Fifty two of a test panel of 175 consumers say that they would buy cereal A if it is presented on the market and 35 of another test panel of 150 consumers say that they would buy cereal B. Construct a 99% confidence interval for the difference between the two cereals. Give the upper limit.

Answer 1

To construct a confidence interval for the difference between the two cereals, we can use the formula:

\[ \text{Confidence Interval} = (\hat{p}_1 - \hat{p}_2) \pm Z \times\sqrt{(\hat{p}_1(1-\hat{p}_1)/n_1) + (\hat{p}_2(1-\hat{p}_2)/n_2)}\]

where:

- \(\hat{p}_1\) is the proportion of consumers who say they would buy cereal A
- \(\hat{p}_2\) is the proportion of consumers who say they would buy cereal B
- \(n_1\) is the size of the test panel for cereal A
- \(n_2\) is the size of the test panel for cereal B
- \(Z\) is the critical z-value for the desired confidence level

In this case, we have:

\(\hat{p}_1 = \frac{52}{175}\), \(\hat{p}_2 = \frac{35}{150}\), \(n_1 = 175\), \(n_2 = 150\), and for a 99% confidence level, the critical z-value is 2.576.

Calculating the confidence interval:

\(\text{Confidence Interval} = \left(\frac{52}{175} - \frac{35}{150}\right) \pm 2.576 \times\sqrt{\left(\frac{52}{175}\times\frac{123}{175}\right)/175 + \left(\frac{35}{150}\times\frac{115}{150}\right)/150)}\)

\(\text{Confidence Interval} = 0.2971 \pm 2.576 \times 0.0663\)

\(\text{Confidence Interval} = 0.2971 \pm 0.1707\)

\(\text{Confidence Interval} \approx (0.1264, 0.4678)\)

Therefore, the upper limit of the 99% confidence interval for the difference between the two cereals is approximately 0.4678.