Confidence Interval for a Population Mean: Apply It 1

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Confidence Interval for a Population Mean: Apply It 1

Check the assumptions for a one-sample [latex]t[/latex] confidence interval for population mean.
Calculate a confidence interval for a population mean and explain what it means.

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There are many situations where you might be interested in estimating a population mean.

For example, you might be interested in collecting data from a random sample of students who graduated from a two-year college in 2020 to learn about their student loans. If you asked each student in the sample the amount of their student loan debt, you could then use the data to estimate the mean student loan debt for two-year college graduates.

The formula for a confidence interval for a population mean is:

[latex]\text{estimate }\pm \text{ margin of error}[/latex]

[latex]\bar{x} \pm (t\text{-critical value})\frac{s}{\sqrt{n}}[/latex]

where [latex]\bar{x}[/latex] is the sample mean and the standard error used is the standard error of the sample mean, [latex]\frac{s}{\sqrt{n}}[/latex].

The [latex]t[/latex]-critical value in the confidence interval will depend on the sample size (degrees of freedom for the [latex]t[/latex]-distribution: [latex]df=n-1[/latex]) and the confidence level.

Before creating a confidence interval:

Use the [latex]t[/latex]-model if [latex]σ[/latex] (the population standard deviation) is unknown. If [latex]σ[/latex] is known, then use the normal ([latex]z[/latex]) model instead of the [latex]t[/latex]-model.
Use the [latex]t[/latex]-model if variable values are normally distributed in the population. If this is not true, then make sure the sample size is large (more than [latex]30[/latex]).
The sample is random. The sample is independent. The sample size is large enough to assume an approximately normal sampling distribution.

This interval is often called a one-sample [latex]t[/latex] interval.

While you will use technology to do the calculations, you can see from this confidence interval formula that you are just taking the sample mean and forming an interval around it by subtracting and adding the margin of error to get an interval of plausible values for the population mean.

Normally, we need to check the random condition each time we complete an inference procedure like creating a confidence interval. However, in these examples, we are given that the large samples are “representative” of their respective populations, so we are making the assumption that the random condition does not need to be met.