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. 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 mu 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 mu. Is this statement correct? Why or why not?

Answer 1

Confidence interval:

A range of values such that the population parameter can expected to contain for the given confidence level is termed as the confidence interval. In other words, it can be defined as an interval estimate of the population parameter which is calculated for the given data based on a point estimate and for the given confidence level.

Concepts and reason

The confidence level indicates the possibility that the confidence interval can contain the population parameter. Usually, the confidence level is denoted by . The value is chosen by the researcher. Some of the most common confidence levels are 90%, 95%, and 99%.

The margin of error is defined as a statistic which gives the amount of sampling error in the given study. Also, the margin of error tells the percentage of points that the obtained results would differ from that of the given population value.

The different situations for the confidence interval for the mean are:

• As the confidence level increases the width of the confidence interval for the mean increases.

• As the standard deviation increases the width of the confidence interval for the mean increases.

• As the sample size increases the width of the confidence interval decreases.

(a)

From the given information, the 95% confidence interval for the average alcohol content is $\left( {7.8,9.4} \right)$ . As the confidence level decreases the margin of error decreases. Then, the width of the confidence interval decreases. That is, as the confidence level decreases from 95% to 90% the confidence interval would be narrower.

(b)

From the given information, the 95% confidence interval for the average alcohol content is $\left( {7.8,9.4} \right)$ . It can be interpreted as that there is 95% confident that the average alcohol content lies between 7.8 and 9.4.

(c)

From the given information, the 95% confidence interval for the average alcohol content is $\left( {7.8,9.4} \right)$ . It can be interpreted as that there is 95% confidence that the average alcohol content lies between 7.8 and 9.4 and it cannot tell about the alcohol content of all the bottles lies in this interval.

(d)

From the given information, the 95% confidence interval for the average alcohol content is $\left( {7.8,9.4} \right)$ . That is, if the sample is repeated infinite number of times the average alcohol content lies in 95% of the intervals.

Ans: Part a

The 90% confidence interval calculated from the sample has been narrower than the given interval.

Part b

The statement “There is a 95% chance that $\mu$ is between 7.8 and 9.4” is not correct.

Part c

The 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 not correct.

Part d

The statement “If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times” is not correct.