Question

the amount of caffeine in a sample of five - ounce servings of brewed coffee is shown in the histogram. make a frequency distribution for the data. then use the table to estimate the sample mean and the sample standard deviation of the data set. click the icon to view the histogram. σf = 50 find the mean of the data set. x = 114.5 (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 - 44 - 22 0 22 44 (x - x)² 1936 484 0 484 1936 (x - x)²f 1936 5808 0 5808 1936 σ(x - x)²f = 15488 σxf = 5725 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 $\sum(x - \bar{x})^2f = 15488$ and $n=\sum f=50$.

Step3: Calculate the sample standard deviation

Substitute the values into the formula:
\[

$$\begin{align*} s&=\sqrt{\frac{15488}{50 - 1}}\\ &=\sqrt{\frac{15488}{49}}\\ &=\sqrt{316.081633}\\ &\approx17.78 \end{align*}$$

\]

Answer:

$17.78$