To construct a 99.5% confidence interval for the difference in the mean BMI between women and men with diabetes, we will use the following formula for the confidence interval for the difference in means:
\[ \bar{X_1} - \bar{X_2} \pm z^* \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]
Where:
- \(\bar{X_1} = 31.40\) is the mean BMI for women,
- \(\bar{X_2} = 30.60\) is the mean BMI for men,
- \(s_1 = 0.3\) is the standard deviation for women,
- \(s_2 = 0.6\) is the standard deviation for men,
- \(n_1 = 1931\) is the sample size for women,
- \(n_2 = 1560\) is the sample size for men,
- \(z^*\) is the z-score corresponding to the desired confidence level.
Step 1: Find the z-score
For a 99.5% confidence interval, we look for the z-score that leaves 0.5% in the tails (0.25% in each tail). The critical value \(z^*\) can be found using the standard normal distribution table or a calculator:
\[ z^* \approx 2.576 \]
Step 2: Calculate the standard error (SE)
Next, we calculate the standard error (SE):
\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.3^2}{1931} + \frac{0.6^2}{1560}} \]
Calculating the individual components:
\[ \frac{0.3^2}{1931} = \frac{0.09}{1931} \approx 0.0000465 \] \[ \frac{0.6^2}{1560} = \frac{0.36}{1560} \approx 0.0002308 \]
Now summing those:
\[ SE = \sqrt{0.0000465 + 0.0002308} = \sqrt{0.0002773} \approx 0.01665 \]
Step 3: Calculate the margin of error (ME)
Now, we calculate the margin of error:
\[ ME = z^* \cdot SE = 2.576 \cdot 0.01665 \approx 0.04288 \]
Step 4: Calculate the confidence interval
Finally, we can find the confidence interval for the difference in means:
\[ \text{Difference} = \bar{X_1} - \bar{X_2} = 31.40 - 30.60 = 0.80 \]
Thus, the confidence interval is:
\[ 0.80 \pm 0.04288 \]
Calculating the lower and upper bounds:
- Lower bound: \(0.80 - 0.04288 \approx 0.75712\)
- Upper bound: \(0.80 + 0.04288 \approx 0.84288\)
Final Result
Rounding the results to two decimal places, we get:
\[ \text{99.5% Confidence Interval} \approx (0.76, 0.84) \]
So the 99.5% confidence interval for the difference in mean BMI between women and men with diabetes is \((0.76, 0.84)\).