2. *Suppose that a random sample of 50 bottles of a particular brand of cough sy

Question

2. *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. Let denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.8, 9.4).
a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning.
b. Consider the following statement: There is a 95% chance that is between 7.8 and 9.4. Is this statement correct? Why or why not?
c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4. Is this statement correct? Why or why not?
d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include . Is this statement correct? Why or why not?

Explanation / Answer

(a)
If 90% confidence interval calculated from this same sample, it would be narrower than (7.8, 9.4). Because the value of z for 90% CI is 1.65 compared to z for 95% of CI is 1.96. Lower the value of z makes ME small which decreases the size of confidence interval.

(b)
The given statement is not correct. 95% CI means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time.

(c)
The given statement is not correct. 95% CI means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time.

(d)
The given statement is correct. 95% CI means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time.

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