To find the confidence intervals for the difference between the population means of the two groups, we can use the following formula for the confidence interval of the difference between two independent means:
\[ CI = (\bar{X}_1 - \bar{X}_2) \pm z \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]
Where:
- \(\bar{X}_1\) and \(\bar{X}_2\) are the sample means of the two groups.
- \(s_1\) and \(s_2\) are the sample standard deviations of the two groups.
- \(n_1\) and \(n_2\) are the sample sizes of the two groups.
- \(z\) is the z-value for the desired confidence level.
Given data:
-
For group 1 (athletic):
- \(\bar{X}_1 = 68.2\) kg
- \(s_1 = 3\) kg
- \(n_1 = 50\)
-
For group 2 (non-athletic):
- \(\bar{X}_2 = 67.5\) kg
- \(s_2 = 2.8\) kg
- \(n_2 = 50\)
Step 1: Calculate the standard error (SE)
\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]
Plugging in the values: \[ SE = \sqrt{\frac{3^2}{50} + \frac{2.8^2}{50}} = \sqrt{\frac{9}{50} + \frac{7.84}{50}} = \sqrt{\frac{16.84}{50}} = \sqrt{0.3368} \approx 0.58 \]
Step 2: Determine the z-scores for the confidence levels
- For a 99% confidence level, \(z \approx 2.576\)
- For a 95% confidence level, \(z \approx 1.96\)
- For a 90% confidence level, \(z \approx 1.645\)
- For an 80% confidence level, \(z \approx 1.28\)
Step 3: Calculate the confidence intervals
-
99% Confidence Interval \[ CI_{99%} = (68.2 - 67.5) \pm 2.576 \cdot 0.58 \] \[ = 0.7 \pm 1.493 \Rightarrow (0.7 - 1.493, 0.7 + 1.493) \] \[ \approx (-0.793, 2.193) \]
-
95% Confidence Interval \[ CI_{95%} = (68.2 - 67.5) \pm 1.96 \cdot 0.58 \] \[ = 0.7 \pm 1.138 \Rightarrow (0.7 - 1.138, 0.7 + 1.138) \] \[ \approx (-0.438, 1.838) \]
-
90% Confidence Interval \[ CI_{90%} = (68.2 - 67.5) \pm 1.645 \cdot 0.58 \] \[ = 0.7 \pm 0.9541 \Rightarrow (0.7 - 0.9541, 0.7 + 0.9541) \] \[ \approx (-0.254, 1.654) \]
-
80% Confidence Interval \[ CI_{80%} = (68.2 - 67.5) \pm 1.28 \cdot 0.58 \] \[ = 0.7 \pm 0.7416 \Rightarrow (0.7 - 0.7416, 0.7 + 0.7416) \] \[ \approx (-0.0416, 1.4416) \]
Summary of Confidence Intervals
- 99% CI: (-0.793, 2.193)
- 95% CI: (-0.438, 1.838)
- 90% CI: (-0.254, 1.654)
- 80% CI: (-0.0416, 1.4416)
These intervals give us a range in which we can be confident the true difference in means lies, for the specified levels of confidence.