Final answer:
The 98% confidence interval for the mean length of carpet from a sample with a mean of 72.2 yards and a population standard deviation of 2.6 yards is approximately 71.5 to 72.9 yards. The Z-score for 98% confidence is 2.33, leading to a margin of error of 0.699 when applied to the given standard deviation and sample size. Therefore, option B) 71.5 μ 72.9 is the correct answer.
Step-by-step explanation:
To find the 98% confidence interval for the mean length of carpet from a sample of 75 bolts with a mean of 72.2 yards and a known population standard deviation of 2.6 yards, we need to use the formula for the confidence interval of the mean with a known population standard deviation:
Confidence Interval = μ ± (Z * (σ / √n))
Where μ is the sample mean, σ is the population standard deviation, n is the sample size, and Z is the Z-score corresponding to the desired confidence level.
For a 98% confidence level, the Z-score is approximately 2.33 (this can be found in Z-score tables or using statistical software). Thus, the margin of error (ME) can be calculated as:
ME = Z * (σ / √n) = 2.33 * (2.6 / √75) = 2.33 * 0.3 ≈ 0.699
Therefore, the confidence interval is:
72.2 ± 0.699
Which gives us the interval:
72.2 - 0.699 = 71.501 (approximately 71.5)
and
72.2 + 0.699 = 72.899 (approximately 72.9)
Thus, the 98% confidence interval for the mean length per bolt of carpet is approximately 71.5 to 72.9 yards.
The correct answer from the options provided would be B) 71.5 μ 72.9.