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On a typical Friday, a local bakery sells 40 out of 50 cakes and 20 out of 30 bread rolls.

Construct a 99% confidence interval for the difference in the proportion of items sold.

Give an interpretation of this interval.

Answer :

Final answer:

A 99% confidence interval for the difference in proportions of sold items is calculated using sample proportions and the corresponding Z-value for the 99% confidence level. The resulting interval provides a range where we believe the true difference in proportions lies with 99% confidence.

Explanation:

The student asked to construct a 99% confidence interval for the difference in the proportion of items sold at a local bakery, where 40 out of 50 cakes and 20 out of 30 bread rolls are sold on a typical Friday. To calculate this, we will use the formula for the confidence interval of the difference between two proportions which is: p1 - p2 ± Z*sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2)), where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and Z is the Z-value corresponding to the 99% confidence level.

First, we compute the sample proportions, p1 = 40/50 = 0.80 for cakes and p2 = 20/30 = 0.6667 for bread rolls. We then find the Z-value for a 99% confidence interval which typically is about 2.576. Plugging these values into the formula, we calculate the margin of error and construct the confidence interval.

The interpretation of this confidence interval is that we are 99% confident that the true difference in the proportion of cakes and bread rolls sold lies within the calculated interval.

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