Forty-nine items are randomly selected from a population of 500 items. The sample mean is 40 and the sample standard deviation 9.
Develop a 99% confidence interval for the population mean
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.005) and the Z score. Insert data into first equation to solve.
To develop a 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean +/- (Critical Value * Standard Error)
1. Calculate the standard error:
Standard Error = Sample Standard Deviation / √Sample Size
In this case, the sample standard deviation is 9 and the sample size is 49.
Standard Error = 9 / √49 = 9 / 7 = 1.286
2. Determine the critical value. Since we want a 99% confidence interval, we need to find the z-score corresponding to a 99% confidence level. The z-score can be obtained using a z-table or a statistical software. In this case, the z-score for a 99% confidence level is approximately 2.576.
3. Calculate the confidence interval:
Confidence Interval = Sample Mean +/- (Critical Value * Standard Error)
Confidence Interval = 40 +/- (2.576 * 1.286)
Lower Limit = 40 - (2.576 * 1.286) = 40 - 3.314 = 36.686
Upper Limit = 40 + (2.576 * 1.286) = 40 + 3.314 = 43.314
Therefore, the 99% confidence interval for the population mean is [36.686, 43.314].