Question

Question 5. A random sample of 50 bottles of a brand of cough syrup is selected, and the alcohol content of each bottle is measured. Let μ be the mean alcohol content for the population of all bottles of the brand under consideration. The 95% confidence interval estimated from the data is (7.8 μ 9.4). (a) would a 90% confidence interval calculated from the same sample be wider or narrower? Explain. (b) Consider the statement: there is a 95% chance that μ is between 7.8 and 9.4. Is this correct? Explain. (c) Consider the statement: we can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content between 7.8 and 9.4. Is this correct? Explain your answer. (d) Consider the statement: if the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, exactly 95 of the resulting intervals will include μ. Is this correct? Explain your answer. (e) Consider the statement: if the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, approximately 95 of the resulting intervals will include μ. Is this correct? Explain your answer.

Answer 1

(a) it will be narrower, higher the confidence level the wider will be confidence interval

(b) yes the statement is true,

A confidence interval gives a range of plausible values for the parameter of interest.

(c) True, statement is true

A confidence interval gives a range of plausible values for the parameter of interest and we have the confidence of 95%.

(d) no, statement is not true

we can expect about 95 of those intervals to contain but not the exact 95

(e) yes, statement is true

The confidence level tells us the long-term capture rate of these intervals over repeated samples.