.
$\sum_{i=1}^{16}(x_{i}-36257.8)^{2}=(47341 - 36257.8)^{2}+(40426 - 36257.8)^{2}+\cdots+(43495 - 36257.8)^{2}$
$=122847030.24 + 17373891.24+\cdots+5244195.84$
$=499793957.44$

Step3: Calculate the sample standard deviation

Substitute into the formula: $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}=\sqrt{\frac{499793957.44}{16 - 1}}=\sqrt{\frac{499793957.44}{15}}\approx\sqrt{33319597.1627}\approx5772.3$

Answer:

$5772.3$