Question

the age of players (in years) in mr. james soccer club are shown below.
27, 26, 25, 27, 27, 25, 26, 26, 25, 27
a new player, who is 17 years old, joins the soccer club. what effect does the new players age have on the center and spread of the soccer clubs age distribution?
○a. the value of the center decreases and the distribution has a larger spread.
○b. the value of the center increases and the distribution has a smaller spread.
○c. the value of the center increases and the distribution has a larger spread.
○d. the value of the center decreases and the distribution has a smaller spread.

Explanation:

Step1: Analyze the center (mean)

The original ages are \(25,26,25,27,27,25,26,26,25,27\).
The sum of original ages \(S_1=(25\times4)+(26\times3)+(27\times3)=100 + 78+81 = 259\).
The number of original players \(n_1 = 10\).
The original mean \(\bar{x}_1=\frac{S_1}{n_1}=\frac{259}{10}=25.9\).
After adding a \(17 -\)year - old player, the new sum \(S_2=259 + 17=276\).
The new number of players \(n_2=11\).
The new mean \(\bar{x}_2=\frac{S_2}{n_2}=\frac{276}{11}\approx25.09\).
Since \(25.09<25.9\), the center (mean) decreases.

Step2: Analyze the spread (range)

The original range: The maximum value in the original data is \(27\) and the minimum value is \(25\), so the original range \(R_1=27 - 25=2\).
After adding \(17\), the new maximum is \(27\) and the new minimum is \(17\), so the new range \(R_2=27 - 17 = 10\).
Since \(10>2\), the spread (range) increases.

Answer:

A. The value of the center decreases and the distribution has a larger spread.