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Example 9.4 Estimating the Response to a New Sandwich

Example 9.4 Estimating the Response to a New Sandwich. Confidence Interval for a Proportion. Objective. To illustrate the procedure for finding a confidence interval for the proportion of customers who rate the new sandwich at least 6 on a 10-point scale. Background Information.

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Example 9.4 Estimating the Response to a New Sandwich

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  1. Example 9.4Estimating the Response to a New Sandwich Confidence Interval for a Proportion

  2. Objective To illustrate the procedure for finding a confidence interval for the proportion of customers who rate the new sandwich at least 6 on a 10-point scale.

  3. Background Information • The fast food manager from Example 9.2 has already sampled 40 customers to estimate the population mean rating of its new sandwich. • Recall that each rating is on a scale of 1 to 10, 10 being best. • The manager would now like to use the same sample to estimate the proportion of customer who rate the sandwich at least 6. • Her thinking is that these are the customers who are likely to purchase the sandwich again.

  4. SANDWICH2.XLS • This file contains the solution. • To create this output, we first count the number of ratings that are at least 6 in cell E6. The easiest way to do this is with the formula =COUNTIF(B4:B43,”>=6”) • Then we calculate the sample proportion in cell E7 with the formula =E6/E3

  5. Results

  6. Results -- continued • The rest is simply a matter of implementing the equations. Specifically, the standard error formula in cell E11 is =SQRT(E7*(1-E7)/E3)and the formula for the lower limit of the confidence interval, in cell E13 is =E7-E12*E11 • Of course, the formula for the upper limit of the confidence interval is the same except with a plus sign.

  7. Results -- continued • Then using the confidence interval limits pL = 0.475 and pU=0.775, we can check the assumption of the large sample size. • With n=40, you can check that npL,n(1-pL), npU, and n(1-pU) are all well above 5, so that the validity of this confidence interval is established.

  8. Conclusions • The output is fairly good news for the manager. • Based on this sample of size 40, she can be 95% confident that the percentage of all customers who would rate the sandwich 6 or higher is somewhere between 47.5% and 77.5%. • This is a large interval so there is a lot of uncertainty. To reduce this interval the manager would have to sample more customers.

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