Briefly explain how increasing sample size influences each of the following. Assume that all other factors are held constant.
a. The size of the z- score in a hypothesis test.
b. The size of Cohen’s d.
c. The power of a hypothesis test.
a. The z-score increases.
b. Cohen's d is not influenced by sample size.
c. Power increases.
a. Increasing sample size will decrease the size of the z-score in a hypothesis test. The z-score is calculated by dividing the difference between the sample mean and the population mean by the standard deviation of the population. As the sample size increases, the standard deviation of the sample mean decreases, which leads to a smaller z-score.
b. Increasing sample size does not directly influence the size of Cohen's d. Cohen's d is a measure of effect size and represents the standardized difference between two means. It is calculated by dividing the difference between the means by the pooled standard deviation. Sample size indirectly affects the standard deviation by increasing the precision of the estimate, but it does not directly impact the size of Cohen's d.
c. Increasing sample size increases the power of a hypothesis test. Power is the probability of correctly rejecting the null hypothesis when it is false. With a larger sample size, the test has more statistical power to detect true differences or effects in the population. As the sample size increases, the standard error decreases, reducing the likelihood of type II errors (false negatives) and increasing the power of the test.
a. The size of the z-score in a hypothesis test:
To understand how increasing sample size influences the size of the z-score in a hypothesis test, we need to consider the concept of standard deviation. The z-score measures the number of standard deviations a data point is from the mean. As the sample size increases, the standard deviation of the sample mean decreases. This means that the difference between the sample mean and the population mean becomes more accurate and reliable. Consequently, as the sample size increases, the z-score becomes smaller because the standard deviation decreases.
b. The size of Cohen's d:
Cohen's d is a measure of the effect size, which indicates the magnitude of the difference between two groups. It is calculated by dividing the difference between the means of two groups by the pooled standard deviation. Increasing the sample size tends to reduce the standard deviation, thus making the difference between the means relatively larger compared to the standard deviation. As a result, increasing the sample size generally increases the size of Cohen's d, indicating a larger effect size.
c. The power of a hypothesis test:
Power in a hypothesis test is the probability of correctly rejecting the null hypothesis when it is false. Increasing the sample size usually increases the power of a hypothesis test. This is because a larger sample size provides a more accurate representation of the population and reduces the impact of random variation. With a larger sample size, the test becomes more sensitive, increasing the chances of detecting a significant effect and rejecting the null hypothesis when it is false. Therefore, increasing the sample size generally increases the power of a hypothesis test.