You are in charge of selling advertising for radio station KGSM. The fee you can set for air time is directly related to the share of the listening market your station reaches. From time to time you conduct surveys to determine KGSM's share of the market. This month, when you contacted 200 randomly selected residential phone numbers, 12 respondents said they listen to KGSM.

a.Compute the 99% confidence interval for the percentage of the market that are listeners of KGSM. Interpret this interval.

b.How many individuals should be sampled in order to be 99% confident of being within 1% of the actual population percentage of the market that are KGSM listeners?

Solution:

a) Compute the 99% confidence interval for the percentage of the market that are listeners of KGSM. Interpret this interval.

The information given in the above problem can be represented with the following notations.

Sample size n = 200

The number of respondents who listen to KGSM = x = 12

Sample Proportion =

The 99 % confidence interval for the population proportion is given by

Z_a/2 = Z_0.005 = 2.58 (by referring Standard normal distribution table)

(0.0167, 0.1033)

The 99 % confidence interval for the percentage of the market that is listeners of KGSM is (0.0167, 0.1033)

Interpretation of Confidence Interval.

If repeated samples were taken for large number of times , in 99 % of the cases the interval (0.0167, 0.1033) will contain the true population parameter.

b) How many individuals should be sampled in order to be 99% confident of being within 1 % the actual population percentage of the market that are KGSM listeners?

The sample size n =

The margin of error E = 1% = 0.01

Thus n =

= = 3754.209

Sample size n = 3754 (approximately)

Thus the number of individuals should be sampled is 3754.