To construct a 99% confidence interval for the mean circumference of soccer balls within 0.13 inches, a minimum sample size of 142 is required, assuming a population standard deviation of 0.6 inches.
To determine the minimum sample size required to construct a 99% confidence interval for the population mean circumference of soccer balls within a margin of error of 0.13 inches, we can use the formula for the confidence interval margin of error:
E = Z * (σ / √n)
Where:
E is the margin of error,
Z is the Z-score associated with the desired confidence level (for 99%, Z is approximately 2.576),
σ is the population standard deviation, and
n is the sample size.
Rearranging the formula to solve for n, we get:
n = ((Z * σ) / E)^2
Plugging in the given values (Z is approximately 2.576, σ is 0.6, and E is 0.13), we can calculate the minimum sample size required:
n = ((2.576 * 0.6) / 0.13)^2
n is approximately ((1.5456) / 0.13)^2
n is approximately (11.8883)^2
n is approximately 141.4147
Since the sample size must be a whole number, we round up to the nearest integer, yielding a minimum sample size of 142.