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.
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.
To construct a confidence interval for the difference between the two cereals, we can use the formula:
CI = (p1 - p2) ± Z * √(p1 * (1 - p1)/n1 + p2 * (1 - p2)/n2)
Where:
- p1 is the proportion of the first test panel who would buy cereal A
- p2 is the proportion of the second test panel who would buy cereal B
- n1 is the size of the first test panel
- n2 is the size of the second test panel
- Z is the critical value for a 99% confidence level (Z = 2.576)
First, we need to calculate the proportions:
p1 = 52/175 = 0.2971
p2 = 35/150 = 0.2333
Next, we substitute the values into the formula:
CI = (0.2971 - 0.2333) ± 2.576 * √((0.2971 * (1 - 0.2971))/175 + (0.2333 * (1 - 0.2333))/150)
Simplifying the equation:
CI = 0.0638 ± 2.576 * √(0.2072/175 + 0.1794/150)
Calculating the values inside the square root:
CI = 0.0638 ± 2.576 * √(0.0012 + 0.0012)
Summing the values inside the square root:
CI = 0.0638 ± 2.576 * √(0.0024)
Taking the square root:
CI = 0.0638 ± 2.576 * 0.0490
Multiplying by 2.576:
CI = 0.0638 ± 0.1260
Calculating the upper limit of the confidence interval:
CI = 0.0638 + 0.1260 = 0.1898
Therefore, the upper limit of the 99% confidence interval for the difference between the two cereals is 0.1898 (or approximately 18.98%).
To construct a confidence interval for the difference between two proportions, we can use the following formula:
CI = (p1 - p2) ± Z * sqrt( (p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2) )
Where:
- CI is the confidence interval
- p1 and p2 are the proportions (success rates) of the two groups
- n1 and n2 are the sample sizes of the two groups
- Z is the Z-score corresponding to the desired confidence level
First, calculate the proportions for each group:
p1 = 52/175 = 0.2971 (proportion for cereal A)
p2 = 35/150 = 0.2333 (proportion for cereal B)
Next, calculate the required Z-score for a 99% confidence level. The Z-score corresponds to the area in the tails of the standard normal distribution for the desired confidence level. For a 99% confidence level, the Z-score is approximately 2.576.
Now, substitute the values into the formula:
CI = (0.2971 - 0.2333) ± 2.576 * sqrt( (0.2971 * (1 - 0.2971) / 175) + (0.2333 * (1 - 0.2333) / 150) )
Calculating the expression inside the square root:
sqrt( (0.2971 * (1 - 0.2971) / 175) + (0.2333 * (1 - 0.2333) / 150) )
sqrt(0.0016601 + 0.001519689)
sqrt(0.003179789)
0.05638 (rounded to five decimal places)
Now, calculate the confidence interval:
CI = (0.2971 - 0.2333) ± 2.576 * 0.05638
CI = 0.0638 ± 0.1454
Finally, calculate the upper limit of the confidence interval:
0.0638 + 0.1454 = 0.2092
Therefore, the upper limit for the 99% confidence interval for the difference between the two cereals is 0.2092.