To determine the 99% confidence interval for the true proportion of times the number cube would land with a six facing up, we can follow these steps:
1. Identify and state the given information:
- The number cube was rolled 45 times.
- It landed on six 12 times.
- We need to determine the 99% confidence interval for the true proportion of sixes.
2. Calculate the sample proportion:
The sample proportion ([tex]\(\hat{p}\)[/tex]) of times the cube landed on six is calculated by dividing the number of times it landed on six by the total number of rolls:
[tex]\[
\hat{p} = \frac{12}{45} = 0.2667
\][/tex]
3. Determine the critical value for a 99% confidence interval:
The z-value for a 99% confidence interval is 2.58.
4. Calculate the margin of error:
The margin of error (E) can be found using the formula:
[tex]\[
E = z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
Substituting the values:
[tex]\[
E = 2.58 \cdot \sqrt{\frac{0.2667 \times (1 - 0.2667)}{45}}
\][/tex]
5. Determine the lower and upper bounds:
- The lower bound of the confidence interval is:
[tex]\[
\hat{p} - E = 0.2667 - 0.1701 = 0.0966
\][/tex]
- The upper bound of the confidence interval is:
[tex]\[
\hat{p} + E = 0.2667 + 0.1701 = 0.4367
\][/tex]
6. Conclusion:
The 99% confidence interval for the true proportion of times the number cube would land with a six facing up is approximately from 0.0966 to 0.4367.
Therefore, the answer is:
[tex]\(0.27 \pm 2.58 \sqrt{\frac{0.27(1-0.27)}{45}}\)[/tex]
This matches the given options and confirms that the confidence interval is correctly calculated. The selection of the correct formula from the options provided also matches with:
[tex]\[
0.27 \pm 2.58 \sqrt{\frac{0.27(1-0.27)}{45}}
\][/tex]
This shows that the student's conclusion about the proportion being different (and potentially biased) is supported by this interval at a 99% confidence level.
