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
a. Increasing the sample size in a hypothesis test has a direct influence on the size of the z-score. The z-score represents how many standard deviations an observed value is from the mean. As the sample size increases, the standard deviation of the sample mean decreases, which results in a smaller z-score. This means that the observed value is less extreme, relative to the population mean. Therefore, increasing the sample size generally leads to a decrease in the size of the z-score.
To find the size of the z-score in a hypothesis test, you would first calculate the standardized test statistic (z-score) using the formula:
z = (x - μ) / (σ / √n)
Where:
- x is the observed value
- μ is the population mean
- σ is the population standard deviation
- n is the sample size
By plugging in the appropriate values into the formula, you can calculate the z-score.
b. Increasing the sample size does not directly influence the size of Cohen's effect size. Cohen's effect size is a measure of the difference between two groups or conditions. It is generally calculated using the difference between the means of the groups divided by the pooled standard deviation.
To calculate Cohen's effect size, you would use the formula:
d = (μ1 - μ2) / σpooled
Where:
- μ1 and μ2 are the means of the two groups or conditions being compared
- σpooled is the pooled standard deviation
The sample size does not appear directly in this formula. However, increasing sample size can indirectly affect Cohen's effect size if it leads to a more precise estimation of the population standard deviation. With a larger sample size, the standard deviation may be estimated more accurately, resulting in a more reliable Cohen's effect size estimate.
To calculate the pooled standard deviation (σpooled), you would use the following formula:
σpooled = √((σ1² + σ2²) / 2)
Where:
- σ1 and σ2 are the standard deviations of the two groups or conditions being compared
By substituting the values into the appropriate formulas, you can calculate Cohen's effect size.