Calculating the Confidence Interval

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 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 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.

There is another probability called alpha ( αα). αα is related to the confidence level CL. αα is the probability that the sample produced a point estimate that is not within the appropriate margin of error of the unknown population parameter.

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 that our sample has a mean of x¯ = 10 x =10 and 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 leave out 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 n σ n . σ n σ 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.

Calculating 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).

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

The z-score that has an area to the right of α2 α2 is denoted by z α 2 z α 2

For example, when CL = 0.95CL=0.95 then α = 0.05α=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

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). "We estimate with ___% confidence that the true population average (include context of the problem) is between ___ and ___ (include appopriate units)."

Changing the Confidence Level or Sample Size

Example 3: Changing the Confidence Level

Problem 1

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.

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.

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.

Figure 1

(a) (b)

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

When we calculate a confidence interval, we find the sample mean and calculate the error bound and use them to calculate the confidence interval. But sometimes when we read statistical studies, the study may state the confidence interval only. 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.

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 EBM = z α 2 ⋅ ( σ n ) EBM= z α 2 ⋅( σ n )

The formula for sample size is n=z2⁢σ2EBM2 n z 2 σ 2 EBM 2 , found by solving the error bound formula for n n

In this formula, z z is z α 2 z α 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.