To construct a confidence interval for a single unknown population mean μμ , where the population standard deviation is known, we need x¯ x as an estimate for μμ and we need the margin of error. Here, the margin of error is called the error bound for a population mean (abbreviated EBM). The sample mean x¯ x is called the point estimate of the unknown population mean μμ

The margin of error depends on the confidence level (abbreviated CL). The confidence level is the probability that the confidence interval estimate that we will calculate will contain the true value of the population parameter. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because he wants to be reasonably certain of his conclusions.

A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose we want to estimate the average time that ALL vehicles wait in the traffic at the toll booth at a certain bridge. Our data for a random sample of vehicles has a sample mean of x¯ = 10 x =10 minutes. Suppose we have constructed the 90% confidence interval (5, 15) where EBM = 5 EBM=5.

To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we have a total of 10% in both tails, or 5% in each tail, of the normal distribution.

To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. 1.645 is the z-score from a Standard Normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.

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 σ X n σ X n . σ X n σ X 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 σ σ.

To find the Confidence Interval:

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:

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

Finding Z for the stated Confidence Level

When we know the population standard deviation σ, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of Z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z~N(0,1). We learned how to do this in chapter 6:

The confidence level, CLCL, is CL = 1 - α CL=1-α. 1 - α1-α is the area in the middle of the standard normal distribution. Therefore α=1 - CLα=1-CL is the area that is split equally between the two tails. Each of the tails contains an area equal to α 2 α 2 .

z α 2 z α 2 is the symbol for the z-score that has an area of α2 α2 to the right and therefore an area of 1 - α2 1-α2 to the left.

For example, if CL = 0.95CL=0.95 then 1 - α= 0.951-α=0.95 , so α = 1 - 0.95 = 0.05α=1-0.95=0.05 and α 2 = 0.025 α 2=0.025 ; we write z α 2 = z .025 z α 2 = z .025

The area to the right of z .025 z .025 is 0.025 and the area to the left of z .025 z .025 is 1-0.025 = 0.975

We find the value z α 2 = z 0.025 = 1.96 z α 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(.975,0,1)=1.96

CALCULATOR NOTE: Remember to use area to the LEFT of z α 2 z α 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

The error bound formula for an unknown population mean μμ when the population standard deviation σσ is known is

The graph gives a picture of the entire situation.

CL + α 2 + α 2 = CL + α = 1 CL+ α 2 + α 2 =CL+α=1.

Writing the Interpretation

The interpretation should clearly state the confidence level (CL), explain what population parameter is being estimated (here, a population mean or average ), and should state the confidence interval (both endpoints). For example: We estimate with 90% confidence that the true average time that all vehicles wait in traffic at the toll booth at this bridge is between 5 and 15 minutes.

Example 2

Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A random sample of 36 scores is taken and gives a sample mean (sample average score) of 68. Find a confidence interval estimate for the population mean exam score (the average score on all exams).

Problem 1

Find a 90% confidence interval for the true (population) mean of statistics exam scores.

• The solution is shown step-by-step.

Solution

To find the confidence interval, you need the sample mean, x¯ x , and the EBM.

• x¯ = 68 x =68
• EBM = z α 2 ⋅ ( σ n ) EBM= z α 2 ⋅( σ n )
• σ = 3 σ=3 ; n = 36 n=36 ; The confidence level is 90% (CL=0.90)

CL = 0.90 CL = 0.90 so α = 1 - CL = 1 - 0.90 = 0.10 α=1-CL=1-0.90=0.10

α 2 = 0.05 z α 2 = z .05 α 2 =0.05 z α 2 = z .05

The area to the right of z.05 z.05 is 0.05 and the area to the left of z.05 z.05 is 1−0.05=0.95

z α 2 = z .05 = 1.645 z α 2 = z .05 =1.645

using invnorm(.95,0,1) 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.

EBM = 1.645 ⋅ ( 3 36 ) = 0.8225 EBM=1.645⋅( 3 36 )=0.8225

x¯ - EBM = 68 - 0.8225 = 67.1775 x -EBM=68-0.8225=67.1775

x¯ + EBM = 68 + 0.8225 = 68.8225 x +EBM=68+0.8225=68.8225

The 90% confidence interval is (67.1775, 68.8225).

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Interpretation

We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82.

Explanation of 90% Confidence Level

Ninety percent of all confidence intervals constructed in this way contain the true average statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.

Problem 2

Changing the Confidence Level

Now let's see how this changes if we increase the confidence level from 90% ro 95%. For the same problem, find a 95% confidence interval for the true (population) mean of statistics exam scores.

Solution

To find the confidence interval, you need the sample mean, x¯ x , and the EBM.

• x¯ = 68 x =68
• EBM = z α 2 ⋅ ( σ n ) EBM= z α 2 ⋅( σ n )
• σ = 3 σ=3 ; n = 36 n=36 ; The confidence level is 95% (CL=0.95)

CL = 0.95 CL = 0.95 so α = 1 - CL = 1 - 0.95 = 0.05 α=1-CL=1-0.95=0.05

α 2 = 0.025 z α 2 = z .025 α 2 =0.025 z α 2 = z .025

The area to the right of z.025 z.025 is 0.025 and the area to the left of z.025 z.025 is 1−0.025=0.975

z α 2 = z .025 = 1.96 z α 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.

EBM = 1.96 ⋅ ( 3 36 ) = 0.98 EBM=1.96⋅( 3 36 )=0.98

x¯ - EBM = 68 - 0.98 = 67.02 x -EBM=68-0.98=67.02

x¯ + EBM = 68 + 0.98 = 68.98 x +EBM=68+0.98=68.98

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Interpretation

We estimate with 95 % confidence that the true population average 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 average statistics exam score.

Effect of Increasing the Confidence Level

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.

Figure 1

(a) (b)

In general, increasing the confidence level increases the error bound. Decreasing the confidence level decreases the error bound.

Working Backwards to find the Error Bound or the Sample Mean

When we do the calculations to find a confidence interval, we find the sample mean and calculate the error bound . We then use the sample mean and error bound to calculate the confidence interval. But sometimes when we read statistical studies, the study may state the confidence interval only, and not state the value of the error bound. A statistical study will usually state the value of the sample mean, but sometimes may not, giving only the confidence interval. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.

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.

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This work is licensed by Roberta Bloom under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Roberta Bloom on Dec 16, 2008 12:53 am US/Central.