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.
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.
To construct a confidence interval for the difference between the proportions of consumers who would buy cereal A and cereal B, you can use the formula:
CI = (p1 - p2) ± Z * √[(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)]
Where:
CI = Confidence Interval
p1 = Proportion of consumers who would buy cereal A
p2 = Proportion of consumers who would buy cereal B
Z = Z-value for the desired confidence level (99% confidence level corresponds to Z = 2.576)
n1 = Size of the test panel for cereal A
n2 = Size of the test panel for cereal B
First, calculate the proportions:
p1 = 52 / 175 = 0.2971 (rounded to four decimal places)
p2 = 35 / 150 = 0.2333 (rounded to four decimal places)
Next, plug the values into the formula:
CI = (0.2971 - 0.2333) ± 2.576 * √[(0.2971 * (1 - 0.2971) / 175) + (0.2333 * (1 - 0.2333) / 150)]
= 0.0638 ± 2.576 * √[(0.2971 * 0.7029 / 175) + (0.2333 * 0.7667 / 150)]
= 0.0638 ± 2.576 * √[0.001189 + 0.000596]
= 0.0638 ± 2.576 * √0.001785
= 0.0638 ± 2.576 * 0.04226
= 0.0638 ± 0.1089
Finally, calculate the lower limit of the confidence interval:
Lower limit = 0.0638 - 0.1089 = -0.0451 (rounded to four decimal places)
Therefore, the lower limit of the 99% confidence interval for the difference between the two cereals is approximately -0.0451.
To construct a confidence interval for the difference between two proportions, you can use the following formula:
CI = (p1 - p2) ± Z * sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
where:
- p1 and p2 are the proportions of consumers who would buy cereal A and cereal B, respectively.
- n1 and n2 are the sample sizes of the two test panels.
- Z is the z-score corresponding to the desired confidence level. For a 99% confidence level, Z = 2.576.
First, calculate the proportions of consumers who would buy cereal A and cereal B:
p1 = 52/175 = 0.2971
p2 = 35/150 = 0.2333
Next, 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))
Simplify the calculation:
CI = 0.0638 ± 2.576 * sqrt((0.2070/175) + (0.1661/150))
CI = 0.0638 ± 2.576 * sqrt(0.0012 + 0.0011)
CI = 0.0638 ± 2.576 * sqrt(0.0023)
CI = 0.0638 ± 2.576 * 0.048
CI = 0.0638 ± 0.1236
Finally, calculate the lower limit of the confidence interval:
Lower limit = 0.0638 - 0.1236 = -0.0598
Therefore, the lower limit of the 99% confidence interval for the difference between the two cereals is approximately -0.0598.