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 [latex]\displaystyle\overline{{x}}={10}[/latex] and we have constructed the 90% confidence interval (5, 15) where the margin of error (EBM) is equal to 5.

Calculating the Confidence Interval

To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need the sample mean [latex]\overline{{x}}[/latex] as the point estimate of the unknown population mean μ.

The confidence interval estimate will have the form:

(point estimate – error bound, point estimate + error bound) or, in symbols,
[latex]\displaystyle{(\overline{{x}}-{E}{B}{M},\overline{{x}}+{E}{B}{M})}[/latex]

or  [latex]\displaystyle{\overline{{x}}-{E}{B}{M} \le \mu \le \overline{{x}}+{E}{B}{M}}[/latex]

The margin of error (EBM) depends on the confidence level (abbreviated CL). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.

There is another probability called alpha (α). α is related to the confidence level, CL. α is the probability that the interval does not contain the unknown population parameter.

Mathematically, α + CL = 1 or 1 – α  = CL

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 [latex]\displaystyle\overline{{x}}={10}[/latex], and we have constructed the 90% confidence interval (5, 15) where 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 α = 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. The value 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[latex]\displaystyle\frac{{\sigma}}{{\sqrt{n}}}[/latex]. The fraction [latex]\displaystyle\frac{{\sigma}}{{\sqrt{n}}}[/latex], is commonly called the “standard error of the mean” in order to distinguish clearly the standard deviation for a mean from the population standard deviation σ.

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

• [latex]\displaystyle\overline{X}[/latex], that is, [latex]\displaystyle\overline{X}{\sim}{N}\left({\mu}_{x}, \frac{{\sigma}}{{\sqrt{n}}}\right)[/latex]
• When the population standard deviation σ is known, we use a normal distribution to calculate the error bound.

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:

• Calculate the sample mean [latex]\displaystyle\overline{{x}}[/latex] from the sample data. Remember, in this section we already know the population standard deviation σ.
• 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 the z-score 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, CL, is the area in the middle of the standard normal distribution. CL = 1 – α, so α is the area that is split equally between the two tails. Each of the tails contains an area equal to α divide by 2 (this is denoted α2).

The z-score that has an area to the right of α2 is denoted by zα2.

For example, when CL = 0.95, α = 0.05 and α2 = 0.025; we write zα2=z0.025.

The area to the right of z_0.025 is 0.025 and the area to the left of z_0.025 is 1 – 0.025 = 0.975.

zα2=z0.025=1.96, using a calculator, computer or a standard normal probability table.

NORM.INV(0.975, 0, 1) or invNorm(0.975, 0, 1) = 1.96

 Note

Remember to use the area to the LEFT of the zα2 you are looking for [CL + (1 – CL)/2]; in this chapter the inputs for the mean and standard deviation are 0 and 1, because you are using a standard normal distribution Z ~ N(0, 1).

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Calculating the Error Bound (EBM)

The error bound formula for an unknown population mean μ when the population standard deviation σ is known is

• EBM = ([latex]\displaystyle{z}_{\frac{{\alpha}}{{2}}})(\frac{{\sigma}}{{\sqrt{n}}})[/latex]

In Excel, the function CONFIDENCE.NORM can be used to find the EBM.  The syntax is CONFIDENCE.NORM(alpha, standard_dev, size).

Constructing the Confidence Interval

• The confidence interval estimate has the format ([latex]\displaystyle\overline{x}[/latex]–EBM,  [latex]\displaystyle\overline{x}[/latex]+EBM).

The graph gives a picture of the entire situation.

CL + [latex]\displaystyle\frac{{\alpha}}{{2}}+\frac{{\alpha}}{{2}}={\text{CL}}+{\alpha}=1[/latex]

Writing the Interpretation

The interpretation should clearly state the confidence level ( CL), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). “We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).”

Before creating confidence intervals, be sure the requirements for the test are met.

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 mean 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.

Notice the difference in the confidence intervals calculated in the Example and the Try It just completed. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.

