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]
