Question

a) compute the mean and standard deviation of the distribution.
the mean is 8.
the standard deviation is 4.85.
(round to the nearest hundredth as needed.)
b) multiply each number in the distribution by 2 and compute the mean and the standard deviation of this new distribution.
the mean is

Explanation:

Step1: Recall the property of mean and standard deviation for scaling

If \(x_1,x_2,\cdots,x_n\) is a data - set with mean \(\bar{x}\) and standard deviation \(s\), and \(y_i = ax_i\) (where \(a\) is a constant), then the mean of the new data - set \(y_1,y_2,\cdots,y_n\) is \(\bar{y}=a\bar{x}\) and the standard deviation is \(s_y = a s\).

Step2: Identify the values of \(a\), \(\bar{x}\) and \(s\)

Here, \(a = 2\), \(\bar{x}=8\), and \(s = 4.85\).
For the mean of the new distribution:
Using the formula \(\bar{y}=a\bar{x}\), substitute \(a = 2\) and \(\bar{x}=8\). So \(\bar{y}=2\times8=16\).
For the standard deviation of the new distribution:
Using the formula \(s_y=a s\), substitute \(a = 2\) and \(s = 4.85\). So \(s_y=2\times4.85 = 9.7\).

Answer:

The mean of the new distribution is \(16\) and the standard deviation is \(9.7\).