Question

a simple random sample of kitchen toasters is to be taken to determine the mean operational lifetime in hours. assume that the lifetimes are normally distributed with population standard deviation 21 hours. find the sample size needed so that a 98% confidence interval for the population mean lifetime will have a margin of error of 4.

Answer 1

To achieve a 98% confidence interval for the mean operational lifetime of kitchen toasters with a margin of error of 4 hours and a known population standard deviation of 21 hours, the required sample size is 150.

Step-by-step explanation:

To determine the necessary sample size for a 98% confidence interval with a margin of error of 4 hours, when the population standard deviation (σ) is 21 hours, we use the formula for sample size in a confidence interval estimation:

n = (Z*σ/E)^2

Where Z is the z-score corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error. For a 98% confidence level, the z-score (Z) is approximately 2.33. Plugging the values into the formula:

n = (2.33*21/4)^2

This simplifies to:

n = (48.93/4)^2

n = (12.2325)^2

n ≈ 149.632.

Since the sample size (n) must be a whole number, we round up to the nearest whole number, which is 150. Therefore, a sample size of 150 kitchen toasters is required to achieve the desired margin of error for the confidence interval.