independent samples t-test: The Average Height of Gnomes versus Dwarves (Testing for Equal Means)

by Kurious Fox

Suppose we want to compare the average height of gnomes and dwarves. We will use an independent samples t-test to determine if there is a significant difference between the two groups. Here is how we set up the hypothesis test and perform the analysis.

Step 1: State the Hypotheses

The null hypothesis H_0 and the alternative hypothesis H_A can be stated as follows:
H_0: \mu_1 = \mu_2 \quad \text{(no difference in average height between gnomes and dwarves)}
H_A: \mu_1 \neq \mu_2 \quad \text{(there is a difference in average height between gnomes and dwarves)}
where \mu_1 is the mean height of gnomes and \mu_2 is the mean height of dwarves.

Step 2: Collect Data

Suppose we have the following sample data for the heights (in cm) of gnomes and dwarves:

Step 3: Calculate the Test Statistic

First, we calculate the sample means and sample standard deviations:
\bar{X}_1 = \frac{112 + 115 + 113 + 118 + 110}{5} = 113.6

\bar{X}_2 = \frac{120 + 123 + 119 + 122 + 121}{5} = 121

s_1 = \sqrt{\frac{\sum{i=1}^n (X_{1i} - \bar{X}1)^2}{n-1}} = \sqrt{\frac{(112-113.6)^2 + (115-113.6)^2 + (113-113.6)^2 + (118-113.6)^2 + (110-113.6)^2}{5-1}} \approx 3.361

s_2 = \sqrt{\frac{\sum{i=1}^n (X_{2i} - \bar{X}_2)^2}{n-1}} = \sqrt{\frac{(120-121)^2 + (123-121)^2 + (119-121)^2 + (122-121)^2 + (121-121)^2}{5-1}} \approx 1.581

Next, we calculate the pooled standard deviation s_p :
s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} = \sqrt{\frac{(5-1)(3.361)^2 + (5-1)(1.581)^2}{5+5-2}} \approx 2.758

The test statistic t is then calculated as:
t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} = \frac{113.6 - 121}{2.758 \sqrt{\frac{1}{5} + \frac{1}{5}}} \approx -4.761

Step 4: Determine the Degrees of Freedom

The degrees of freedom for the independent samples t-test is:
df = n_1 + n_2 - 2 = 5 + 5 - 2 = 8

Step 5: Compare the Test Statistic to the Critical Value

Using a t-table or calculator, find the critical t-value for a two-tailed test with \alpha = 0.05 and df = 8 . The critical t-value is approximately \pm 2.306 .

Since t = -4.761 is less than -2.306 , we reject the null hypothesis.

Conclusion

At the \alpha = 0.05 significance level, we have enough evidence to conclude that there is a significant difference in average height between gnomes and dwarves. Therefore, we reject the null hypothesis that there is no difference in average height between the two groups.

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