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.

For a smaller sample size, the margin of error would increase, as we can see from it's equation that it is inversely proportional to the square root of n, hence the confidence interval would become larger.

With a higher level of confidence, the margin of error would increase, as the value of z would increase and so would the margin of error, as we can see from the margin of error equation, hence the confidence interval would become larger.

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

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