Answer :
Final answer:
To estimate the population mean with a confidence of 99.7% and an error tolerance of ±0.5, a minimum sample size of approximately 302400 is needed.
Explanation:
To estimate the population mean of a random variable to ±0.5 with a confidence of 99.7%, we need to determine the minimum sample size. We can use the formula:
n = (zα/2 * σ / E)2
Where n is the sample size, zα/2 is the critical value corresponding to the desired confidence level (in this case, for a 99.7% confidence level, zα/2 is approximately 2.748), σ is the population standard deviation (given as √100 = 10), and E is the maximum error tolerance (0.5).
Substituting the values into the formula, we get:
n = (2.748 * 10 / 0.5)2 ≈ 2.7482 * 102 / 0.52 ≈ 75.6 * 100 / 0.25 ≈ 75600 / 0.25 ≈ 302400
Therefore, the minimum sample size needed to estimate the population mean with a confidence of 99.7% and an error tolerance of ±0.5 is approximately 302400.
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