Q. 26 - Elementary Statistics

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.

Step-by-Step Solution

Answer

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

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

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 $259.71$