Suppose that 177

 out of a random sample of 250

 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 95

%

 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 95

%

 confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)

Lower limit:

Upper limit:





To compute the 95% confidence interval for the proportion of letters delivered the day after they were mailed, we use the formula for a confidence interval for a population proportion:

\[
CI = \hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion of success (letters delivered the next day),
- \(z\) is the z-score corresponding to the confidence level (for 95%, \(z = 1.96\)),
- \(n\) is the sample size.

### Step 1: Calculate the sample proportion (\(\hat{p}\)).

\[
\hat{p} = \frac{177}{250} = 0.708
\]

### Step 2: Calculate the standard error (SE).

\[
SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.708(1 - 0.708)}{250}} = \sqrt{\frac{0.708 \times 0.292}{250}} = \sqrt{\frac{0.206736}{250}} = \sqrt{0.000827} \approx 0.02876
\]

### Step 3: Calculate the margin of error (ME).

\[
ME = z \times SE = 1.96 \times 0.02876 = 0.05636
\]

### Step 4: Calculate the confidence interval.

\[
CI = \hat{p} \pm ME = 0.708 \pm 0.05636
\]

- **Lower limit** = \( 0.708 - 0.05636 = 0.6516 \) → Rounded to **0.65**
- **Upper limit** = \( 0.708 + 0.05636 = 0.7644 \) → Rounded to **0.76**

### Final Answer:
- **Lower limit**: 0.65
- **Upper limit**: 0.76

Would you like to see more details on any specific step or have any questions?

Here are some follow-up questions:
1. What is the importance of using a confidence interval?
2. How does increasing the sample size affect the confidence interval?
3. Why do we use the z-score in confidence intervals?
4. How would the confidence interval change with a different confidence level, like 99%?
5. How does the sample proportion impact the width of the confidence interval?

**Tip:** Always ensure that the sample size is sufficiently large when applying the formula for confidence intervals in proportion problems.

Suppose that 177 out of a random sample of 250 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 95% 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 95% confidence interval. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places.

Learn how to compute a 95% confidence interval for the proportion of letters delivered the next day, based on a sample of 250 letters, with 177 arriving within 24 hours. This problem covers concepts like sample proportion, standard error, and margin of error. Suitable for high school statistics students.

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