To calculate the 95% confidence interval for the difference of two sample means, you can follow these steps:
### Step 1: Identify the given values
- Difference of sample means ([tex]\( \bar{X}_1 - \bar{X}_2 \)[/tex]): 22
- Standard deviation of the difference of sample means ([tex]\( \sigma_{\bar{X}_1 - \bar{X}_2} \)[/tex]): 10
- Confidence level: 95%
### Step 2: Determine the critical value (z-score) for a 95% confidence interval
A 95% confidence interval corresponds to a z-score that captures the middle 95% of the distribution. This is the value of [tex]\( z \)[/tex] such that:
[tex]\[ P\left( -z < Z < z \right) = 0.95 \][/tex]
Since we have a two-tailed test (dividing the 5% equally on both tails of the distribution), we need to find the z-score for 97.5% ([tex]\(1 - 0.025\)[/tex]) of the cumulative distribution function (CDF).
The critical value (z-score) from standard normal distribution tables for 97.5% is approximately:
[tex]\[ z = 1.96 \][/tex]
### Step 3: Calculate the margin of error (ME)
The margin of error can be calculated using the formula:
[tex]\[ ME = z \times \sigma_{\bar{X}_1 - \bar{X}_2} \][/tex]
Substitute the known values:
[tex]\[ ME = 1.96 \times 10 \][/tex]
[tex]\[ ME = 19.6 \][/tex]
### Step 4: Determine the confidence interval
Now, we can calculate the lower and upper bounds of the confidence interval using the following formulas:
- Lower bound: [tex]\( \text{Lower bound} = (\bar{X}_1 - \bar{X}_2) - \text{ME} \)[/tex]
- Upper bound: [tex]\( \text{Upper bound} = (\bar{X}_1 - \bar{X}_2) + \text{ME} \)[/tex]
Substitute the known values:
- Lower bound: [tex]\( 22 - 19.6 = 2.4 \)[/tex]
- Upper bound: [tex]\( 22 + 19.6 = 41.6 \)[/tex]
### Step 5: State the confidence interval
The 95% confidence interval for the difference of the means of the two populations is:
[tex]\[ (2.4, 41.6) \][/tex]
