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Let's solve the question step-by-step.

### Given Information:
- Total candies sampled (\( n \)) = 43
- Number of red candies in the sample (\( x \)) = 9
- Claimed percentage of red candies = 29%
- Confidence level = 90%

### Step 1: Calculate the Sample Proportion
The sample proportion (\( \hat{p} \)) of red candies can be calculated as:
\[
\hat{p} = \frac{x}{n} = \frac{9}{43} \approx 0.2093
\]

### Step 2: Find the Z-score for a 90% Confidence Interval
For a 90% confidence interval, the Z-score (\( z \)) is approximately 1.645.

### Step 3: Calculate the Margin of Error (E)
The margin of error \( E \) can be calculated using the formula:
\[
E = z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substitute the values:
\[
E = 1.645 \times \sqrt{\frac{0.2093 \times (1 - 0.2093)}{43}}
\]
\[
E \approx 1.645 \times \sqrt{\frac{0.2093 \times 0.7907}{43}}
\]
\[
E \approx 1.645 \times \sqrt{\frac{0.1656}{43}}
\]
\[
E \approx 1.645 \times \sqrt{0.00385}
\]
\[
E \approx 1.645 \times 0.062
\]
\[
E \approx 0.102
\]

### Step 4: Construct the Confidence Interval
The 90% confidence interval for the population proportion \( p \) is:
\[
\hat{p} - E < p < \hat{p} + E
\]
\[
0.2093 - 0.102 < p < 0.2093 + 0.102
\]
\[
0.1073 < p < 0.3113
\]

### Conclusion
The 90% confidence interval estimate of the percentage of red candies is approximately **10.7% to 31.1%**. Since the company's claim of 29% falls within this interval, we do not have enough evidence to reject the claim.

Would you like further details on any part of this calculation?

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### Follow-Up Questions:
1. How would the interval change with a higher confidence level (e.g., 95%)?
2. What would be the margin of error if the sample size were doubled?
3. How does the Z-score affect the width of the confidence interval?
4. What are the implications if the confidence interval did not contain the claimed percentage?
5. How is the sample proportion an estimate of the population proportion?

### Tip:
Remember, increasing the sample size reduces the margin of error, leading to a narrower confidence interval.

The data given includes information from 43 candies, with 9 of them being red. The company claims that 29% of its candies are red. Use the sample data to construct a 90% confidence interval estimate of the percentage of red candies. What do you conclude about the claim of 29%?

Learn how to construct a 90% confidence interval for the percentage of red candies in a sample of 43, with a company-claimed red candy percentage of 29%. Includes calculations using the sample proportion and z-score for 90% confidence.

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