Question

A random sample of 25 professional basketball players shows a mean height of 6 feet, 5 inches with a 95% confidence interval of 0.4 inches. Explain what this indicates. If the sample were smaller, would the confidence interval become smaller or larger? Explain. If you wanted a higher level of confidence (99%) would the confidence interval become smaller or larger? Explain.

Answer 1

The confidence interval indicates that we are 95% sure that the mean height is within 0.4 inches of the sample mean of 6 feet and 5 inches.

For a smaller sample size, the confidence interval would become larger.

With a higher level of confidence, the confidence interval would become larger.

What is a z-distribution confidence interval?

The confidence interval is:

$\overline{x} \pm z\frac{\sigma}{\sqrt{n}}$

The margin of error is given by:

$M = z\frac{\sigma}{\sqrt{n}}$

In which:

• $\overline{x}$ is the sample mean.

• z is the critical value.

• n is the sample size.

• $\sigma$ is the standard deviation for the population.

The interpretation of an interval is that we are x% sure that the population mean is in that interval, in which x% is the confidence interval. Hence, considering the margin of error and the sample mean for this problem, the confidence interval indicates that we are 95% sure that the mean height is within 0.4 inches of the sample mean of 6 feet and 5 inches.

The margin of error equation means that:

• A decrease in n leads to a higher margin of error, hence a larger confidence interval.

• An increase in the confidence level leads to a higher value of z, so there is a higher margin of error, hence a larger confidence interval.

More can be learned about confidence intervals at https://brainly.com/question/25890103

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