Question

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?

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

Answer 1

The conditions for inference, including the randomness condition, the 10% condition, and the Large Counts condition, are all met for constructing the 95% confidence interval for the proportion of red beads.

Step-by-step explanation:

To determine if the conditions for inference are met for constructing a 95% confidence interval for the true proportion of red beads in the container, we must look at the three main conditions:

Randomness condition: The sample should be randomly selected.

10% condition: The sample size should be no more than 10% of the whole population.

Large Counts condition: The sample follows a binomial distribution, where np (number of successes) and n(1-p) (number of failures) are both greater than 10.

In the given scenario, assuming the student shook the container and selected beads at random, the randomness condition is met. The 10% condition is typically irrelevant in this context because most containers of beads contain more than 500 beads, and selecting 50 would be less than 10% of all beads. The condition most in question would be the Large Counts condition, which, given that there are 19 red beads (successes) out of 50, and 31 other beads (failures), both numbers are greater than 10 - hence, the Large Counts Condition is met. Therefore, we can conclude that the conditions for inference are met.

Answer 2

A

Step-by-step explanation: