To find the 94% confidence interval for the difference between the two population means [tex]\((\mu_1 - \mu_2)\)[/tex], we will follow these steps:
1. Calculate the Standard Error (SE):
The standard error for the difference between two means is calculated using the formula:
[tex]\[
SE = \sqrt{\left(\frac{s_1^2}{n_1}\right) + \left(\frac{s_2^2}{n_2}\right)}
\][/tex]
Given:
[tex]\[
\bar{x}_1 = 34, \quad \bar{x}_2 = 24
\][/tex]
[tex]\[
n_1 = 65, \quad n_2 = 68
\][/tex]
[tex]\[
s_1 = 2, \quad s_2 = 5
\][/tex]
Plugging these values into the formula:
[tex]\[
SE = \sqrt{\left(\frac{2^2}{65}\right) + \left(\frac{5^2}{68}\right)} = \sqrt{\left(\frac{4}{65}\right) + \left(\frac{25}{68}\right)} \approx 0.6551
\][/tex]
2. Determine the Critical Value (Z) for a 94% Confidence Level:
For a 94% confidence level, we look up the critical value (Z) from the standard normal distribution table. The critical value that corresponds to a confidence level of 94% is approximately 1.8808.
3. Calculate the Margin of Error (ME):
The margin of error is found by multiplying the critical value by the standard error:
[tex]\[
ME = Z \times SE
\][/tex]
[tex]\[
ME = 1.8808 \times 0.6551 \approx 1.2322
\][/tex]
4. Construct the Confidence Interval:
The confidence interval for the difference between the two means is given by:
[tex]\[
(\bar{x}_1 - \bar{x}_2) \pm ME
\][/tex]
Calculate the point estimate:
[tex]\[
\bar{x}_1 - \bar{x}_2 = 34 - 24 = 10
\][/tex]
Determine the lower and upper bounds of the confidence interval:
[tex]\[
\text{Lower bound} = 10 - 1.2322 \approx 8.77
\][/tex]
[tex]\[
\text{Upper bound} = 10 + 1.2322 \approx 11.23
\][/tex]
Thus, the 94% confidence interval for the difference between the population means [tex]\((\mu_1 - \mu_2)\)[/tex] is approximately:
[tex]\[
8.77 < \mu_1 - \mu_2 < 11.23
\][/tex]
