A random sample of 33 meals offered at the Macquarie Centre is selected and tasted by a food critic, and 16 are found to be delicious.

Compute a 95% confidence interval for the true proportion of delicious meals among those sold at the Macquarie Centre. Give your final answer with at least 33 dp.

To compute a 95% confidence interval for the true proportion of delicious meals at the Macquarie Centre, we can follow these steps:

### Step 1: Identify the sample proportion (\( \hat{p} \))
The sample proportion \( \hat{p} \) is calculated as:
\[
\hat{p} = \frac{x}{n}
\]
where:
- \( x = 16 \) is the number of delicious meals.
- \( n = 33 \) is the total number of meals sampled.

\[
\hat{p} = \frac{16}{33} \approx 0.484848484848484848484848484848485
\]

### Step 2: Determine the critical value for a 95% confidence interval
For a 95% confidence level, the critical value \( z^* \) from the standard normal distribution is approximately 1.96.

### Step 3: Calculate the standard error (SE)
The standard error \( SE \) is given by:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substituting the values:
\[
SE = \sqrt{\frac{0.484848484848484848484848484848485 \times (1 - 0.484848484848484848484848484848485)}{33}}
\]

### Step 4: Compute the margin of error (ME)
The margin of error is given by:
\[
ME = z^* \times SE
\]

### Step 5: Calculate the confidence interval
The confidence interval is given by:
\[
\hat{p} \pm ME
\]

Let's go through the calculations.The 95% confidence interval for the true proportion of delicious meals at the Macquarie Centre is approximately \([0.314330726954189600000000000000000, 0.655366242742780100000000000000000]\). This means we are 95% confident that the true proportion of delicious meals lies within this interval.

Do you want any more details or have any questions? 

Here are 5 related questions:
1. How would the confidence interval change with a larger sample size?
2. What happens to the confidence interval if the confidence level is increased to 99%?
3. How can we interpret the margin of error in this context?
4. How does the choice of \( z^* \) affect the confidence interval?
5. What assumptions are we making about the sampling distribution in this calculation?

**Tip:** The width of the confidence interval is influenced by both the sample size and the confidence level. Larger samples and lower confidence levels typically result in narrower intervals.

Learn how to compute a 95% confidence interval for the true proportion of delicious meals at the Macquarie Centre using statistical methods. This problem involves estimating proportions, calculating standard errors, and understanding confidence intervals in statistics.

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