Final answer:
The 98% confidence interval for the mean difference in calories between grade AA and grade A eggs is approximately (8.704, 11.296) calories.
Step-by-step explanation:
To find a 98% confidence interval for the mean difference in the number of calories between grade AA and grade A eggs, we will use the formula for the confidence interval of the difference between two independent means with known standard deviations. Since the sample sizes are large (n1 = 90 and n2 = 85 respectively), we can assume the distribution of the sample means is approximately normal by the Central Limit Theorem.
The formula for the confidence interval is:
(μ1 - μ2) ± z*(σpooled/√n1 + σpooled/√n2)
Where:
- μ1 is the mean of the first sample (grade A eggs), which is 70.
- μ2 is the mean of the second sample (grade AA eggs), which is 80.
- σpooled is the pooled standard deviation. For large samples, we can approximate σpooled by taking a weighted average of the sample standard deviations:
σpooled = √[(σ²[n1-1] + σ²[n2-1]) / (n1 + n2 - 2)]
Plugging in our values, σpooled = √[(2.5²[90-1] + 2.7²[85-1]) / (90 + 85 - 2)] = 2.603.
The z* value is the z-score that corresponds to the desired confidence level. For a 98% confidence interval, z* is approximately 2.33.
Thus, the confidence interval for the difference in means is:
(80 - 70) ± 2.33*(2.603/√90 + 2.603/√85)
After calculating, we get:
(10 ± 2.33*(0.274 + 0.282)) = (10 ± 2.33*0.556) = (10 ± 1.296) = (8.704, 11.296)
Therefore, the 98% confidence interval for the mean difference in calories between grade AA and grade A eggs is approximately (8.704, 11.296) calories.