8.1 Confidence intervals for a population mean, population standard  (Page 2/7)

It is important that the "standard deviation" used must be appropriate for the parameter we are estimating. So in this section, we need to use the standard deviation that applies to sample means, which is $\frac{\sigma }{\sqrt{n}}$ . $\frac{\sigma }{\sqrt{n}}$ is commonly called the "standard error of the mean" in order to clearly distinguish the standard deviation for a mean from the population standard deviation $\sigma$ .

In summary, as a result of the central limit theorem:

• $\overline{X}$ is normally distributed, that is, $\overline{X}$ ~ $\mathrm{N}({\mu }_X,\frac{\sigma }{\sqrt{n}})$
• When the population standard deviation $\sigma$ is known, we use a Normal distribution to calculate the error bound.

Calculating the confidence interval:

• Calculate the sample mean $\overline{X}$ from the sample data. Remember, in this section, we already know the population standard deviation $\sigma$ .
• Find the Z-score that corresponds to the confidence level.
• Calculate the error bound EBM
• Construct the confidence interval
• Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)

We will first examine each step in more detail, and then illustrate the process with some examples.

Finding z for the stated confidence level

The confidence level, $\mathrm{CL}$ , is he area in the middle of the standard normal distribution. $\mathrm{CL}=1-\alpha$ . So $\alpha$ is the area that is split equally between the two tails. Each of the tails contains an area equal to $\frac{\alpha }{2}$ .

The z-score that has an area to the right of $\frac{\mathrm{\alpha }}{2}$ is denoted by $z_{\frac{\alpha }{2}}$

For example, when $\mathrm{CL}=0.95$ then $\alpha =0.05$ and $\frac{\alpha }{2}=0.025$ ; we write $z_{\frac{\alpha }{2}}=z_{.025}$

The area to the right of $z_{.025}$ is 0.025 and the area to the left of $z_{.025}$ is 1-0.025 = 0.975

$z_{\frac{\alpha }{2}}=z_{0.025}=1.96$ , using a calculator, computer or a Standard Normal probability table.

Using the TI83, TI83+ or TI84+ calculator: invNorm $(.975,0,1)=1.96$

CALCULATOR NOTE: Remember to use area to the LEFT of $z_{\frac{\alpha }{2}}$ ; in this chapter the last two inputs in the invnorm command are 0,1 because you are using a Standard Normal Distribution Z~N(0,1)

Ebm: error bound

• $\mathrm{EBM}=z_{\frac{\alpha }{2}}\cdot \frac{\sigma }{\sqrt{n}}$

Constructing the confidence interval

• The confidence interval estimate has the format $(\overline{x}-\mathrm{EBM},\overline{x}+\mathrm{EBM})$ .

The graph gives a picture of the entire situation.

$\mathrm{CL}+\frac{\alpha }{2}+\frac{\alpha }{2}=\mathrm{CL}+\alpha =1$ .

Writing the interpretation