To determine the 95% confidence interval for the difference of the means of two populations, follow these steps:
1. Identify the given information:
- Difference of sample means ([tex]\(\bar{X}_1 - \bar{X}_2\)[/tex]) = 22
- Standard deviation of the difference ([tex]\(s_{\bar{X}_1 - \bar{X}_2}\)[/tex]) = 10
- Confidence level = 95%
2. Find the Z-critical value for a 95% confidence interval:
- For a 95% confidence level, the Z-critical value can be found using standard normal distribution tables, or it is commonly known to be approximately 1.96.
3. Calculate the margin of error (ME):
- Margin of error = [tex]\( Z_{\text{critical}} \times s_{\bar{X}_1 - \bar{X}_2} \)[/tex]
- Here, [tex]\( Z_{\text{critical}} = 1.96 \)[/tex] and [tex]\( s_{\bar{X}_1 - \bar{X}_2} = 10 \)[/tex]
- Therefore, [tex]\( \text{ME} = 1.96 \times 10 = 19.6 \)[/tex]
4. Interpret the margin of error in the context of the confidence interval:
- The confidence interval for the difference of means is calculated as:
[tex]\[
\left( \bar{X}_1 - \bar{X}_2 \right) \pm \text{ME}
\][/tex]
- Substitute the values:
[tex]\[
22 \pm 19.6
\][/tex]
Thus, the difference of the means of the two populations at a 95% confidence interval is ± 19.6.
