Question

A sample of 49 observations is taken from a normal population with a standard deviation of 10. The sample mean is 55. Determine the 99 percent confidence interval for the population mean. Give the lower limit.

Answer 1

To calculate the confidence interval for the population mean, we will use the formula:

Confidence Interval = sample mean ± (critical value * standard deviation / √sample size)

The critical value can be found using the z-table for a 99% confidence level. Since we want to find the lower limit, we will subtract the margin of error from the sample mean.

First, let's find the critical value:
The 99% confidence level will leave 1% for each tail, so we divide it by 2 to get 0.5% in each tail.
Looking up the z-value in the z-table for 0.5% (or 0.005), we find a value of -2.576.

Now, we can calculate the confidence interval:
Confidence Interval = 55 ± (-2.576 * 10 / √49)
Confidence Interval = 55 ± (-2.576 * 10 / 7)
Confidence Interval = 55 ± (-3.668)

To find the lower limit, we subtract the margin of error from the sample mean:
Lower Limit = 55 - 3.668
Lower Limit = 51.332

Therefore, the lower limit of the 99% confidence interval for the population mean is 51.332.