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The difference of two sample means is 22, and the standard deviation of the difference of the sample means is 10.
The difference of the means of the two populations at a 95% confidence interval is ±



Answer :

GinnyAnswer
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.

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