A teacher has two large containers filled with blue, red, and green beads. He wants his students to estimate the difference in the proportion of red beads in each container. Each student shakes the first container, randomly selects 50 beads, counts the number of red beads, and returns the beads to the container. The student repeats this for the second container. One student sampled 13 red beads from the first container and 16 red beads from the second container.

Assuming the conditions for inference are met, what is the 95% confidence interval for the difference in the proportions of red beads in each container?

A. [tex]$(0.26 - 0.32) \pm 1.96 \sqrt{\frac{0.26(1-0.26)}{50} + \frac{0.32(1-0.32)}{50}}$[/tex]

B. [tex]$(0.26 - 0.32) \pm 1.65 \sqrt{\frac{0.26(1-0.26)}{50} + \frac{0.32(1-0.32)}{50}}$[/tex]

C. [tex]$(0.74 - 0.68) \pm 1.96 \sqrt{\frac{0.74(1-0.74)}{50} + \frac{0.68(1-0.68)}{50}}$[/tex]

D. [tex]$(0.74 - 0.68) \pm 1.65 \sqrt{\frac{0.74(1-0.74)}{50} + \frac{0.68(1-0.68)}{50}}$[/tex]

To find the 95% confidence interval for the difference in the proportions of red beads in each container, follow these steps:

1. Calculate the sample proportions:
- The first sample has 13 red beads out of 50, so the proportion [tex]$ \hat{p}_1 $[/tex] is:
[tex]\[ \hat{p}_1 = \frac{13}{50} = 0.26 \][/tex]

- The second sample has 16 red beads out of 50, so the proportion [tex]$ \hat{p}_2 $[/tex] is:
[tex]\[ \hat{p}_2 = \frac{16}{50} = 0.32 \][/tex]

2. Calculate the difference in proportions:
[tex]\[ \hat{p}_1 - \hat{p}_2 = 0.26 - 0.32 = -0.06 \][/tex]

3. Calculate the standard error of the difference in proportions:
[tex]\[ \text{Standard Error} = \sqrt{\frac{0.26 \times (1 - 0.26)}{50} + \frac{0.32 \times (1 - 0.32)}{50}} \][/tex]

Let's compute each part inside the square root separately:

- For [tex]$ \hat{p}_1 $[/tex]:
[tex]\[ \frac{0.26 \times 0.74}{50} = \frac{0.1924}{50} = 0.003848 \][/tex]

- For [tex]$ \hat{p}_2 $[/tex]:
[tex]\[ \frac{0.32 \times 0.68}{50} = \frac{0.2176}{50} = 0.004352 \][/tex]

Adding these together:
[tex]\[ 0.003848 + 0.004352 = 0.0082 \][/tex]

Taking the square root:
[tex]\[ \sqrt{0.0082} \approx 0.09055 \][/tex]

4. Determine the margin of error using the Z-value for a 95% confidence interval (which is 1.96):
[tex]\[ \text{Margin of Error} = 1.96 \times 0.09055 \approx 0.17749 \][/tex]

5. Calculate the confidence interval:
- Lower limit:
[tex]\[ -0.06 - 0.17749 \approx -0.23749 \][/tex]

- Upper limit:
[tex]\[ -0.06 + 0.17749 \approx 0.11749 \][/tex]

Therefore, the 95% confidence interval for the difference in proportions of red beads in the two containers is approximately:
[tex]\[ (-0.23749, 0.11749) \][/tex]