One Sample Z Test Calculator

One Sample Z-Test Procedure

Step 1: Define the Null and Alternative Hypotheses

The structure of the null and alternative hypotheses depends on whether the test is left-tailed, right-tailed, or two-tailed:

• Left-Tailed Test:
  • Null Hypothesis (H₀): The population mean is equal to the hypothesized value.
    $$ H_0: \mu = \mu_0 $$
  • Alternative Hypothesis (H_a): The population mean is less than the hypothesized value.
    $$ H_a: \mu < \mu_0 $$
• Right-Tailed Test:
  • Null Hypothesis (H₀): The population mean is equal to the hypothesized value.
    $$ H_0: \mu = \mu_0 $$
  • Alternative Hypothesis (H_a): The population mean is greater than the hypothesized value.
    $$ H_a: \mu > \mu_0 $$
• Two-Tailed Test:
  • Null Hypothesis (H₀): The population mean is equal to the hypothesized value.
    $$ H_0: \mu = \mu_0 $$
  • Alternative Hypothesis (H_a): The population mean is not equal to the hypothesized value.
    $$ H_a: \mu \neq \mu_0 $$

Step 2: Calculate the Z-Test Statistic

Use the formula below to compute the z-statistic:

$$ Z_{stat} = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}, \text{ where:} $$

• \( \bar{X} \) is the sample mean
• \( \mu_0 \) is the hypothesized population mean
• \( \sigma \) is the population standard deviation
• \( n \) is the sample size

Step 3: Determine the P-Value

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

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

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.