Two Sample Z-Test Calculator

Two Sample Z-Test Procedure

Step 1: Define the Null and Alternative Hypotheses.

The formulation of these hypotheses depends on the direction of the test: left-tailed, right-tailed, or two-tailed. Below are the null and alternative hypotheses for each case:

• Left-tailed Test:
  • Null Hypothesis (H₀): The difference between the population means is equal to the hypothesized value.
    $$ H_0: \mu_1 - \mu_2 = \mu_0 $$
  • Alternative Hypothesis (H_a): The difference between the population means is less than the hypothesized value.
    $$ H_a: \mu_1 - \mu_2 < \mu_0 $$
• Right-tailed Test:
  • Null Hypothesis (H₀): The difference between the population means is equal to the hypothesized value.
    $$ H_0: \mu_1 - \mu_2 = \mu_0 $$
  • Alternative Hypothesis (H_a): The difference between the population means is greater than the hypothesized value.
    $$ H_a: \mu_1 - \mu_2 > \mu_0 $$
• Two-tailed Test:
  • Null Hypothesis (H₀): The difference between the population means is equal to the hypothesized value.
    $$ H_0: \mu_1 - \mu_2 = \mu_0 $$
  • Alternative Hypothesis (H_a): The difference between the population means is not equal to the hypothesized value.
    $$ H_a: \mu_1 - \mu_2 \neq \mu_0 $$

Step 2: Calculate the Z-Test Statistic

Use the formula below to compute the z-statistic:

$$ Z_{stat} = \frac{(\bar{X}_1 - \bar{X}_2) - \mu_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}, \text{ where:} $$

• \( \bar{X}_1 \) is the sample mean of the first sample
• \( \bar{X}_2 \) is the sample mean of the second sample
• \( \mu_0 \) is the hypothesized difference between the population means
• \( \sigma_1 \) and \( \sigma_2 \) are the population standard deviations of the two samples
• \( n_1 \) and \( n_2 \) are the sample sizes of the two samples

Step 3: Determine the P-Value

Find the p-value based on the type of test:

• Left-tailed: $$ p-value = P(z < z_{\text{stat}}) $$
• Right-tailed: $$ p-value = P(z > z_{\text{stat}}) $$
• Two-tailed: $$ p-value = 2 \times P(z > |z_{\text{stat}}|) $$

Step 4: Make a Decision

Compare the p-value to your chosen significance level, \( \alpha \):

• If \( p \leq \alpha \): Reject the null hypothesis.
• If \( p > \alpha \): Do not reject the null hypothesis.

Step 5: Draw a Conclusion

Based on your decision:

• Reject \( H_0 \): There is significant evidence to support the alternative hypothesis.
• Fail to reject \( H_0 \): There is no significant evidence to support the alternative hypothesis.