(i) In a study on household income in a small town, 64% of families have monthly household incomes of less than RM 2 500. Find a 90% confidence interval for the percentage of all families in the town with household incomes less than RM 2 500 per month if n = 550

To find a 90% confidence interval for the percentage of all families in the town with household incomes less than RM 2,500 per month, we use the formula for the confidence interval of a population proportion:

\[ \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:
- \(\hat{p}\) is the sample proportion (0.64)
- \(n\) is the sample size (550)
- \(Z_{\alpha/2}\) is the Z-value for the desired confidence level (90%)

For a 90% confidence level, the Z-value (\(Z_{\alpha/2}\)) is approximately 1.645.

### Step-by-Step Calculation:

1. **Calculate the standard error (SE):**

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.64(1 - 0.64)}{550}} = \sqrt{\frac{0.64 \times 0.36}{550}} = \sqrt{\frac{0.2304}{550}} = \sqrt{0.000419636} \approx 0.0205
\]

2. **Calculate the margin of error (ME):**

\[
ME = Z_{\alpha/2} \times SE = 1.645 \times 0.0205 \approx 0.0337
\]

3. **Calculate the confidence interval:**

\[
\hat{p} \pm ME = 0.64 \pm 0.0337
\]

This gives us:

\[
0.64 - 0.0337 = 0.6063
\]

\[
0.64 + 0.0337 = 0.6737
\]

### Confidence Interval:

The 90% confidence interval for the percentage of all families in the town with household incomes less than RM 2,500 per month is approximately:

\[ 60.63\% \text{ to } 67.37\% \]

Would you like more details or have any questions?

### Related Questions:

1. How would you interpret the confidence interval in the context of this study?
2. What is the meaning of a 90% confidence level?
3. How would the confidence interval change if the sample size was larger?
4. How would the confidence interval change if the sample proportion was different?
5. How can you use the confidence interval to make decisions about household income in the town?
6. What are some potential sources of error in this study?
7. How would you calculate a confidence interval for a different confidence level, say 95%?
8. What assumptions must be met for this confidence interval to be valid?

### Tip:
Remember that the width of the confidence interval depends on the sample size and the variability in the data. Larger sample sizes result in narrower confidence intervals.

Learn how to calculate a 90% confidence interval for the percentage of families with household incomes below RM 2,500 in a small town. This problem involves statistics and population proportion concepts, including the formula for confidence interval calculation. Suitable for university-level statistics students.

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