Math / Data & Statistics

QuestionFrom a random sample of 80 women, 48 say they have been sexually harassed. From another random sample of 65 men, 13 say they have been sexually harassed. Construct a $99 \%$ confidence interval for the difference between the proportions of women and men who say they have been sexually harassed.

Studdy Solution

STEP 1

1. The samples are random and independent.
2. The sample sizes are large enough for the normal approximation to be valid.
3. We are using a two-sample proportion confidence interval.
4. The confidence level is 99%.

STEP 2

1. Calculate the sample proportions for women and men.
2. Calculate the pooled proportion.
3. Calculate the standard error of the difference in proportions.
4. Determine the critical value for 99% confidence level.
5. Calculate the margin of error.
6. Construct the confidence interval.

STEP 3

Calculate the sample proportions for women and men:
For women: $\hat{p}_w = \frac{48}{80} = 0.6$ For men: $\hat{p}_m = \frac{13}{65} = 0.2$

STEP 4

Calculate the pooled proportion:
$\hat{p} = \frac{n_w\hat{p}_w + n_m\hat{p}_m}{n_w + n_m} = \frac{80(0.6) + 65(0.2)}{80 + 65} = \frac{48 + 13}{145} = \frac{61}{145} \approx 0.4207$

STEP 5

Calculate the standard error of the difference in proportions:
$SE = \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_w} + \frac{1}{n_m})}$ $SE = \sqrt{0.4207(1-0.4207)(\frac{1}{80} + \frac{1}{65})}$ $SE \approx 0.0816$

STEP 6

Determine the critical value for 99% confidence level:
For a 99% confidence level, the z-score is 2.576.

STEP 7

Calculate the margin of error:
$ME = z_{0.995} \times SE = 2.576 \times 0.0816 \approx 0.2100$

STEP 8

Construct the confidence interval:
$(\hat{p}_w - \hat{p}_m) \pm ME$ $(0.6 - 0.2) \pm 0.2100$ $0.4 \pm 0.2100$
Lower bound: $0.4 - 0.2100 = 0.1900$ Upper bound: $0.4 + 0.2100 = 0.6100$
The 99% confidence interval for the difference between the proportions of women and men who say they have been sexually harassed is:
$\boxed{(0.1900, 0.6100)}$
This means we can be 99% confident that the true difference in proportions of women and men who have been sexually harassed is between 0.1900 and 0.6100.

Was this helpful?

Studdy Solution

STEP 1

1. The samples are random and independent.
2. The sample sizes are large enough for the normal approximation to be valid.
3. We are using a two-sample proportion confidence interval.
4. The confidence level is 99%.

STEP 2

1. Calculate the sample proportions for women and men.
2. Calculate the pooled proportion.
3. Calculate the standard error of the difference in proportions.
4. Determine the critical value for 99% confidence level.
5. Calculate the margin of error.
6. Construct the confidence interval.

STEP 3

Calculate the sample proportions for women and men:
For women: $\hat{p}_w = \frac{48}{80} = 0.6$ For men: $\hat{p}_m = \frac{13}{65} = 0.2$

STEP 4

Calculate the pooled proportion:
$\hat{p} = \frac{n_w\hat{p}_w + n_m\hat{p}_m}{n_w + n_m} = \frac{80(0.6) + 65(0.2)}{80 + 65} = \frac{48 + 13}{145} = \frac{61}{145} \approx 0.4207$

STEP 5

Calculate the standard error of the difference in proportions:
$SE = \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_w} + \frac{1}{n_m})}$ $SE = \sqrt{0.4207(1-0.4207)(\frac{1}{80} + \frac{1}{65})}$ $SE \approx 0.0816$

STEP 6

Determine the critical value for 99% confidence level:
For a 99% confidence level, the z-score is 2.576.

STEP 7

Calculate the margin of error:
$ME = z_{0.995} \times SE = 2.576 \times 0.0816 \approx 0.2100$

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The samples are random and independent.2. The sample sizes are large enough for the normal approximation to be valid.3. We are using a two-sample proportion confidence interval.4. The confidence level is 99%.

STEP 2

1. Calculate the sample proportions for women and men.2. Calculate the pooled proportion.3. Calculate the standard error of the difference in proportions.4. Determine the critical value for 99% confidence level.5. Calculate the margin of error.6. Construct the confidence interval.

STEP 3

Calculate the sample proportions for women and men:For women: p^w=4880=0.6 \hat{p}_w = \frac{48}{80} = 0.6 p^​w​=8048​=0.6 For men: p^m=1365=0.2 \hat{p}_m = \frac{13}{65} = 0.2 p^​m​=6513​=0.2

STEP 4

STEP 5

STEP 6

Determine the critical value for 99% confidence level:For a 99% confidence level, the z-score is 2.576.

STEP 7

Calculate the margin of error:ME=z0.995×SE=2.576×0.0816≈0.2100 ME = z_{0.995} \times SE = 2.576 \times 0.0816 \approx 0.2100 ME=z0.995​×SE=2.576×0.0816≈0.2100

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The samples are random and independent.
2. The sample sizes are large enough for the normal approximation to be valid.
3. We are using a two-sample proportion confidence interval.
4. The confidence level is 99%.

1. Calculate the sample proportions for women and men.
2. Calculate the pooled proportion.
3. Calculate the standard error of the difference in proportions.
4. Determine the critical value for 99% confidence level.
5. Calculate the margin of error.
6. Construct the confidence interval.

Calculate the sample proportions for women and men:
For women: $\hat{p}_w = \frac{48}{80} = 0.6$ For men: $\hat{p}_m = \frac{13}{65} = 0.2$

Determine the critical value for 99% confidence level:
For a 99% confidence level, the z-score is 2.576.

Calculate the margin of error:
$ME = z_{0.995} \times SE = 2.576 \times 0.0816 \approx 0.2100$