Question

Members at a popular fitness club currently pay $40 per month to be a member of the club. The owner of the club wants to raise the fee to $50 but is concerned that some members will quit if the fee increases. To investigate, the owner plans to survey a random sample of the members and construct a confidence interval for the proportion of all members who would quit if the fee was increased to $50.

(a) Explain the meaning of "95% confidence" in the context of the study.
(b) After the owner conducted the survey, he calculated the confidence interval to be 0.18 0.075. Interpret this interval in the context of the study.
(c) According to the club's accountant, the fee increase will be worthwhile if fewer than 20% of the members quit. According to the interval from part (b), can the owner be confident that the fee increase will be worthwhile? Explain.
(d) One of the conditions for calculating the confidence interval in part (b) is that and. Explain why it is necessary to check this condition

Answer 1

A 95% confidence level means that 95% of the intervals calculated from many samples will contain the actual proportion of members who would quit. The interval calculated suggests that the owner cannot be confident that less than 20% of members will quit. Conditions ensure at least five successes and failures for the normal approximation to be valid.

Step-by-step explanation:

(a) A 95% confidence level in the context of this study means that if the owner takes many random samples of the fitness club members and calculates the confidence interval for each sample, 95% of those intervals will contain the true proportion of members who would quit due to the fee increase.

(b) The calculated confidence interval of 0.18 ± 0.075 implies that the owner is 95% confident that the true proportion of members who would quit if the monthly fee increased to $50 is between 10.5% (0.18 - 0.075) and 25.5% (0.18 + 0.075).

(c) Based on this confidence interval, the owner cannot be certain that less than 20% of members will quit, as the upper bound of the interval (25.5%) is greater than 20%. Therefore, the fee increase may not be worthwhile according to the accountant's criterion.

(d) The condition n π ≥ 5 and n (1 - π) ≥ 5 ensures that there are at least five expected successes and five expected failures in the sample, which is necessary for the normal approximation to be valid when constructing a confidence interval for a proportion.