- 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.
When estimating a population mean, you usually start with data from a sample from the population of interest. To estimate the population mean, you start by calculating the sample mean. The sample mean can then be used to construct a confidence interval for the population mean in the same way that a sample proportion is used to construct a confidence interval estimate of a population proportion.
The form of a confidence interval is estimate [latex]\pm[/latex] margin of error, where the margin of error was calculated by multiplying the standard error of the estimate by a [latex]z[/latex]-critical value corresponding to the desired confidence level.
In a previous section, we constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions or conditions were met.
Let’s look at the sampling distribution of the sample mean to help us construct a confidence interval to estimate a population mean when conditions are met and interpret the confidence interval in context.
Sampling Distribution of the Sample Mean
When taking many random samples of size [latex]n[/latex] from a population distribution with mean [latex]\mu[/latex] and sample standard deviation [latex]s[/latex]:
- The mean of the distribution of the sample means is [latex]\mu[/latex].
- The standard error of the distribution of the sample means is [latex]\dfrac{s}{\sqrt{n}}[/latex].
- If the population distribution is normal or if the sample size is large ([latex]n \ge 30[/latex]), the distribution of the sample means follows an approximate normal distribution.