To solve this problem, let's break down each part using the information provided.

Given:

• Sample size, $n = 400$
• Sample proportion, $\hat{p} = 0.77$ (or 77% who like dogs)
• Confidence level, $95\%$

The Theory-Based Inference applet typically uses the standard normal distribution for calculating the confidence interval for a proportion.

Step-by-Step Solution:

Part (a): Confidence Interval Calculation

The formula for a 95% confidence interval for a proportion $p$ is:

$\hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p} = 0.77$
• $z = 1.96$ for a 95% confidence level (from the standard normal table)
• $n = 400$

Let's calculate the margin of error first:

$\text{Margin of Error} = 1.96 \cdot \sqrt{\frac{0.77 \times (1 - 0.77)}{400}}$

Calculating the term inside the square root:

$\sqrt{\frac{0.77 \times 0.23}{400}} \approx 0.0203$

Then:

$\text{Margin of Error} = 1.96 \times 0.0203 \approx 0.0398$

Now, the confidence interval is:

$0.77 \pm 0.0398$

So:

• Lower limit = $0.77 - 0.0398 \approx 0.7302$
• Upper limit = $0.77 + 0.0398 \approx 0.8098$

To express this in percentage terms:

• Lower limit: $73.0\%$
• Upper limit: $81.0\%$

Answer for part (a): We are 95% confident that the proportion of all people who like dogs is between 73.0% and 81.0%.

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Part (b): The Sample Statistic

The sample statistic is simply the sample proportion $\hat{p}$.

Answer for part (b): $\hat{p} = 0.77$

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Part (c): Margin of Error

We've already calculated the margin of error in part (a) as part of the confidence interval calculation.

Answer for part (c): The margin of error is $0.0398$.

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Part (d): Confidence Interval as $\hat{p} \pm \text{Margin of Error}$

Using the values from parts (b) and (c):

Answer for part (d): The confidence interval can be written as $0.77 \pm 0.0398$.

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Summary of Answers:

• (a) 95% confidence interval for the proportion: 73.0% and 81.0%
• (b) Sample statistic: 0.77
• (c) Margin of error: 0.0398
• (d) Confidence interval in form $\hat{p} \pm \text{Margin of Error}$: 0.77 ± 0.0398

Would you like more details on any step? Here are some related questions:

1. What is the significance of the 95% confidence level?
2. How would the confidence interval change if the sample size were larger?
3. What is the purpose of the margin of error in a confidence interval?
4. Why is a z-value of 1.96 used for a 95% confidence level?
5. How would you interpret the confidence interval in practical terms?

Tip: When calculating confidence intervals, always check that sample sizes are large enough for the approximation (using the rule of thumb $np \geq 10$ and $n(1 - p) \geq 10$).