Question

find the mean of the data set. x = 110.1 (round to the nearest tenth as needed.) complete the table. round values to the nearest tenth as needed. midpoint x 70.5 92.5 114.5 136.5 158.5 x - x -39.6 -17.6 4.4 26.4 48.4 (x - x)^2 1568.2 309.8 19.4 697.0 2342.6 (x - x)^2f 4704.6 4337.2 504.4 2788.0 7027.8 ∑(x - x)^2f = 19362.0 ∑f = 5505 2977 546 475.5 find the sample standard deviation of the data set. s = (round to the nearest hundredth as needed.)

Explanation:

Step1: Recall sample - standard - deviation formula

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum(x - \bar{x})^2f}{n - 1}}$, where $\sum(x - \bar{x})^2f$ is the sum of the squared deviations from the mean weighted by the frequencies, and $n=\sum f$ is the total number of data points.

Step2: Identify given values

We are given that $\sum(x - \bar{x})^2f = 19362.0$ and $n=\sum f=50$.

Step3: Calculate the sample standard deviation

Substitute the values into the formula:
\[

$$\begin{align*} s&=\sqrt{\frac{19362.0}{50 - 1}}\\ &=\sqrt{\frac{19362.0}{49}}\\ &=\sqrt{395.142857}\\ &\approx19.88 \end{align*}$$

\]

Answer:

$19.88$