Question
Construct the 99.5% confidence interval for the difference pv1-pv2 when xv1=40, nv1=70, xv2=15 and nv2= 50. Round the answers to three decimal places.
Answers
GPT-4o mini
To construct the 99.5% confidence interval for the difference between two population proportions \( p_{v1} - p_{v2} \), we first need to calculate the sample proportions \( p_{v1} \) and \( p_{v2} \).
Given:
- \( x_{v1} = 40 \) (number of successes in sample 1)
- \( n_{v1} = 70 \) (size of sample 1)
- \( x_{v2} = 15 \) (number of successes in sample 2)
- \( n_{v2} = 50 \) (size of sample 2)
### Step 1: Calculate sample proportions
The sample proportions are calculated as follows:
\[
p_{v1} = \frac{x_{v1}}{n_{v1}} = \frac{40}{70} = 0.5714
\]
\[
p_{v2} = \frac{x_{v2}}{n_{v2}} = \frac{15}{50} = 0.3
\]
### Step 2: Calculate the difference in sample proportions
\[
\hat{p} = p_{v1} - p_{v2} = 0.5714 - 0.3 = 0.2714
\]
### Step 3: Calculate the standard error
The standard error (SE) for the difference in proportions is calculated using:
\[
SE = \sqrt{\frac{p_{v1}(1 - p_{v1})}{n_{v1}} + \frac{p_{v2}(1 - p_{v2})}{n_{v2}}}
\]
Calculating each component:
\[
SE = \sqrt{\frac{0.5714(1 - 0.5714)}{70} + \frac{0.3(1 - 0.3)}{50}}
\]
\[
= \sqrt{\frac{0.5714 \times 0.4286}{70} + \frac{0.3 \times 0.7}{50}}
\]
\[
= \sqrt{\frac{0.2453}{70} + \frac{0.21}{50}}
\]
\[
= \sqrt{0.003504 + 0.0042}
\]
\[
= \sqrt{0.007704} \approx 0.0878
\]
### Step 4: Find the critical value for 99.5% confidence interval
For a 99.5% confidence level, we look at the z-score corresponding to \( \alpha = 0.005 \) (two-tailed) which is \( z = 2.576 \) (using standard normal distribution).
### Step 5: Calculate the margin of error
\[
ME = z \times SE = 2.576 \times 0.0878 \approx 0.2262
\]
### Step 6: Construct the confidence interval
Now we construct the confidence interval for \( p_{v1} - p_{v2} \):
\[
(p_{v1} - p_{v2}) \pm ME
\]
Substituting the values:
\[
0.2714 \pm 0.2262
\]
Calculating the lower and upper bounds:
\[
Lower\:bound = 0.2714 - 0.2262 \approx 0.0452
\]
\[
Upper\:bound = 0.2714 + 0.2262 \approx 0.4976
\]
### Step 7: Final Result
Thus, the 99.5% confidence interval for \( p_{v1} - p_{v2} \) is approximately:
\[
(0.045, 0.498)
\]
Rounded to three decimal places, the final confidence interval is:
\[
\boxed{(0.045, 0.498)}
\]
Given:
- \( x_{v1} = 40 \) (number of successes in sample 1)
- \( n_{v1} = 70 \) (size of sample 1)
- \( x_{v2} = 15 \) (number of successes in sample 2)
- \( n_{v2} = 50 \) (size of sample 2)
### Step 1: Calculate sample proportions
The sample proportions are calculated as follows:
\[
p_{v1} = \frac{x_{v1}}{n_{v1}} = \frac{40}{70} = 0.5714
\]
\[
p_{v2} = \frac{x_{v2}}{n_{v2}} = \frac{15}{50} = 0.3
\]
### Step 2: Calculate the difference in sample proportions
\[
\hat{p} = p_{v1} - p_{v2} = 0.5714 - 0.3 = 0.2714
\]
### Step 3: Calculate the standard error
The standard error (SE) for the difference in proportions is calculated using:
\[
SE = \sqrt{\frac{p_{v1}(1 - p_{v1})}{n_{v1}} + \frac{p_{v2}(1 - p_{v2})}{n_{v2}}}
\]
Calculating each component:
\[
SE = \sqrt{\frac{0.5714(1 - 0.5714)}{70} + \frac{0.3(1 - 0.3)}{50}}
\]
\[
= \sqrt{\frac{0.5714 \times 0.4286}{70} + \frac{0.3 \times 0.7}{50}}
\]
\[
= \sqrt{\frac{0.2453}{70} + \frac{0.21}{50}}
\]
\[
= \sqrt{0.003504 + 0.0042}
\]
\[
= \sqrt{0.007704} \approx 0.0878
\]
### Step 4: Find the critical value for 99.5% confidence interval
For a 99.5% confidence level, we look at the z-score corresponding to \( \alpha = 0.005 \) (two-tailed) which is \( z = 2.576 \) (using standard normal distribution).
### Step 5: Calculate the margin of error
\[
ME = z \times SE = 2.576 \times 0.0878 \approx 0.2262
\]
### Step 6: Construct the confidence interval
Now we construct the confidence interval for \( p_{v1} - p_{v2} \):
\[
(p_{v1} - p_{v2}) \pm ME
\]
Substituting the values:
\[
0.2714 \pm 0.2262
\]
Calculating the lower and upper bounds:
\[
Lower\:bound = 0.2714 - 0.2262 \approx 0.0452
\]
\[
Upper\:bound = 0.2714 + 0.2262 \approx 0.4976
\]
### Step 7: Final Result
Thus, the 99.5% confidence interval for \( p_{v1} - p_{v2} \) is approximately:
\[
(0.045, 0.498)
\]
Rounded to three decimal places, the final confidence interval is:
\[
\boxed{(0.045, 0.498)}
\]