Question

In a sample of 90 large grade A eggs, the mean number of calories was 70 with a standard deviation of 2.5. In a sample of 85 large grade AA eggs, the mean number of calories was 80 with a standard deviation of 2.7. Find a 98% confidence interval for the mean difference in the number of calories between grade AA and grade A eggs.

Answer 1

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.