Question

use the data 0, 2, 3, 5, 5, 7. add 3 to each of the numbers. how does this affect the mean? how does this affect the standard deviation? choose the correct answer below. o a. the mean will be increased by 3, and the standard deviation will remain the same. o b. both of the mean and the standard deviation will be increased. o c. both of the mean and the standard deviation will remain the same. o d. the standard deviation will be increased by 3, and the mean will remain the same.

Explanation:

Step1: Analyze the effect on the mean

Let the original data be \( x_1, x_2, \dots, x_n \), and the mean be \( \bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n} \). After adding 3 to each data point, the new data is \( y_i=x_i + 3 \), and the new mean \( \bar{y}=\frac{\sum_{i = 1}^{n}(x_i + 3)}{n}=\frac{\sum_{i = 1}^{n}x_i+3n}{n}=\bar{x}+3 \). So the mean increases by 3.

Step2: Analyze the effect on the standard deviation

The standard deviation \( s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}} \). For the new data, \( y_i=x_i + 3 \), and the new mean \( \bar{y}=\bar{x}+3 \). Then \( y_i-\bar{y}=(x_i + 3)-(\bar{x}+3)=x_i-\bar{x} \). So the standard deviation \( s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n}}=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}} = s \). So the standard deviation remains the same.

Answer:

A. The mean will be increased by 3, and the standard deviation will remain the same.