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A student believes that a certain number cube is unfair and is more likely to land with a six facing up. The student rolls the number cube 45 times, and the cube lands with a six facing up 12 times. Assuming the conditions for inference have been met, what is the 99% confidence interval for the true proportion of times the number cube would land with a six facing up?

A. [tex]\(0.27 \pm 2.58 \sqrt{\frac{0.27(1-0.27)}{45}}\)[/tex]
B. [tex]\(0.73 \pm 2.33 \sqrt{\frac{0.73(1-0.73)}{45}}\)[/tex]
C. [tex]\(0.27 \pm 2.33 \sqrt{\frac{0.27(1-0.27)}{45}}\)[/tex]
D. [tex]\(0.73 \pm 2.58 \sqrt{\frac{0.73(1-0.73)}{45}}\)[/tex]

Sagot :

GinnyAnswer
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.
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