NCAA Football Coach Salaries A simple random sample of 40 salaries of NCAA football coaches has a mean of \$ 415,953. Assume that $\sigma=$ \$463,364.
Find the best point estimate of the mean salary of all NCAA football coaches.
Construct a 95% confidence interval estimate of the mean salary of an NCAA football coach.
Solution
Given that sample size $n = 40$, sample mean $\overline{X}= 415953$ and population standard deviation $\sigma = 463364$.
The best point estimate of the mean salary of all NCAA football coaches is $\overline{X} = 415953$.
Step 1 Specify the confidence level $(1-\alpha)$
Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.
Step 2 Given information
Given that sample size $n =40$, sample mean $\overline{X}=415953$ and population standard deviation is $\sigma = 463364$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval for the population mean $\mu$ is
$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$
where $E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$, and $Z_{\alpha/2}$ is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.
Step 4 Determine the critical value
The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.
Thus $Z_{\alpha/2} = Z_{0.025} = 1.96$.
Step 5 Compute the margin of error
The margin of error for mean is
$$ \begin{aligned} E & = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\\ & = 1.96 \frac{463364}{\sqrt{40}} \\ & = 143597.991. \end{aligned} $$
Step 6 Determine the confidence interval
$95$ % confidence interval estimate for population mean is
$$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 415953 - 143597.991 & \leq \mu \leq 415953 + 143597.991\\ 272355.009 & \leq \mu \leq 559550.991. \end{aligned} $$
Thus, $95$% confidence interval for population mean is $(272355.009,559550.991)$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
