In a survey of 729 lesbian, gay, bisexual, or transgender adults in a certain country, 410 said that they consider themselves bisexual.
Construct a 99% 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 99% nfidence interval for the population proportion is ( , ).
(Round to three decimal places as needed.)
Interpret your results. Choose the correct answer below.
A. With confidence, it can be said that the population proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between the endpoints of the given confidence interval.
B. The endpoints of the given confidence interval show that of lesbian, gay, bisexual, or transgender adults consider themselves bisexual.
C. With probability, the population proportion of lesbian, gay, bisexual, or transgender adults who do not consider themselves bisexual is between the endpoints of the given confidence interval.
D. With confidence, it che said that the sample proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between the endpoints of the given confidence interval.

Ask by Haynes Ross. in the United States

Mar 26,2025

Question

In a survey of 729 lesbian, gay, bisexual, or transgender adults in a certain country, 410 said that they consider themselves bisexual.
Construct a 99% 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 99% nfidence interval for the population proportion is ( , ).
(Round to three decimal places as needed.)
Interpret your results. Choose the correct answer below.
A. With confidence, it can be said that the population proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between the endpoints of the given confidence interval.
B. The endpoints of the given confidence interval show that of lesbian, gay, bisexual, or transgender adults consider themselves bisexual.
C. With probability, the population proportion of lesbian, gay, bisexual, or transgender adults who do not consider themselves bisexual is between the endpoints of the given confidence interval.
D. With confidence, it che said that the sample proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between the endpoints of the given confidence interval.

Ask by Haynes Ross. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

Let 
$$
n=729,\quad x=410,\quad \hat{p}=\frac{410}{729}\approx 0.563.
$$

**Step 1. Find the sample proportion**

$$
\hat{p}=\frac{410}{729}\approx 0.563.
$$

**Step 2. Compute the standard error**

The standard error (SE) for a proportion is given by

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

First, compute $$\hat{p}(1-\hat{p})$$:

$$
0.563\,(1-0.563)=0.563\times 0.437\approx 0.246.
$$

Then,

$$
\mathrm{SE}=\sqrt{\frac{0.246}{729}}\approx \sqrt{0.000337}\approx 0.0184.
$$

**Step 3. Identify the critical value**

For a $$99\%$$ confidence interval, the critical value for a two-tailed test is approximately

$$
z^*=2.576.
$$

**Step 4. Compute the margin of error**

The margin of error $$\mathrm{E}$$ is

$$
\mathrm{E}=z^*\times \mathrm{SE}\approx 2.576\times 0.0184\approx 0.0474.
$$

Rounded to three decimal places,

$$
\mathrm{E}\approx 0.047.
$$

**Step 5. Construct the confidence interval**

The confidence interval is given by

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

This yields

$$
(0.563-0.047,\ 0.563+0.047)=(0.516,\ 0.610).
$$

Rounded to three decimal places, the interval is

$$
(0.516, 0.610).
$$

**Step 6. Interpretation**

The correct interpretation is:

A. With $$99\%$$ confidence, it can be said that the population proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between the endpoints of the given confidence interval.

Thus, the answers are:

- The margin of error: $$\mathrm{E} \approx 0.047$$.
- A $$99\%$$ confidence interval for the population proportion is $$(0.516, 0.610)$$.
The margin of error is approximately 0.047. A 99% confidence interval for the population proportion is (0.516, 0.610). Interpretation: With 99% confidence, it can be said that the population proportion of lesbian, gay, bisexual, or transgender adults who consider themselves bisexual is between 0.516 and 0.610.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Statistics Questions