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 lower limit.

Answer 1

To construct a 99% confidence interval for the difference between the two cereals, we can use the following formula:

CI = (p1 - p2) ± Z * √[ (p1 * (1-p1))/n1 + (p2 * (1-p2))/n2 ]

Where:
p1 = proportion of consumers who would buy cereal A
p2 = proportion of consumers who would buy cereal B
n1 = size of the test panel for cereal A
n2 = size of the test panel for cereal B
Z = Z-score for the desired confidence level

First, we need to calculate the proportion (p) for each cereal:

p1 = 52/175 ≈ 0.297
p2 = 35/150 ≈ 0.233

Next, we need to determine the Z-score for a 99% confidence level. The Z-score can be obtained from a standard normal distribution table or using statistical software. For a 99% confidence level, the Z-score is approximately 2.576.

Now, we can plug in the values into the formula:

CI = (0.297 - 0.233) ± 2.576 * √[ (0.297 * (1-0.297))/175 + (0.233 * (1-0.233))/150 ]

Simplifying the calculation:

CI = 0.064 ± 2.576 * √[ 0.207/175 + 0.179/150 ]

Calculating the value inside the square root:

√[ 0.207/175 + 0.179/150 ] ≈ √0.0011816 ≈ 0.03438

Plugging in the value:

CI = 0.064 ± 2.576 * 0.03438

Calculating the upper and lower limits:

Lower limit = 0.064 - (2.576 * 0.03438) ≈ 0.064 - 0.08870 ≈ -0.0247

Therefore, the lower limit of the 99% confidence interval for the difference between the two cereals is approximately -0.0247.