Homework: HW QR GM

In a survey of 2769 adults, 1483 say they have started paying bills online in the last year.
Construct a confidence interval for the population proportion. Interpret the results. Being by finding the margin of error.

The margin of error for the confidence interval for the population proportion is . (Round to three decimal places as needed.)

A confidence interval for the population proportion is . ). (Round to three decimal places as needed.)
Interpret your results. Choose the correct answer below.
A. The endpoints of the given confidence interval show that adults pay bills online of the time.
B. With 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.
C. With confidence, it ch 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.

Ask by Mitchell Cervantes. in the United States

Mar 26,2025

Question

Homework: HW QR GM

In a survey of 2769 adults, 1483 say they have started paying bills online in the last year.
Construct a confidence interval for the population proportion. Interpret the results. Being by finding the margin of error.

The margin of error for the confidence interval for the population proportion is . (Round to three decimal places as needed.)

A confidence interval for the population proportion is . ). (Round to three decimal places as needed.)
Interpret your results. Choose the correct answer below.
A. The endpoints of the given confidence interval show that adults pay bills online of the time.
B. With 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.
C. With confidence, it ch 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.

Ask by Mitchell Cervantes. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

**Step 1. Calculate the Sample Proportion**

The number of adults in the survey is $$ n = 2769 $$ and the number of adults who started paying bills online is $$ x = 1483 $$. The sample proportion is

$$
\hat{p} = \frac{x}{n} = \frac{1483}{2769} \approx 0.536.
$$

**Step 2. Determine the $$ z $$-value for a $$ 99\% $$ Confidence Level**

For a $$ 99\% $$ confidence level, the critical value is

$$
z^{*} \approx 2.576.
$$

**Step 3. Compute the Standard Error**

The standard error (SE) for the population proportion is calculated using

$$
\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.
$$

Substitute the values:

$$
\text{SE} = \sqrt{\frac{0.536(1-0.536)}{2769}} = \sqrt{\frac{0.536 \times 0.464}{2769}} \approx \sqrt{\frac{0.248704}{2769}} \approx \sqrt{0.00008975} \approx 0.00949.
$$

**Step 4. Calculate the Margin of Error**

The margin of error is

$$
E = z^{*} \cdot \text{SE} \approx 2.576 \times 0.00949 \approx 0.024.
$$

(Rounded to three decimal places, $$ E = 0.024 $$.)

**Step 5. Construct the $$ 99\% $$ Confidence Interval**

The confidence interval is given by

$$
\hat{p} \pm E.
$$

Thus, the endpoints are:

$$
\text{Lower bound} = 0.536 - 0.024 = 0.512,
$$
$$
\text{Upper bound} = 0.536 + 0.024 = 0.560.
$$

A $$ 99\% $$ confidence interval for the population proportion is

$$
(0.512, 0.560).
$$

**Step 6. Interpretation**

The correct interpretation is:

C. 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.
**Margin of Error:** 0.024 **99% Confidence Interval:** (0.512, 0.560) **Interpretation:** With 99% confidence, it can be said that the population proportion of adults who have started paying bills online in the last year is between 51.2% and 56.0%.

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