In a survey of

32773277

​adults,

15001500

say they have started paying bills online in the last year.

Construct a​ 99% confidence interval for the population proportion. Interpret the results.

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Part 1

A​ 99% confidence interval for the population proportion is

left parenthesis 0.435 comma 0.480 right parenthesis0.4350.435,0.4800.480.

​(Round to three decimal places as​ needed.)

Part 2

Interpret your results. Choose the correct answer below.

A.

With​ 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.

B.

The endpoints of the given confidence interval show that adults pay bills online​ 99% of the time.

C.

With​ 99% confidence, it can be said that the sample proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.

### Part 1: Constructing the 99% Confidence Interval

To construct a 99% confidence interval for the population proportion, we can use the following formula for a confidence interval for a population proportion \( p \):

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

Where:
- \( \hat{p} \) is the sample proportion (calculated as the number of successes divided by the total sample size),
- \( Z_{\alpha/2} \) is the critical value corresponding to the 99% confidence level (which is 2.576 for a 99% confidence interval),
- \( n \) is the sample size.

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

\[
\hat{p} = \frac{1500}{32773} \approx 0.04578
\]

#### Step 2: Apply the confidence interval formula

The margin of error is calculated as:

\[
\text{Margin of Error} = Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 2.576 \cdot \sqrt{\frac{0.04578(1-0.04578)}{32773}}
\]

First, compute the standard error:

\[
\text{Standard Error} = \sqrt{\frac{0.04578(1-0.04578)}{32773}} \approx 0.00116
\]

Then, compute the margin of error:

\[
\text{Margin of Error} = 2.576 \cdot 0.00116 \approx 0.00299
\]

#### Step 3: Compute the confidence interval

Now, calculate the lower and upper bounds of the confidence interval:

\[
\text{Lower Bound} = \hat{p} - \text{Margin of Error} = 0.04578 - 0.00299 \approx 0.043
\]
\[
\text{Upper Bound} = \hat{p} + \text{Margin of Error} = 0.04578 + 0.00299 \approx 0.049
\]

Thus, the 99% confidence interval is approximately \( (0.043, 0.049) \).

### Part 2: Interpretation of Results

The correct interpretation of the confidence interval is:

**A.** With​ 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval.

This interpretation correctly refers to the population proportion being estimated, as the confidence interval provides a range for the true proportion of all adults in the population who have started paying bills online.

### Follow-up Questions:

1. What is the formula for the margin of error in a confidence interval for a proportion?
2. Why do we use a Z-score in the calculation of the confidence interval?
3. How does increasing the confidence level affect the width of the confidence interval?
4. What would happen to the confidence interval if the sample size were larger?
5. How would you interpret a 90% confidence interval instead of a 99% one?

**Tip:** Always double-check your rounding when interpreting statistical results, especially when it's a key factor like confidence intervals.

In a survey of 32,773 adults, 1,500 say they have started paying bills online in the last year. Construct a 99% confidence interval for the population proportion. Interpret the results.

Learn how to construct and interpret a 99% confidence interval for the population proportion from survey data. In this example, 1,500 out of 32,773 adults reported starting to pay bills online. Explore the steps involved in calculating the interval using statistical formulas, along with an interpretation of the results.

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