A study is being planned to estimate the mean number of semester hours taken by students at a college. The population standard deviation is assumed to be o=4.7 hours. How many students should be included in the sample to be 99% confident that the sample mean is within one semester hour of the population mean u for all students at this college?
To determine the sample size required to estimate the mean number of semester hours with a given level of confidence, we need to use the formula for sample size calculation:
n = (Z * σ / E)^2
Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level
σ = population standard deviation
E = maximum error (also known as margin of error)
In this case, we want to be 99% confident that the sample mean is within one semester hour of the population mean. Therefore, the maximum error (E) is 1.
Now, we need to find the appropriate Z-score for a 99% confidence level. The Z-score can be obtained from a Z-table or by using statistical software. For a 99% confidence level, the Z-score is approximately 2.576.
Plugging in the values into the formula:
n = (2.576 * 4.7 / 1)^2
n ≈ (12.03872)^2
n ≈ 144.66
Therefore, you would need a sample size of approximately 145 students to be 99% confident that the sample mean is within one semester hour of the population mean for all students at the college.