From a large number of actuarial exam scores, a random sample of 300 scores is selected, and it is found that 225 of these 300 are passing scores. Based on this sample, find a 95% confidence interval for the proportion of all scores that are passing. Then find the lower limit and upper limit of the 95% confidence interval.
Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)

Lower limit:	
 
Upper limit:	


To find a 95% confidence interval for the population proportion \(p\) of passing scores, we can use the following formula:

\[
\hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \(\hat{p}\) = sample proportion = \( \frac{225}{300} \)
- \(n = 300\) is the sample size
- \(Z\) is the Z-value for a 95% confidence level, which is 1.96.

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### Step 1: Calculate the sample proportion \(\hat{p}\)
\[
\hat{p} = \frac{225}{300} = 0.750
\]

### Step 2: Calculate the standard error (SE)
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.750(1 - 0.750)}{300}}
\]
\[
SE = \sqrt{\frac{0.750 \times 0.250}{300}} = \sqrt{\frac{0.1875}{300}} = \sqrt{0.000625} = 0.025
\]

### Step 3: Calculate the margin of error (ME)
\[
ME = Z \times SE = 1.96 \times 0.025 = 0.049
\]

### Step 4: Calculate the confidence interval
\[
\hat{p} \pm ME = 0.750 \pm 0.049
\]

- **Lower limit**: \(0.750 - 0.049 = 0.701\)
- **Upper limit**: \(0.750 + 0.049 = 0.799\)

---

### Final Answers:
- **Lower limit**: \(0.70\)  
- **Upper limit**: \(0.80\)  

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Would you like further clarifications or have any additional questions?  

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Here are 5 related questions you could explore:  
1. What happens to the confidence interval if the confidence level is increased to 99%?  
2. How does sample size impact the width of the confidence interval?  
3. What is the interpretation of a confidence interval in the context of a population proportion?  
4. Can we assume normality when constructing confidence intervals for proportions?  
5. How would the interval change if only 150 scores were passing instead of 225?  

**Tip:** Confidence intervals provide a range of plausible values for a population parameter, but they are not guarantees!

From a large number of actuarial exam scores, a random sample of 300 scores is selected, and it is found that 225 of these 300 are passing scores. Based on this sample, find a 95% confidence interval for the proportion of all scores that are passing. Then find the lower limit and upper limit of the 95% confidence interval. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places.

Find a 95% confidence interval for the proportion of passing scores from a sample of actuarial exam results. This solution explains step-by-step how to compute the sample proportion, standard error, and margin of error, with final answers rounded to two decimal places. Ideal for undergraduate statistics students.

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