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.

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 [latex]\overline{{x}}[/latex] as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean [latex]\overline{{x}}[/latex] is 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]

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 their 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.

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] is normally distributed, 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, we already know the population standard deviation σ in this section.
• 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 [latex]\dfrac{α}{2}[/latex].

The z-score that has an area to the right of [latex]\dfrac{α}{2}[/latex] is denoted by [latex]z_\dfrac{α}{2}[/latex].

For example, when CL = 0.95, α = 0.05 and [latex]\dfrac{α}{2}[/latex] = 0.025; we write [latex]z_\dfrac{α}{2}[/latex] = z_0.025.

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.

[latex]z_\dfrac{α}{2}[/latex] = z_0.025 = 1.96, using a calculator, computer or a standard normal probability table.

invNorm(0.975, 0, 1) = 1.96

Note

Remember to use the area to the LEFT of [latex]z_\dfrac{α}{2}[/latex]; 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).

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]

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).”

Notice the difference in the confidence intervals calculated in Example 3 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.