To construct a 95% confidence interval for the mean difference in weight losses between two groups (Group 1 with exercise and Group 2 without exercise), we follow these steps:
Let's denote:
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\( \bar{x}_1 = 26 \) (mean weight loss of the exercise group)
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\( s_1 = 10 \) (standard deviation of the exercise group)
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\( n_1 = 71 \) (sample size of the exercise group)
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\( \bar{x}_2 = 15 \) (mean weight loss of the non-exercise group)
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\( s_2 = 8 \) (standard deviation of the non-exercise group)
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\( n_2 = 42 \) (sample size of the non-exercise group)
We want to find the confidence interval for the difference in means \( \mu_1 - \mu_2 \).
The formula for the standard error (SE) of the difference in means is:
\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]
Calculating the standard error:
\[ SE = \sqrt{\frac{10^2}{71} + \frac{8^2}{42}} = \sqrt{\frac{100}{71} + \frac{64}{42}} \]
Calculating each term:
\[ \frac{100}{71} \approx 1.40845 \] \[ \frac{64}{42} \approx 1.52381 \]
Adding these together:
\[ SE = \sqrt{1.40845 + 1.52381} = \sqrt{2.93226} \approx 1.711 \]
Next, we find the mean difference:
\[ \bar{x}_1 - \bar{x}_2 = 26 - 15 = 11 \]
Now we will calculate the critical value for a 95% confidence interval using the t-distribution. Since the sample sizes are 71 and 42, we can use the formula to calculate the degrees of freedom using the Welch-Satterthwaite equation:
\[ df \approx \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]
Calculating the numerator:
\[ \left( \frac{10^2}{71} + \frac{8^2}{42} \right)^2 = (1.40845 + 1.52381)^2 = (2.93226)^2 \approx 8.5904 \]
Calculating the denominator:
\[ \frac{\left( \frac{10^2}{71} \right)^2}{70} + \frac{\left( \frac{8^2}{42} \right)^2}{41} \approx \frac{(1.40845)^2}{70} + \frac{(1.52381)^2}{41} \] \[ = \frac{1.98463}{70} + \frac{2.32264}{41} \approx 0.02835 + 0.05694 \approx 0.08529 \]
Thus,
\[ df \approx \frac{8.5904}{0.08529} \approx 100.79 \text{ (round down to 100)} \]
Now we can find the critical value \( t^* \) for \( df \approx 100 \) at a 95% confidence level. The critical \( t \) value for a two-tailed test is approximately \( t^* \approx 1.984 \) (you may look this up in a t-table).
Now we can construct the confidence interval:
\[ CI = \left( (\bar{x}_1 - \bar{x}_2) - t^* \cdot SE, (\bar{x}_1 - \bar{x}_2) + t^* \cdot SE \right) \]
Calculating the margin of error:
\[ t^* \cdot SE = 1.984 \cdot 1.711 \approx 3.3976 \]
Finally, we can construct the confidence interval:
\[ CI = \left( 11 - 3.3976, 11 + 3.3976 \right) = \left( 7.6024, 14.3976 \right) \]
So the 95% confidence interval for the mean difference in weight losses is approximately:
\[ \boxed{(7.60, 14.40)} \]