Q 8.112. - Elementary Statistics

Question

Batting Averages. An issue of Scientific American revealed that batting averages, $x$ of major-league baseball players are normally

distributed and have a mean of $0.270$ and a standard deviation of $0.031$ For samples of $20$ batting averages, identify the distribution of each variable.

a. $\frac{\overline{x}-0.270}{0.031/\sqrt{20}}$

b. $\frac{\overline{x}-0.270}{s/\sqrt{20}}$

Step-by-Step Solution

Answer

Part (a) Follows the conventional normal distribution, which has a mean of $0$ and a standard deviation of $1$

Part (b) With $(n-1)$ degrees of freedom, it follows the $t-$distribution.

Part (a) Step 1: Given information

Batting average, $x$ of major - league baseball players are normally distributed and have a mean $\mu =0.270$ and S.D $\sigma =0.031$

Sample size $n=20$

Part (a) Step 2: Concept

The formula used: $z=\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$

Part (a) Step 3: Calculation

$\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}=\frac{\overline{x}-0.270}{\frac{0.031}{\sqrt{20}}}$ Follows standard normal distribution i.e. normal distribution with mean $0$ and S.D $1$

Part (b) Step 1: Calculation

$\frac{\frac{\overline{x}-\mu }{s}}{\frac{s}{\sqrt{n}}}=\frac{\overline{x}-0.270}{\frac{0.031}{\sqrt{20}}}$ Follows $t$-distribution with $(n-1)$ degrees of freedom.