To determine the number of days a researcher should measure the water temperature to estimate the true mean within a specified margin of error at a certain confidence level, we can use the formula for sample size in hypothesis testing:
n = (z^2 * variance) / error_margin^2
Here, 'n' is the sample size, 'z' is the z-score corresponding to the desired confidence level, 'variance' is the variance of the water temperature, and 'error_margin' is the desired margin of error.
Given:
- variance = 30 degrees^2
- desired error margin (error_margin) = ±4 degrees
- confidence level = 99%
Step 1: Find the z-score corresponding to the 99% confidence level.
For a 99% confidence level, the z-score is typically around 2.576. However, we will calculate it precisely using the z-distribution.
Using statistical tables or a calculator equipped with a cumulative distribution function for the standard normal distribution, we find:
z = Z(0.995) since the confidence level is 99% and we want the z-score that leaves 0.5% in one tail (1 - 0.99 = 0.01, and 0.01/2 = 0.005). The area to the left of the z-score is 1 - 0.005 = 0.995.
Typical statistical tables or calculators give z ≈ 2.576 (rounded to three decimal places).
Step 2: Substitute the values into the formula and solve for 'n'.
n = (z^2 * variance) / error_margin^2
n = ((2.576)^2 * 30) / (4)^2
Step 3: Calculate
n = (6.635776 * 30) / 16
n = 198.07328 / 16
n ≈ 12.37958
Since we cannot collect a fraction of a day's worth of data, we need to round up to the next whole number.
n ≈ 13
Therefore, the researcher needs to measure the water temperature for at least 13 days to estimate the true mean temperature within a margin of error of 4 degrees with 99% confidence.
