Suppose that 283 out of a random sample of 350 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 90% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed. Then find the lower limit and upper limit of the 90% confidence interval

We are asked to compute a 90% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed, based on a sample. Here are the steps:

### 1. **Determine the sample proportion:**
The sample proportion \(\hat{p}\) is calculated as:

\[
\hat{p} = \frac{x}{n}
\]

Where:
- \(x = 283\) (the number of letters delivered the day after being mailed)
- \(n = 350\) (the total sample size)

\[
\hat{p} = \frac{283}{350} \approx 0.8086
\]

### 2. **Find the critical value (z-value):**
For a 90% confidence interval, the critical value \(z_{\alpha/2}\) corresponds to the middle 90% of the standard normal distribution, leaving 5% in each tail. The critical value for 90% confidence is approximately:

\[
z_{\alpha/2} = 1.645
\]

### 3. **Compute the standard error:**
The standard error (SE) of the sample proportion is given by:

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Substitute the values:

\[
SE = \sqrt{\frac{0.8086(1 - 0.8086)}{350}} = \sqrt{\frac{0.8086 \times 0.1914}{350}} \approx 0.0217
\]

### 4. **Calculate the margin of error:**
The margin of error (ME) is the critical value multiplied by the standard error:

\[
ME = z_{\alpha/2} \times SE = 1.645 \times 0.0217 \approx 0.0357
\]

### 5. **Compute the confidence interval:**
The confidence interval is calculated as:

\[
\hat{p} \pm ME
\]

Lower limit:

\[
\hat{p} - ME = 0.8086 - 0.0357 \approx 0.7729
\]

Upper limit:

\[
\hat{p} + ME = 0.8086 + 0.0357 \approx 0.8443
\]

### Conclusion:
The 90% confidence interval for the proportion of letters delivered the day after being mailed is approximately:

\[
(0.7729, 0.8443)
\]

### Lower limit: 0.7729
### Upper limit: 0.8443

Would you like further details on any of the steps? 

Here are five related questions to deepen understanding:

1. How does increasing the sample size affect the confidence interval width?
2. Why is a 90% confidence interval wider than a 95% confidence interval?
3. What assumptions are made about the population when constructing this interval?
4. How would the confidence interval change if the proportion of letters delivered on time was lower?
5. How can the margin of error be reduced in future studies?

**Tip:** Confidence intervals estimate population parameters but do not guarantee the true parameter is inside the interval for any specific sample.

Suppose that 283 out of a random sample of 350 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 90% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed. Then find the lower limit and upper limit of the 90% confidence interval.

Learn how to calculate a 90% confidence interval for the proportion of letters delivered on time, based on a sample. This problem explores the key statistical concepts of confidence intervals, margin of error, and standard error. Suitable for high school statistics students.

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