To evaluate the effect of a treatment, a sample of n=8 is...

To evaluate the effect of a treatment, a sample of $n=8$ is obtained from a population with a mean of $\mu=50,$ and the treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be $M=55$ a. Assuming that the sample variance is $s^{2}=32$, use a two-tailed hypothesis test with $\alpha=.05$ to determine whether the treatment effect is significant and compute both Cohen's $d$ and $r^{2}$ to measure effect size. b. Assuming that the sample variance is $s^{2}=72$, repeat the test and compute both measures of effect size. c. Comparing your answers for parts a and b, how does the variability of the scores in the sample influence the outcome of a hypothesis test and measures of effect size?

To evaluate the effect of a treatment, a sample of $n=8$ is obtained from a population with a mean of $\mu=50,$ and the treatment is administered to the individuals in the sample. After treatment, the sample mean is found to be $M=55$
a. Assuming that the sample variance is $s^{2}=32$, use a two-tailed hypothesis test with $\alpha=.05$ to determine whether the treatment effect is significant and compute both Cohen's $d$ and $r^{2}$ to measure effect size.
b. Assuming that the sample variance is $s^{2}=72$, repeat the test and compute both measures of effect size.
c. Comparing your answers for parts a and b, how does the variability of the scores in the sample influence the outcome of a hypothesis test and measures of effect size?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. In the context of the problem, a two-tailed test at a significance level of 0.05 evaluates whether the observed sample mean significantly deviates from the population mean, implying that the treatment might have an effect.

t-Test

A t-test is used when comparing a sample mean to a known or hypothesized population mean, especially when the population standard deviation is unknown. It involves calculating a t-statistic using the difference between the sample mean and population mean, adjusted for the sample variance and sample size (degrees of freedom), which is then compared against a critical value from the t-distribution.

Effect Size (Cohen's d)

Cohen's d is a standardized measure of the magnitude of the difference between two means, expressed in terms of the pooled standard deviation. It quantifies the practical significance of a treatment effect by showing how many standard deviations the means differ, which helps interpret the effectiveness of the treatment beyond p-values.

Effect Size (r^2)

The r^2 statistic represents the proportion of the total variance in the dependent variable that is explained by the independent variable (or treatment effect). It provides a measure of how well the data can be explained by the tested model, indicating the strength of the relationship between the treatment and the outcome.

Impact of Variability

Variability, as indicated by the sample variance or standard deviation, plays a crucial role in hypothesis testing and effect size estimation. High variability can dilute the apparent effect by inflating the standard error, leading to smaller t-statistics and reduced effect size measures like Cohen's d and r^2, whereas lower variability relative to the mean difference can enhance the statistical power and perceived magnitude of the treatment effect.

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