Question

The weights of running shoes are normally distributed with a population standard deviation of 3 ounces and an unknown population mean. If a random sample of 23 running shoes is taken and results in a sample mean of 18 ounces, find a 90% confidence interval for the population mean.

Answer 1

A 90% confidence interval for the population mean weight of running shoes, with a sample mean of 18 ounces, and population standard deviation of 3 ounces, from a sample size of 23, is approximately (16.97 ounces, 19.03 ounces).

Step-by-step explanation:

The student is asking about constructing a 90% confidence interval for the population mean weight of running shoes, based on a sample mean and known population standard deviation. To calculate this, we use the formula for a confidence interval when the population standard deviation (σ) is known:

Confidence Interval = Sample Mean ± (Z-score * (Population Standard Deviation / sqrt(Sample Size)))

Since the standard deviation is 3 ounces, the sample mean is 18 ounces, and the sample size is 23, we can plug these values into the formula. The Z-score for a 90% confidence interval is approximately 1.645 (which can be found on a standard normal distribution table or using statistical software). Hence, the confidence interval is given by:

18 ± (1.645 * (3 / sqrt(23)))

Calculating the margin of error:

1.645 * (3 / sqrt(23)) ≈ 1.03 ounces

Thus, the 90% confidence interval for the mean weight of running shoes is 18 ± 1.03 ounces, or:

(16.97 ounces, 19.03 ounces)

Answer 2

The 90% confidence interval for the population mean of running shoe weights is 17.03 ounces to 18.97 ounces.

Step-by-step explanation:

Calculate the standard error of the mean:

Standard error of the mean (SE) = Population standard deviation / √n

SE = 3 ounces / √23 ≈ 0.36 ounces

Find the critical value for a 90% confidence interval:

For a 90% confidence level with 22 degrees of freedom (n-1), the critical value from the t-distribution table is ≈ 1.717.

Calculate the margin of error:

Margin of error (E) = critical value * standard error of the mean

E ≈ 1.717 * 0.36 ounces ≈ 0.62 ounces

Construct the confidence interval:

Lower limit = sample mean - margin of error = 18 ounces - 0.62 ounces ≈ 17.03 ounces

Upper limit = sample mean + margin of error = 18 ounces + 0.62 ounces ≈ 18.97 ounces

Therefore, we can be 90% confident that the true population mean of running shoe weights falls between 17.03 ounces and 18.97 ounces.