Question

A swim coach is planning to run an experiment in which he will randomly select some of his swimmers to wear fins during swim practice for a week. He will randomly select a second group of swimmers to practice for the week witho wearing fins. At the end of the week, the coach plans to time the swimmers to see whether the improvements in the times for a 25-yard freestyle sprint as compared to the previous week differ for the two groups The coach was planning on using a sample size of 16 swimmers who wear fins and 20 swimmers who do not wear fins, but he wanted to calculate the power of the study with those sample sizes before beginning the study. He does not, however, have a good guess for what to use for an effect size, so he decides to use Cohen's convention for a large effect size (d 0.80). The coach plans to use an t test for independent samples to test the following hypotheses: To calculate the power of the coach's experiment he needs to calculate the delta,8. To calculate for an t test for independent samples with unequal sample sizes, he first needs to calculate the harmonic mean of the sample sizes The harmonic mean is,h , and o is The coach plans to use a significance level of .05, using the table below, the power of the test is Note: To see the entire power table below, click on the downward-pointing arrow to the right of the box that says Table: Power. Table: Power To calculate the power of the coach's experiment he needs to calculate the delta, δ. To calculate δ for an t test for independent samples with unequal sample sizes, he first needs to calculate the harmonic mean of the sample sizes. The harmonic mean is ih , and 8 is The coach plans to use a significance level of .05. Using the table below, the power of the test is Note: To see the entire power table below, click on the downward-pointing arrow to the right of the box that says Table: Power. Table: Power Using the same significance level and effect size, the harmonic mean of the sample sizes required to achieve a power of 0.80 is

Answer 1

The harmonic mean of the sample sizes required to achieve a power of 0.80 is 17.39.

Explanation:

The coach wants to calculate the power of his experiment before beginning it. The power of a study measures its ability to detect an effect if there is one. The coach plans to use Cohen's convention for large effect size (d 0.80) and an independent samples t-test to test the hypotheses. To calculate the power of the test, we first need to calculate delta (δ), which is the effect size we want to detect. Since the coach plans to use a large effect size (d 0.80), δ is also 0.80.

To calculate the delta, we need to calculate the pooled standard deviation (sp). We can use the following formula to find sp:

sp = sqrt(((n₁ - 1)s₁² + (n₂ - 1)s₂²) / (n₁ + n₂ - 2))

where n₁ and n₂ are the sample sizes, and s₁ and s₂ are the standard deviations of the two groups. Since we don't have any data yet, we can't calculate sp directly. However, we can estimate it using Cohen's convention for a large effect size, where we assume that the population standard deviations are equal. In this case, we can use the following formula to estimate sp:

sp = sqrt(((n₁ - 1) + (n₂ - 1)) / (n₁ + n₂ - 2)) * d

where d is the effect size, which is 0.80 in this case. Plugging in the values, we get:

sp = sqrt(((16 - 1) + (20 - 1)) / (16 + 20 - 2)) * 0.80

sp = 0.565

Next, we need to calculate the t-value for the desired significance level (α) and sample sizes (n₁ and n₂). In this case, the coach plans to use a significance level of .05, so α = 0.05, and the degrees of freedom (df) = (n₁ + n₂ - 2) = 34. Using a t-table with 34 degrees of freedom and a two-tailed test, we find the t-value to be 1.691.

Finally, we can calculate the power of the test using the following formula:

power = 1 -