Question

A teacher has two large containers filled with blue, red, and green beads, and claims the proportion of red beads is the same for both containers. The students believe the proportions are different. Each student shakes the first container, selects 50 beads, counts the number of red beads, and returns the beads to the container. The student repeats this process for the second container. One student's samples contained 10 red beads from the first container and 16 red beads from the second container. Let p1= the true proportion of red beads in container 1 and p2= the true proportion of red beads in container 2. Which of the following is the correct standardized test statistic and p-value for the hypotheses?

Answer 1

The standardized test statistic for two independent sample proportions is calculated using a formula that includes the sample proportions from each container and the pooled sample proportion. The p-value is determined based on this statistic using the standard normal distribution.

Step-by-step explanation:

The problem presented involves hypothesis testing for two independent sample proportions, specifically comparing the true proportion of red beads in two containers. To compute the standardized test statistic, we will use the formula for comparing two proportions, z = (pˉ₁ - pˉ₂) / sqrt(p·c (1 - p·c) (1/n₁ + 1/n₂)), where p·c is the pooled sample proportion. We determine p·₁ and pˉ₂ from the sample, which are the proportions of red beads in the samples from containers 1 and 2 respectively. Since the exact numbers for the proportions and total bead count in each container aren't given, we cannot calculate the exact statistic and p-value here. However, once these are computed, the p-value is found using the standard normal distribution to see how extreme the test statistic is, considering the null hypothesis that the proportions are equal.