Calculate the confidence interval for the difference between two means.

## Understanding the Confidence Interval for the Difference Between Means

The confidence interval for the difference between two means provides a range within which the true difference between two population means is likely to fall. This interval depends on sample data, sample sizes, standard deviations, and a specified confidence level. When calculating the interval, the method depends on whether the population standard deviations are known or need to be estimated from the sample and whether variances are assumed to be equal or unequal.

### Example Values Used in Calculations

Let's assume:

- Group A has a sample mean \( \bar{x}_1 = 78 \), sample size \( n_1 = 50 \), and a sample standard deviation \( s_1 = 10 \).
- Group B has a sample mean \( \bar{x}_2 = 74 \), sample size \( n_2 = 60 \), and a sample standard deviation \( s_2 = 12 \).
- We desire a 95% confidence interval.

### When Population Standard Deviations are Known

When population standard deviations are known (which is rare in practice), we use the normal distribution, and the confidence interval is calculated as:

$$ CI = (\bar{x}_1 - \bar{x}_2) \pm Z \cdot \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} $$

where \( Z \) is the Z-score (1.96 for a 95% confidence level) and assuming \( \sigma_1 = 10 \), \( \sigma_2 = 12 \).

#### Step-by-Step Calculation

**Step 1:**Calculate the standard error: \[ SE = \sqrt{\frac{10^2}{50} + \frac{12^2}{60}} = \sqrt{\frac{100}{50} + \frac{144}{60}} = \sqrt{2 + 2.4} = \sqrt{4.4} \approx 2.0976177 \]**Step 2:**Calculate the margin of error: \[ MOE = Z \times SE = 1.96 \times 2.0976177 \approx 4.1113307 \]**Step 3:**Construct the confidence interval: \[ CI = (\bar{x}_1 - \bar{x}_2) \pm MOE = (78 - 74) \pm 4.1113307 = [-0.111, 8.111] \]

The 95% confidence interval for the difference between means, assuming known population standard deviations, is approximately \([-0.111, 8.111]\).

### When Population Standard Deviations are Unknown (Equal Variances Assumed)

With unknown population standard deviations and equal variances, we use the pooled standard deviation \( S_p \) and the t-distribution:

$$ CI = (\bar{x}_1 - \bar{x}_2) \pm t \cdot \sqrt{\frac{S_p^2}{n_1} + \frac{S_p^2}{n_2}} $$

where \( S_p^2 \) is the pooled variance.

#### Step-by-Step Calculation

**Step 1:**Calculate the pooled variance \( S_p^2 \): \[ S_p^2 = \frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2} = \frac{(49 \times 100) + (59 \times 144)}{108} \approx 124.0370370 \]**Step 2:**Calculate the standard error: \[ SE = \sqrt{\frac{S_p^2}{n_1} + \frac{S_p^2}{n_2}} = \sqrt{\frac{124.0370370}{50} + \frac{124.0370370}{60}} \approx 2.1326561 \]**Step 3:**Find the t-score for 95% confidence with \( df = 108 \), which is 1.9821.**Step 4:**Calculate the margin of error: \[ MOE = t \times SE = 1.9821 \times 2.1326561 \approx 4.2272290 \]**Step 5:**Construct the confidence interval: \[ CI = (\bar{x}_1 - \bar{x}_2) \pm MOE = (78 - 74) \pm 4.2272290 = [-0.227, 8.227] \]

The 95% confidence interval for the difference between means, assuming equal variances, is approximately \([-0.227, 8.227]\).

### When Population Standard Deviations are Unknown (Unequal Variances)

When variances are unequal, we calculate the standard error for each sample separately and use the Welch-Satterthwaite equation for degrees of freedom:

$$ CI = (\bar{x}_1 - \bar{x}_2) \pm t \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} $$

#### Step-by-Step Calculation

**Step 1:**Calculate the standard error: \[ SE = \sqrt{\frac{10^2}{50} + \frac{12^2}{60}} = \sqrt{\frac{100}{50} + \frac{144}{60}} = \sqrt{4.4} \approx 2.0976177 \]**Step 2:**Calculate the exact degrees of freedom using the Welch-Satterthwaite formula: \[ df = \frac{(s_1^2/n_1 + s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)} \approx 93.6394589 \] Note: We use the exact df value rather than rounding down for more precision**Step 3:**Find the t-score for 95% confidence with \( df = 93.6394589 \), which is 1.9853.**Step 4:**Calculate the margin of error: \[ MOE = t \times SE = 1.9853 \times 2.0976177 \approx 4.1644 \]**Step 5:**Construct the confidence interval: \[ CI = (\bar{x}_1 - \bar{x}_2) \pm MOE = (78 - 74) \pm 4.1644 = [-0.158, 8.158] \]

The 95% confidence interval for the difference between means, assuming unequal variances, is \([-0.158, 8.158]\).

### Comparing the Results

All three methods yield similar but slightly different confidence intervals:

- Known population SDs: \([-0.111, 8.111]\)
- Equal variances: \([-0.227, 8.227]\)
- Unequal variances: \([-0.158, 8.158]\)

Important observations:

- All intervals include zero, indicating we cannot conclude there is a statistically significant difference between the means at the 95% confidence level.
- The equal variances method produces the widest interval, reflecting additional uncertainty from pooling the variances.
- Using exact degrees of freedom and avoiding intermediate rounding leads to more precise results.
- The unequal variances method (Welch's t-test) is generally recommended unless you have strong evidence that the population variances are equal.

## Glossary of Terms

**\( \bar{x}_1 \), \( \bar{x}_2 \):**Sample means of groups 1 and 2, respectively. They represent the average value in each sample.**\( n_1 \), \( n_2 \):**Sample sizes for groups 1 and 2, respectively. They represent the number of observations in each sample.**\( \sigma_1 \), \( \sigma_2 \):**Population standard deviations for groups 1 and 2, respectively. These values indicate the amount of variation or spread in the population. Assumed known in cases where population standard deviations are specified.**\( s_1 \), \( s_2 \):**Sample standard deviations for groups 1 and 2, respectively. They estimate the variation within each sample when population standard deviations are unknown.**Standard Error of the Mean Difference (σ**The standard deviation of the sampling distribution of the difference between two sample means, which indicates the expected variability in the mean difference._{x̄1 - x̄2}or S_{x̄1 - x̄2}):**Confidence Interval (CI):**The range within which the true difference between population means is likely to fall, based on the sample data and the specified confidence level.**Margin of Error (MOE):**The maximum expected difference between the true population mean difference and the observed sample mean difference at the specified confidence level. Calculated as the product of the critical value (Z or t) and the standard error.**Pooled Variance (S**A weighted average of the sample variances from each group, assuming equal population variances. Used when population standard deviations are unknown but variances are assumed equal.^{2}_{p}):**Pooled Standard Deviation (S**The square root of the pooled variance, used to compute the standard error when variances are assumed equal._{p}):**Degrees of Freedom (df):**The number of values that are free to vary in the final calculation, affecting the critical value in t-distributions. Calculated differently depending on whether variances are assumed equal or unequal.**Z-score:**The critical value from the standard normal distribution, used when population standard deviations are known. For example, Z = 1.96 for a 95% confidence level.**t-score:**The critical value from the t-distribution, used when population standard deviations are unknown. This value varies based on the specified confidence level and degrees of freedom.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.