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# Sample size and confidence interval calculation

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Consumer products companies design different promotional activities to accomplish different goals. According to a spokesperson for General Mills, the company's cents-off coupon offers are designed to convince people to buy its products, while its refund offers (money returned with proof of repeated purchases) are designed to encourage people to continue buying its products. However, these promotional activities are only effective if consumers participate.

In a survey of 562 randomly selected households in the Research Triangle Area (Raleigh, Durham, and Chapel Hill) in 2003, 427 principal shoppers (the person in the household who does the majority of the grocery shopping) indicated that they used cents-off coupons when they went grocery shopping.
a) What is your best estimate of the proportion of households throughout the Research Triangle Area who use cents-off coupons when grocery shopping?
b) Develop a 99% confidence interval for the proportion of households throughout the Research Triangle Area who use cents-off coupons when grocery shopping.
c) Suppose it is desired that the 99% confidence interval be of the form "plus or minus 0.03." How large a sample is required?

https://brainmass.com/statistics/confidence-interval/sample-size-confidence-interval-calculation-145072

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Consumer products companies design different promotional activities to accomplish different goals. According to a spokesperson for General Mills, the company's cents-off coupon offers are designed to convince people to buy its products, while its refund offers (money returned with proof of repeated purchases) are designed to encourage people to continue buying its products. However, these promotional activities are only effective if consumers participate.
In a survey of 562 randomly selected households in the Research Triangle Area (Raleigh, Durham, and Chapel Hill) in 2003, 427 principal shoppers (the person in the household who does the majority of the grocery shopping) indicated that they used cents-off coupons when they went grocery shopping.
a) What is your best estimate of the proportion of households throughout the Research Triangle Area who use cents-off coupons when grocery shopping?
The best estimate of proportion of households throughout the Research Triangle Area who use cents-off coupons when grocery shopping is the sample proportion .
Sample proportion =427/562 = 0.759786477
b) Develop a 99% confidence interval for the proportion of households throughout the Research Triangle Area who use cents-off coupons when grocery shopping.
Confidence interval for proportion is given by
Details

Confidence Interval Estimate for the Mean
Data
Sample Size 562
Number of Successes 427
Confidence Level 99%

Intermediate Calculations
Sample Proportion 0.759786477
Z Value -2.5758293
Standard Error of the Proportion 0.018020895
Interval Half Width 0.046418749

Confidence Interval
Interval Lower Limit 0.713367728
Interval Upper Limit 0.806205225
c) Suppose it is desired that the 99% confidence interval be of the form "plus or minus 0.03." How large a sample is required?
The minimum sample size n is given by the formula where p is the sample proportion and E is the margin of Error
Details
Sample Size Determination
Data
Estimate of True Proportion 0.759786477
Sampling Error 0.03
Confidence Level 99%

Intermediate Calculations
Z Value -2.5758293
Calculated Sample Size 1345.490581

Result
Sample Size Needed 1346

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