Sure! You are trying to calculate the 95% confidence interval for the proportion of students who have a pet. Here’s how to do it step by step.

Step 1: Identify the key variables

• Sample size (n) = 250 students
• Number of students with pets (x) = 197 students
• Sample proportion (p̂) = $\frac{x}{n} = \frac{197}{250} = 0.788$

Step 2: Find the z-score for a 95% confidence level

For a 95% confidence interval, the z-score corresponding to 95% confidence is approximately 1.96. This comes from the standard normal distribution.

Step 3: Calculate the standard error (SE) for the proportion

The standard error (SE) of a proportion is given by the formula:

$SE = \sqrt{\frac{p̂(1 - p̂)}{n}}$

Substitute the values:

$SE = \sqrt{\frac{0.788(1 - 0.788)}{250}} = \sqrt{\frac{0.788 \times 0.212}{250}} \approx \sqrt{\frac{0.167}{250}} \approx \sqrt{0.000668} \approx 0.0258$

Step 4: Calculate the margin of error (ME)

The margin of error is:

$ME = z \times SE = 1.96 \times 0.0258 \approx 0.0505$

Step 5: Calculate the confidence interval

Now we can calculate the confidence interval:

$CI = p̂ \pm ME = 0.788 \pm 0.0505$

So, the 95% confidence interval for the proportion of students who have a pet is approximately:

$(0.7375, 0.8385)$

Step 6: Interpret the results

This means we are 95% confident that the true proportion of students who have a pet is between 73.75% and 83.85%.

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Would you like further details on any of the steps or have additional questions? Here's a set of related questions for you:

1. How does changing the confidence level affect the margin of error?
2. What happens to the confidence interval if the sample size is increased?
3. How would you calculate the confidence interval if you had multiple categories (e.g., more than one pet type)?
4. What are the assumptions behind using a normal approximation for the confidence interval?
5. How do you interpret the margin of error in practical terms for surveys?

Tip: Increasing your sample size decreases the margin of error, making your confidence interval more precise.