A hardware store surveyed 600 consumers and found that 414 were enthusiastic about new home décor they plan to show in their stores. Construct a 90% confidence interval for the proportion of individuals in the population who would be enthusiastic about the new décor.

To construct a 90% confidence interval for the proportion of individuals in the population who would be enthusiastic about the new décor, we follow these steps:

### 1. Identify the sample proportion (\( \hat{p} \))
Given:
- Number of enthusiastic consumers = 414
- Total number of consumers surveyed = 600

The sample proportion \( \hat{p} \) is calculated as:
\[
\hat{p} = \frac{414}{600} = 0.69
\]

### 2. Find the critical value (z-value)
For a 90% confidence level, the critical value \( z^* \) corresponds to the value such that the area under the standard normal curve is 0.90. This value is approximately 1.645.

### 3. Calculate the standard error (SE)
The standard error for the sample proportion is calculated using the formula:
\[
\text{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
Substituting the values:
\[
\text{SE} = \sqrt{\frac{0.69 \times (1 - 0.69)}{600}} = \sqrt{\frac{0.69 \times 0.31}{600}} \approx \sqrt{\frac{0.2139}{600}} \approx \sqrt{0.0003565} \approx 0.01888
\]

### 4. Calculate the margin of error (ME)
The margin of error is given by:
\[
\text{ME} = z^* \times \text{SE}
\]
Substituting the values:
\[
\text{ME} = 1.645 \times 0.01888 \approx 0.03106
\]

### 5. Construct the confidence interval
The confidence interval is calculated as:
\[
\hat{p} \pm \text{ME}
\]
Substituting the values:
\[
0.69 \pm 0.03106
\]

This gives us the interval:
\[
[0.65894, 0.72106]
\]

### Conclusion
The 90% confidence interval for the proportion of individuals in the population who would be enthusiastic about the new décor is approximately \([0.659, 0.721]\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the confidence interval change if the confidence level were 95%?
2. What happens to the confidence interval if the sample size increases?
3. How does the margin of error relate to the width of the confidence interval?
4. What assumptions are required for this confidence interval to be valid?
5. How would you interpret this confidence interval in the context of decision-making for the hardware store?

**Tip:** When calculating a confidence interval, the width of the interval depends on both the sample size and the confidence level chosen. Larger sample sizes and lower confidence levels result in narrower intervals.

Learn how to construct a 90% confidence interval for the proportion of individuals enthusiastic about new home décor based on survey data. This problem involves calculating sample proportions, standard errors, and margin of error, applying the Central Limit Theorem. Suitable for advanced high school or college students.

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