2. A magazine reports that the average bowling score for league bowlers in the United States is 157 with a standard deviation of 12, and that the scores are approximately normally distributed. State the shape, the mean, and the standard deviation of the mean x of 15 bowlers selected at random. Justify each statement by showing your work or identifying the theorem that tells you your answer is correct.

The shape is going to be bell-shaped(normally distributed), with mean of 157 and standard deviation of [tex]\frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{15}} = 3.1[/tex]

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].

So, in this problem:

The shape is going to be bell-shaped(normally distributed), with mean of 157 and standard deviation of [tex]\frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{15}} = 3.1[/tex]