In a random sample of 450 cases of juice, 3 of the 48 chosen cases had major defects. The sampling method had a margin of error of 0.04.

What is the interval estimate for the proportion that would have defects in the form (lower limit, upper limit)?



(0.1,0.9)

(0.6,0.14)

(0.02,0.10)

(0.4,0.12) <<my choice

To determine the interval estimate for the proportion of cases with defects, you can use the formula for a confidence interval for a proportion.

First, you need to calculate the sample proportion, which is the number of cases with major defects divided by the total sample size:
Sample proportion (p̂) = (number of cases with defects) / (total sample size)
= 3 / 48
= 0.0625

Next, calculate the standard error, which is the square root of [(sample proportion) * (1 - sample proportion)] divided by the square root of the sample size:
Standard error = √[(p̂ * (1 - p̂)) / n]
= √[(0.0625 * (1 - 0.0625)) / 48]
= √[(0.05859375) / 48]
≈ 0.06471

The margin of error is given as 0.04, which means that the interval estimate will be within 0.04 of the sample proportion.

Finally, you can calculate the lower and upper limits of the interval:
Lower limit = sample proportion - margin of error
Upper limit = sample proportion + margin of error

Lower limit = 0.0625 - 0.04
= 0.0225

Upper limit = 0.0625 + 0.04
= 0.1025

So, the interval estimate for the proportion of cases with defects is approximately (0.0225, 0.1025).

Therefore, the closest option to the correct answer is (0.02, 0.10).

To calculate the interval estimate for the proportion of cases with defects, you can use the formula:

(lower limit, upper limit) = (sample proportion - margin of error, sample proportion + margin of error)

The sample proportion is calculated by dividing the number of cases with defects by the total number of cases:

sample proportion = (number of cases with defects) / (total number of cases)

In this case, the number of cases with defects is 3 out of 48, and the total number of cases is 450.

sample proportion = 3 / 48 = 0.0625

The margin of error given is 0.04.

Now, let's plug these values into the formula:

(lower limit, upper limit) = (0.0625 - 0.04, 0.0625 + 0.04)

Simplifying:

(lower limit, upper limit) = (0.0225, 0.1025)

Therefore, the interval estimate for the proportion of cases with defects is (0.0225, 0.1025).

The correct answer is (0.02, 0.10), so your choice of (0.4, 0.12) is incorrect.