Question

Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. The resulting 95% confidence interval for the true population mean alcohol content is (8.4, 10.2)

a. Find a point estimate for the population mean.

b. What is the margin of error for the confidence interval?

b. Would a 98% confidence interval calculated from the same sample have a larger or smaller margin of error? Explain your reasoning.

c. Consider the following statement: There is a 95% chance that the true mean is between 8.4 and 10.2. Is this statement correct? Why or why not?

d. If the critical value for this confidence interval was Z, what was the population standard deviation?

Answer 1

Sample size: n = 50

95% confidence interval is (8.4, 10.2)

a) Point estimate for the population mean is denoted by .

Adding equations (1) and (2) we get,

Thus point estimate for population mean is 9.3

b) Margin of error is denoted by M.E.

We know that,

Thus, marginal error for the confidence interval is 0.9

b. As we increase confidence level margin of error goes on increasing. For large confidence level critical value becomes large and hence it increases value of margin of error.

Critical value Z at 95% confidence interval is 1.96

Critical value Z at 98% confidence interval is 2.33

Margin of error formula:

We can clearly see that,

Thus, 98% confidence interval calculated from same sample will have larger margin of error.

c)

Meaning of 95% confidence interval (8.4, 10.2)

It interpretes that we are 95% certain that true mean is between 8.4 and 10.2

Thus the given statement is correct.

d) Population standard deviation is denoted by

We know that,

Plug in values into the formula:

Thus population standard deviation at 95% confidence level is 3.2469