Known-Standard-Deviation Confidence Intervals for One Mean

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What is a population mean?

The average value of a variable across an entire population, denoted by $\mu$.

What is the sample mean?

The average calculated from a sample, denoted by $\bar{x}$; it serves as a point estimate of the population mean.

What does EBM mean?

Error bound for a population mean, also called the margin of error. It is the distance from the sample mean to either endpoint of a symmetric confidence interval.

What is the standard error when $\sigma$ is known?

It is $\sigma/\sqrt{n}$, the standard deviation of the sampling distribution of the sample mean.

How are confidence level and alpha related?

They satisfy $CL=1-\alpha$. Alpha is the total area outside a two-sided confidence interval.

Quiz questions

1. For a two-sided 95% confidence interval using the standard normal distribution, what is the positive critical value approximately?Answer: 1.96
2. When the population standard deviation is known, which formula gives the error bound for a two-sided confidence interval for a mean?Answer: $z_{\alpha/2}\sigma/\sqrt{n}$
3. A study uses $z_{\alpha/2}=1.96$, known population standard deviation $\sigma=15$, and target error bound $EBM=2$. What minimum sample size meets the target?Answer: 217
4. A symmetric confidence interval has endpoints 14 and 22. What are its midpoint and half-width?Answer: Midpoint 18; half-width 4.
5. If all other factors are held constant, what happens to the width of a confidence interval when its confidence level increases?Answer: The interval becomes wider.