Answer :
Final answer:
To estimate the mean inflation pressure to within 0.025 pounds with 99 percent confidence, the manufacturer should use a sample size of 107 footballs, calculated using the Z-score for a 99% confidence level and the given standard deviation of 0.1.
Explanation:
To determine the necessary sample size for the manufacturer's quality control procedures, we use the formula for estimating the sample size in a normal distribution:
n = (Z*σ/E)²Where:
- ‘n’ is the sample size needed
- ‘Z’ is the Z-score corresponding to the desired confidence level
- ‘σ’ is the population standard deviation
- ‘E’ is the margin of error
Given a 99% confidence level, we look up the Z-score, which is approximately 2.576. The standard deviation (σ) is given as 0.1 pounds, and the desired margin of error (E) is 0.025 pounds.
Plugging those values into the formula, we get:
n = (2.576 * 0.1 / 0.025)²Performing the calculations:
n = (2.576 * 4)²= (10.304)²
= 106.232
We round up to the next whole number. Therefore, the manufacturer should use a sample size of 107 footballs to estimate the mean inflation pressure to within 0.025 pounds with 99 percent confidence.
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