A pilot sample of 50 measurements of the breaking strength of cotton threads gave S_(pr) = 18 grams. Determine the sample size needed to obtain a 90% Confidence Interval of length 4. I tried using the sample size formula but that did not gave me the right answer, according to the book the answer is n = 220.
Figured out the answer:
n = (1.645(18/2))^2 = 219.188 = 220
To determine the sample size needed to obtain a 90% confidence interval of length 4, we need to find the appropriate formula and apply it. The formula used to calculate the sample size for estimating a population mean is:
n = [Z * (S/4)]^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the level of confidence (90% confidence interval corresponds to a Z-value of approximately 1.645)
- S is the standard deviation of the pilot sample (given as S_(pr) = 18 grams)
- 4 is the desired length of the confidence interval
Applying these values to the formula, we have:
n = [1.645 * (18/4)]^2
n = [1.645 * 4.5]^2
n = (7.4025)^2
n ≈ 54.75
The formula calculates a sample size of approximately 54.75. However, since sample sizes must be whole numbers, we need to round up to the nearest integer. Hence, n = 55.
It seems there is a discrepancy between the calculated sample size (55) and the answer provided in the book (220). It's possible that either the formula or the answer in the book is incorrect. I recommend double-checking the requirements of the question and verifying the sample size formula provided in the book.