Question

An automobile club comes to the aid of stranded motorists who are members. Over the long run, about 5% of the members utilize this service in any 12 month period. A small town has 670 members of the club. Consider the 670 members to be representative of the larger club membership. Let p^​ be the proportion of them who require aid in a 12 month period. Based on the Empirical Rule, what interval should p^​ fall about 95% of the time? (Round all answers to the nearest hundredth.) (Enter your answer in the form: lower limit to upper limit. Include the word "to.")

Answer 1

The interval in which the proportion of members requiring aid in the automobile club will fall about 95% of the time is approximately 4.58% to 5.42%.

Step-by-step explanation:

The problem states that about 5% of the automobile club members require aid in a 12 month period. We need to find the interval in which this proportion will fall about 95% of the time based on the Empirical Rule. The Empirical Rule is also known as the 68-95-99.7 rule, which states that for a normally distributed data set, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

To find the interval, we can use the mean and standard deviation. The mean, or average, is equal to the proportion of members who require aid, which is 5%. The standard deviation can be calculated as the square root of the product of the proportion of members who require aid and the proportion of members who do not require aid:

Standard deviation = sqrt(5% * 95%) = 0.21

Now we can calculate the interval. Two standard deviations from the mean on either side will cover about 95% of the data. So the interval is:

Mean - 2 * standard deviation = 5% - 2 * 0.21 = 4.58%

Mean + 2 * standard deviation = 5% + 2 * 0.21 = 5.42%

Therefore, the interval in which the proportion will fall about 95% of the time is approximately 4.58% to 5.42%.