A random sample of n=16 scores is obtained from a population with a mean of µ= 80 and a treatment is administered to the sample. After treatment, the sample mean is found to be M=82.
a)Assuming the sample contained n=16 individuals, measure the size of the treatment effect by computing the estimated d and r squared.
b) Assuming the sample contained n=25 individuals, measure the size of the treatment effect by computing the estimated d and rsquared.
c)comparing your answers for parts (a)and (b) , how does sample variance influence measures of effect size?
To compute the estimated d and r squared, we need to calculate the effect size. Here's how you can do it:
a) For n=16 individuals:
Step 1: Calculate the difference between the sample mean and the population mean: d = M - µ = 82 - 80 = 2.
Step 2: Compute the sample variance (s^2) using the formula: s^2 = Σ(x - M)^2 / (n - 1). Assume the sample has a standard deviation, s=5. Therefore, the sample variance is s^2 = 5^2 = 25.
Step 3: Estimate Cohen's d as the difference divided by the standard deviation: d = 2 / 5 = 0.4.
Step 4: Compute r squared using the formula: r^2 = d^2 / (d^2 + 4) = 0.4^2 / (0.4^2 + 4) = 0.16 / 16.16 ≈ 0.0099.
b) For n=25 individuals:
Step 1: Calculate the difference between the sample mean and the population mean: d = M - µ = 82 - 80 = 2.
Step 2: Compute the sample variance (s^2) using the formula: s^2 = Σ(x - M)^2 / (n - 1). Assume the sample has a standard deviation, s=5. Therefore, the sample variance is s^2 = 5^2 = 25.
Step 3: Estimate Cohen's d as the difference divided by the standard deviation: d = 2 / 5 = 0.4.
Step 4: Compute r squared using the formula: r^2 = d^2 / (d^2 + 4) = 0.4^2 / (0.4^2 + 4) = 0.16 / 4.16 ≈ 0.0385.
c) Comparing the answers for parts (a) and (b), we can observe that the sample variance remains the same (25) for both cases even when the sample size changes. The effect size (Cohen's d) remains the same (0.4) as well since it is calculated as the difference between the sample mean and the population mean divided by the sample standard deviation. Thus, the sample variance does not influence the measure of effect size (Cohen's d).
However, the r squared value does change as the sample size increases (from 0.0099 to 0.0385). The r squared value indicates the proportion of variance in the dependent variable that can be explained by the treatment. In this case, a larger sample size allows for a more accurate estimation of the proportion of variance explained by the treatment, leading to a higher r squared value. Therefore, sample variance indirectly influences the measure of effect size when considering the explained variance.