Question

There are 1,000 golden delicious and 1,000 red delicious apples in a cooler. In a random sample of 75 of the golden delicious apples, 48 had blemishes. In a random sample of 75 of the red delicious apples, 42 had blemishes. Assume all conditions for inference have been met. Which of the following is closest to the interval estimate of the difference in the numbers of apples with blemishes (golden delicious minus red delicious) at a 98 percent level of confidence?A. (–0.076,0.236)

B. (–0.105,0.265)
C. (–10.5,26.5)
D. (–76,236)
E. (−105,265)

Answer 1

E. (−105,265)

Explanation:

The confidence interval for the difference between 2 population proportions is:

(p₁ − p₂) ± t √(p₁q₁/n₁ + p₂q₂/n₂)

where p₁ and p₂ are the population proportions,

q is 1 − p,

n is the sample size,

and t is the critical value for the confidence level.

First, calculate the proportions:

p₁ = 48/75 = 0.64

p₂ = 42/75 = 0.56

Now calculate the standard error:

SE = √(0.64×0.36/75 + 0.56×0.44/75)

SE = 0.0797

For large sample sizes, we can approximate the critical value t using z. At 98% confidence, z = 2.326.

Therefore, the confidence level of the difference in proportions is:

(0.64 − 0.56) ± 2.326 × 0.0797

(-0.105, 0.265)

Since the population is 1000 apples, the confidence interval for number of apples is (-105, 265).