A teacher has a large container filled with blue, red, and green beads. She wants her students to estimate the proportion of red beads. Each student shakes the container, selects 50 beads, counts the number of red beads, and returns the beads to the container. One student’s sample contained 19 red beads. The students are asked to construct a 95% confidence interval for the true proportion of red beads in the container. Are the conditions for inference met?

A. Yes, the conditions for inference are met.
B. No, the 10% condition is not met.
C. No, the randomness condition is not met.
D. No, the Large Counts Condition is not met.

To determine if the conditions for inference are met when constructing a 95% confidence interval for the true proportion of red beads, we need to check the following conditions:

Randomness Condition: The sample must be randomly selected. Since each student shakes the container and selects 50 beads, this condition seems to be met.
10% Condition: The sample size should be less than 10% of the population. Given that there are many beads (the problem states the container is large), choosing 50 beads likely satisfies this condition.
Large Counts Condition: Both np and n(1 - p) should be at least 10.

Here, p = 19/50 = 0.38, so np = 50 × 0.38 = 19 and n(1 - p) = 50 × 0.62 = 31, which are both greater than 10.

Since all of the conditions are met, we can say that the conditions for inference are met.

DiBotcher DiBotcher · Answer 2 · 2023-10-22T20:50:50-04:00

The conditions for constructing a 95% confidence interval for the proportion of red beads are met. The randomness condition is assumed to be satisfied by random selection, the 10% condition is met as the sample is likely less than 10% of the population, and the Large Counts condition is met with enough successes and failures in the sample. The correct answer is option a) that is, Yes, the conditions for inference are met.

To determine whether the conditions for inference are met for constructing a 95% confidence interval for the true proportion of red beads in the teacher's container, we need to check three conditions: randomness, 10% condition, and Large Counts condition.

The randomness condition is met if each student shakes the container and selects beads at random. It's assumed that the students did this, so we can say this condition is met.

The 10% condition requires that the sample size be no more than 10% of the population. This is to ensure that individual selections are independent of each other. Since this is a large container of beads, it's reasonable to assume that sampling 50 beads does not exceed 10% of the total beads in the container, so this condition is met.

Lastly, the Large Counts condition (or the success-failure condition) requires that we have at least 10 successes (in this case, red beads) and 10 failures (non-red beads) in our sample for normal approximation to be valid. With 19 red beads out of 50, we have more than 10 red beads and more than 10 non-red beads, hence this condition is met as well.

Considering all the conditions are met, we can say that it is appropriate to construct a confidence interval for the proportion of red beads in the container.

Therefore, option a) is correct that is, Yes, the conditions for inference are met.