4.1 Confidence intervals: confidence interval, single population  (Page 2/5)

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 the 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 $(0.975,0,1)=1.96$

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

Changing the confidence level or sample size

Changing the confidence level

Suppose we change the original problem by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.

To find the confidence interval, you need the sample mean, $\overline{x}$ , and the EBM.

• $\overline{x}=68$
• $\mathrm{EBM}=z_{\frac{\alpha }{2}}\cdot \left(\frac{\sigma }{\sqrt{n}}\right)$
• $\sigma =3$ ; $n=36$ ; The confidence level is 95% (CL=0.95)

$\mathrm{CL\; =\; 0.95}$ so $\alpha =1-\mathrm{CL}=1-0.95=0.05$

$\frac{\alpha }{2}=0.025z_{\frac{\alpha }{2}}=z_{.025}$

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

$z_{\frac{\alpha }{2}}=z_{.025}=1.96$

using invnorm(.975,0,1) on the TI-83,83+,84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the Standard Normal distribution.)

$\mathrm{EBM}=1.96\cdot \left(\frac{3}{\sqrt{36}}\right)=0.98$

$\overline{x}-\mathrm{EBM}=68-0.98=67.02$

$\overline{x}+\mathrm{EBM}=68+0.98=68.98$

Interpretation

We estimate with 95 % confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.

Explanation of 95% confidence level

95% of all confidence intervals constructed in this way contain the true value of the population meanstatistics exam score.

Comparing the results

The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider.

Summary: effect of changing the confidence level

• Increasing the confidence level increases the error bound, making the confidence interval wider.
• Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

Working backwards to find the error bound or sample mean

Working bacwards to find the error bound or the sample mean

Finding the error bound

• From the upper value for the interval, subtract the sample mean
• OR, From the upper value for the interval, subtract the lower value. Then divide the difference by 2.

Finding the sample mean

• Subtract the error bound from the upper value of the confidence interval
• OR, Average the upper and lower endpoints of the confidence interval

Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.

Calculating the sample size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The error bound formula for a population mean when the population standard deviation is known is $\mathrm{EBM}=z_{\frac{\alpha }{2}}\cdot \left(\frac{\sigma }{\sqrt{n}}\right)$

The formula for sample size is n=\frac{z^2\sigma ^2}{\mathrm{EBM}^2} , found by solving the error bound formula for $n$

In this formula, $z$ is $z_{\frac{\alpha }{2}}$ , corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

**With contributions from Roberta Bloom