Question

Procedures for constructing a confidence interval for a sample mean are given in section 7-2 on page 319. Example 2 worked on pages 320 and 321 can guide us.

In my homework problem 40, the scenario is data on the salaries of 61 players on a football team. We are given that we are interested in the 95% confidence level and the population standard deviation is 3723 thousand dollars. From section 7-2, we know that to be able to use either the normal or the t​ distribution, either the sample must come from a normally distributed population or the sample size must be greater than 30. In this case, the histogram provided does not seem to have a normal distribution, but the sample size is well above 30.

When the population standard​ deviation is​ known, we use the normal distribution, and thus use z_α/2. For 95% confidence level, α = 0.05 and

z_α/2 = z_0.05/2 = z_0.025 = 1.96

Please explain how you would then proceed to determine the 95% confidence interval.

Answer 1

Using normal

To use it we need to have a normal sample and have known population standard deviation. Then using the following formula we can get the confidence interval

(1 - $\alpha$) confidence interval for population mean

$(\bar{x}\bar{+}Z_{\alpha/2}\frac{\sigma}{\sqrt{n}})$

Where $\bar{x}$ = Sample Mean

$Z_{\alpha/2}$ = Critical value (found using normal percentage tables or excel function 'normsinv'

$\sigma$ = Population standard deviation

$n$ = Sample size

Margin of error = $Z_{\alpha/2}$ * $\frac{\sigma}{\sqrt{n}}$

Using t-dist

We use t-dist to approximate the confidence level when we have no population standard deviation and the sample size > 30 (large).

(1 - $\alpha$) confidence interval for population mean

$(\bar{x}\bar{+}t_{\alpha/2,n-1}\frac{S_{x}}{\sqrt{n}})$

Where $\bar{x}$ = Sample Mean

$t_{\alpha/2,n-1}$ = Critical value (found using t-dist tables or excel function 'tinv')

  $S_{x}$= Sample standard deviation

  $n$ = Sample size

Margin of error = $t_{\alpha/2,n-1}$ * $\frac{S_{x}}{\sqrt{n}}$

We use confidence interval to find a range for the population parameter with (1 - $\alpha$) certainty that the range will contain the parameter.