Two-Sample Z-Test

Definition

The two-sample z-test is a statistical hypothesis test used to determine if the means of two independent populations with known standard deviations are significantly different. It is commonly applied when comparing the means of two groups or samples to assess if they come from the same underlying population.

5 Must Know Facts For Your Next Test

1. The two-sample z-test assumes that the populations have known and equal standard deviations, and the sample sizes are large enough for the central limit theorem to apply.
2. The test statistic for the two-sample z-test is calculated as the difference between the two sample means divided by the standard error of the difference.
3. The null hypothesis for the two-sample z-test is that the means of the two populations are equal, while the alternative hypothesis is that the means are not equal.
4. The two-sample z-test can be used to compare the means of two independent samples, such as the average test scores of two different teaching methods or the average incomes of two different regions.
5. The p-value from the two-sample z-test is used to determine the statistical significance of the difference between the two means, with a lower p-value indicating stronger evidence against the null hypothesis.

Review Questions

• Explain the purpose and key assumptions of the two-sample z-test.
  • The purpose of the two-sample z-test is to determine if the means of two independent populations with known standard deviations are significantly different. The key assumptions are that the populations have equal and known standard deviations, and the sample sizes are large enough for the central limit theorem to apply, ensuring the sample means follow a normal distribution. This allows the test statistic to be calculated using the z-distribution, which is the basis for the two-sample z-test.
• Describe the test statistic and hypotheses used in the two-sample z-test.
  • The test statistic for the two-sample z-test is calculated as the difference between the two sample means divided by the standard error of the difference. The null hypothesis is that the means of the two populations are equal, while the alternative hypothesis is that the means are not equal. The test statistic follows a standard normal distribution, allowing the p-value to be determined from the z-distribution and used to assess the statistical significance of the difference between the two means.
• Discuss the practical applications and interpretation of the results from a two-sample z-test.
  • The two-sample z-test is commonly used to compare the means of two independent samples, such as the average test scores of two different teaching methods or the average incomes of two different regions. The p-value from the test is used to determine the statistical significance of the difference between the two means, with a lower p-value indicating stronger evidence against the null hypothesis that the means are equal. The results of the two-sample z-test can inform decision-making, policy changes, or further research, depending on the context of the study and the practical implications of the findings.

Related terms

Hypothesis Testing: The process of using statistical analysis to determine whether a hypothesis about a parameter of a population is likely to be true or false.

Standard Error: The standard deviation of the sampling distribution of a statistic, such as the mean, which provides a measure of the precision of the estimate.

Z-Score: A standardized measure that expresses the number of standard deviations a data point is from the mean of a distribution, allowing for comparison across different data sets.