Known-Standard-Deviation Confidence Intervals for Women's Mean Height

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What is the purpose of a confidence interval for women's heights?

It uses sample information to estimate the population mean height, $\mu$, by giving a range of plausible values for that mean.

What is the formula for a z-confidence interval for a mean when $\sigma$ is known?

The interval is $\bar{x} \pm z^*\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z^*$ is the critical value, $\sigma$ is the population standard deviation, and $n$ is the sample size.

What does the 90% confidence level describe?

It describes the long-run success rate of the procedure: across repeated random samples, about 90% of intervals constructed this way contain the fixed population mean.

Why is the sample standard deviation not used for the margin of error in this activity?

The activity states that the population standard deviation is known to be 3.3 inches, so the standard error uses $\sigma=3.3$ rather than the sample standard deviation $s$.

Does a 90% confidence interval contain 90% of the individual heights?

No. It estimates the population mean. The percentage of individual heights within its endpoints is a different quantity and is not what the confidence level describes.

Quiz questions

1. What does the sample mean represent in the confidence interval formula?Answer: It is the center of the interval and an estimate of the population mean.
2. Which standard deviation should be used to calculate the standard error in this activity?Answer: The known population standard deviation, $\sigma=3.3$ inches.
3. Approximately what margin of error is used for a 90% z-interval with $n=10$ and $\sigma=3.3$?Answer: About 1.72 inches.
4. What does a 90% confidence level mean when the sampling procedure is repeated many times?Answer: About 90% of intervals constructed by the method will contain the fixed population mean.
5. Why is counting how many of the 100 heights fall within one confidence interval not a check of its 90% confidence level?Answer: The interval estimates the population mean, not the proportion of individual heights that lie within its endpoints.