Question

Power $90$Extreme. Beachbody, L.L. provides fitness programs, including home workout videos and nutrition. P90X, or Power $90$Extreme, is a home exercise program that consists of an intense series of workout DVDs. It is a $90$-day program that uses the term "muscle confusion" to refer to their training methods. There are a total of $12$workout DVDs included with the program. each targeting a different muscle group. According to their official website, the Iengths , in minutes, of the $12$ DVDs are as follows.
a. Obtain and interpret the population mean length of DVDs.
b. Obtain and interpret the population standard deviation lengths of the P90X DVDs.

Short answer

Part(a) The population mean length is $54.92$

Part(b) The population standard deviation is $259.71$

Step-by-step solution

Part(a) Step 1: Given information

We are given data of P90X lenghts. We have to find mean length of DVDs.

Part(a) Step 2: Simplify

Mean of sample is sum of all lenghts divided by total number of lenghts given.

$$\begin{gathered} \overline{x}=\frac{53+58+60+92+59+58+57+57+55+51+43+16}{12} \\ \overline{x}=54.92 \end{gathered}$$

Part(b) Step 1: Given information

We are given data of P90X lenghts. We have to find standard deviation length of DVDs.

Part(b) Step 2: Simplify

Standard deviation is given by

$$\begin{gathered} \sigma =\frac{{\displaystyle \sum _{}}{x}^2}{n}-{\left(\overline{x}\right)}^2 \\ \frac{\sum _{}{x}^2}{n}=\frac{{53}^2+{58}^2+{60}^2+{92}^2+{59}^2+{58}^2+{57}^2+{57}^2+{55}^2+{51}^2+{43}^2+{16}^2}{12} \\ \frac{\sum _{}{x}^2}{n}=3275.92 \\ {\left(\overline{x}\right)}^2=3016.21 \\ \sigma =3275.92-3016.21=259.71 \end{gathered}$$

The population standard deviation length of DVDs is $259.71$