Question

Using Grocery Rewards ~ A Gallup Poll of 500 randomly selected American adults resulted in 405 stating they use a grocery stores’ rewards card to save money on their grocery purchases. A local grocery store manager comes to you for help to use these results to provide an estimate of the population proportion of all American adults who use a grocery stores’ rewards card to save money on their grocery purchases. You ask the manager what confidence level should be used ~ 90%, 95%, 98% or 99%. The manager says “I am not sure what the confidence level really means, but use 98% I guess”.

You create the following list to help you remember the various items you need to include in your response back to the manager:

Assumptions: You know you may assume a random sample of 500 American adults was selected. You plan to both state and check the remaining necessary assumption.
Computations: You will compute and report the 98% confidence interval, showing all work so the manager can follow your computations.
Level Explanation: Since the manager was unclear what the confidence level means, you will include a sentence to explain it, that is, provide an interpretation of the 98% confidence level in context.

Answer 1

State and check the remaining necessary assumption.

The necessary assumption to compute the confidence interval of proportion p is

np > 10 and n(1-p) > 10

Sample proportion, p = 405 / 500 = 0.81

np = 500 * 0.81 = 405  

n(1-p) = 500 * (1 - 0.81) = 95

Since np > 10 and n(1-p) > 10, we can compute the confidence interval of proportion.

Computations:

z value for 98% confidence interval is 2.326

Standard error of proportion, SE =

= 0.0175

Margin of error = z * SE = 2.326 * 0.0175 = 0.04

98% confidence interval of proportion of American adults who use a grocery stores’ rewards card to save money on their grocery purchases is

(0.81 - 0.04, 0.81 + 0.04)

(0.77, 0.85)

Level Explanation: We're 95% confident that the interval (0.77, 0.85) captured the true proportion of American adults who use a grocery stores’ rewards card to save money on their grocery purchases.