Question

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the mean height of a group of students is 65.4 inches. a student joins the group who is 73 inches tall. how does this students height affect the mean?
a. the new mean height will still be 64.5 inches.
b. the new mean height will be greater than 65.4 inches.
c. the new mean height will be 73 inches.
d. the new mean height will be less than 65.4 inches.
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Explanation:

Step1: Recall the formula for the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n$ is the number of data - points and $\sum_{i=1}^{n}x_{i}$ is the sum of the data - points.

Let the original number of students be $n$, and the original sum of heights be $S$. So, the original mean $\bar{x}_{old}=\frac{S}{n}=65.4$. Then $S = 65.4n$.

After a new student of height $x = 73$ joins, the new number of students is $n + 1$, and the new sum of heights is $S+x=65.4n + 73$.

The new mean $\bar{x}_{new}=\frac{65.4n+73}{n + 1}=\frac{65.4n+65.4+7.6}{n + 1}=\frac{65.4(n + 1)+7.6}{n + 1}=65.4+\frac{7.6}{n + 1}$

Since $n\geq1$ (because there was a non - empty group of students originally), $\frac{7.6}{n + 1}>0$.

Answer:

B. The new mean height will be greater than 65.4 inches.