Changing the Confidence Level or Sample Size

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

Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean 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. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be 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.

 

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Working Backwards to Find the Error Bound or Sample Mean

When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, 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.

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 two.

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.

 

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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 = [latex]\displaystyle{z}_{\frac{{\alpha}}{{\sqrt{n}}}}[/latex]

The formula for sample size is found by solving the error bound formula for n.

n = [latex]\displaystyle\frac{{{z}^{2}{\sigma}^{2}}}{{{EBM}^{2}}}[/latex]

In this formula, z is [latex]z_{\frac{\alpha}{2}}[/latex], 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.

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References

“American Fact Finder.” U.S. Census Bureau. Available online at http://factfinder2.census.gov/faces/nav/jsf/pages/searchresults.xhtml?refresh=t (accessed July 2, 2013).

“Disclosure Data Catalog: Candidate Summary Report 2012.” U.S. Federal Election Commission. Available online at http://www.fec.gov/data/index.jsp (accessed July 2, 2013).

“Headcount Enrollment Trends by Student Demographics Ten-Year Fall Trends to Most Recently Completed Fall.” Foothill De Anza Community College District. Available online at http://research.fhda.edu/factbook/FH_Demo_Trends/FoothillDemographicTrends.htm (accessed September 30,2013).

Kuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. “2000 CDC Growth Charts for the United States: Methods and Development.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/growthcharts/2000growthchart-us.pdf (accessed July 2, 2013).

La, Lynn, Kent German. “Cell Phone Radiation Levels.” c|net part of CBX Interactive Inc. Available online at http://reviews.cnet.com/cell-phone-radiation-levels/ (accessed July 2, 2013).

“Mean Income in the Past 12 Months (in 2011 Inflaction-Adjusted Dollars): 2011 American Community Survey 1-Year Estimates.” American Fact Finder, U.S. Census Bureau. Available online at http://factfinder2.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_11_1YR_S1902&prodType=table (accessed July 2, 2013).

“Metadata Description of Candidate Summary File.” U.S. Federal Election Commission. Available online at http://www.fec.gov/finance/disclosure/metadata/metadataforcandidatesummary.shtml (accessed July 2, 2013).

“National Health and Nutrition Examination Survey.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/nchs/nhanes.htm (accessed July 2, 2013).

Concept Review

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean ( EBM). A confidence interval has the general form:

(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)

The calculation of EBM depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding EBM increases as well. As the sample size increases, the EBM decreases.

Given a confidence interval, you can work backwards to find the error bound ( EBM) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.

Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the EBM formula for n to discover the size of the sample that is needed to achieve this goal:

Formula Review

[latex]\displaystyle\overline{X}{\sim}{N}\left({\mu}_{x}, \frac{{\sigma}}{{\sqrt{n}}}\right)[/latex]. The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by

(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)

=([latex]\displaystyle\overline{x}[/latex] – EBM, [latex]\displaystyle\overline{x}[/latex]+EBM)

EBM = ([latex]\displaystyle{z}_{\frac{{\alpha}}{{2}}})(\frac{{\sigma}}{{\sqrt{n}}})[/latex] = the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.

CL = confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter

α = 1 – CL = the proportion of confidence intervals that will not contain the population

[latex]\displaystyle{z}_{\frac{{\alpha}}{{\sqrt{n}}}}[/latex]= the z-score with the property that the area to the right of the z-score is  ∝2 this is the z-score used in the calculation of EBM where α = 1 – CL.

n = [latex]\displaystyle\frac{{{z}^{2}{\sigma}^{2}}}{{{EBM}^{2}}}[/latex] = the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence

General form of a confidence interval: (lower value, upper value) = (point estimate−error bound, point estimate + error bound)

To find the error bound when you know the confidence interval

error bound = upper value−point estimate OR error bound =[latex]\displaystyle\frac{{{upper}{value}-{lower}{value}}}{{2}}[/latex]