# Hypothesis Testing for Means and Proportions

1. Think about a real world: Diagnosis Congestive heart failure (CHF), non compliance with daily weight or diet/situation. Address using an independent or related samples t test. 2. Identify the independent (grouping) and dependant (response) variables important to study 3. Explain whether an independent sample or related sample t test is appropriate and why

4. Generate a hypothesis, including null and alternative hypothesis 5. Describe what information the effect size will tell you and what information the effect size will tell you and what information the p value or critical value approach will not 6. Using realistic numbers for the degrees of freedom, sample size and t statistic, report hypothetical results in 2-3 sentences

Solution: (1) Let’s consider the following research situation: The incidence of Congestive heart failure (CHF) is going to be studied based on two different diet groups: one group receives a special diet (a diet designed for preventing CHF), and a control group (which doesn’t receive any diet). We are interested in assessing whether there is a difference in the incidence of CHF for these two groups. In order to perform the analysis, a two-independent t-test will be used.

(2) In this case, the independent (grouping) variable is DIET, and the dependent (response) variable is CHF incidence rate.

(3) This analysis corresponds to an independent-samples design, because the treatments (diet/no diet) are applied to different subjects.

(4) We are interested in the following research question: Is there a difference in the incidence of CHF for the diet and no-diet group?

The following hypotheses are used: where [pic]represents the mean CHF incidence rate.

(5) The information given by the p-value is about SIGNIFICANCE, which means the probability of getting sampling results as extreme or more extreme than the ones obtained, under the assumption that the null hypothesis true. The problem with this information is that a very small effect could be found to be “significant” with an extremely large sample size (which would be an overly sensitive test).

On the other hand, the effect size provides information about the practical significance of the effect of a treatment.

(6) The following hypothetical results could be obtained:

“A sample of n = 25 subjects was used for each treatment. For the diet group, the mean CHF incidence is M = 2.3% (SD = 0.9%), and for the no-diet group, the mean CHF incidence is M = 3.1% (SD = 1.1%). There is enough evidence to support the claim that there is a significant difference in the mean incidence for the diet and no-diet group, t(48) = -2.81, p = 0.0071 < 0.05. The effect size is d = 0.796”

Appendix: Calculations of the hypothetical analysis

We are interested in testing which corresponds to a two-tailed independent samples t-test. Before performing a t-test, we need to test whether the variances can be assumed to be equal or not. We need to test

The F-statistics is computed as The lower and upper critical values for[pic] and df1 = 24 and df2 = 24 are which means that we fail to reject the null hypothesis of equal variances. Observe that we are assuming that the variances are equal, so the t-statistics is computed as: where the pooled standard deviation is computed as

This means that the t-statistics is The critical value for[pic] and for[pic] degrees of freedom for this two-tailed test is[pic]. The rejection region is given by  Since[pic], then we reject the null hypothesis H0.

Hence, we have enough evidence to support the claim that there is a significant difference in the mean incidence for the diet and no-diet group.

The following effect size estimates are obtained: which means that 14.16% of the variation is explained. Also, we compute Cohen’s D: which means that we have a medium size effect.