How to find power test ?

Question

A past study on the reaction time of children with cerebral palsy reports a mean of 1.6 seconds on a particular task. A

researcher believes that the mean reaction time can be reduced by using a motivating set of directions. A set of ten

equivalent children is randomly selected and they complete the same task with the motivating set of directions. Assume

the rection time is normally distributed. The reaction times for the ten children are recorded as follows:

1.4, 1.8, 1.1, 1.4, 1.3, 1.6, 0.9, 1.5, 1.9, and 1.2 seconds.

Test if there is sufficient evidence in the sample data to support the claim of the researcher at significance level of 0.01.

(1) Set up the null and alternative hypothesis and construct an appropriate confidence interval to answer the hypothesis

test question.

(2) Choose an appropriate OC curve to determine the approximate power of test if the true mean reaction time is 1.45

seconds.

Answer 1

Hi Vincent,

I recently joined the platform so I just saw your question. If my answer below is helpful, I would greatly appreciate your feedback!

To test whether there is sufficient evidence in the sample data to support the claim of the researcher, you can perform a hypothesis test and calculate the power of the test. Here are the steps:

(1) Set up the null and alternative hypothesis and construct an appropriate confidence interval:

Null Hypothesis (H0): The mean reaction time with the motivating set of directions is equal to the mean reaction time without the motivating set of directions, μ = 1.6 seconds.

Alternative Hypothesis (Ha): The mean reaction time with the motivating set of directions is less than the mean reaction time without the motivating set of directions, μ < 1.6 seconds.

You will perform a one-tailed test because you are interested in whether the mean reaction time has decreased.

Now, calculate the sample mean and standard error of the mean (SEM) from the provided data:

Sample Mean (X̄) = (1.4 + 1.8 + 1.1 + 1.4 + 1.3 + 1.6 + 0.9 + 1.5 + 1.9 + 1.2) / 10 = 1.35 seconds

SEM = Standard Deviation (σ) / √n

Standard Deviation (σ) can be calculated from the sample data, and n is the sample size (n = 10).

Calculate σ:

σ = √[Σ(xi - X̄)² / (n - 1)] = √[Σ(1.4 - 1.35)² + (1.8 - 1.35)² + ... + (1.2 - 1.35)² / 9] ≈ 0.2909 seconds

Now, calculate the SEM:

SEM = 0.2909 / √10 ≈ 0.0919 seconds

Next, construct a confidence interval at a significance level of 0.01. Since you are interested in whether the mean reaction time is less than 1.6 seconds, this is a one-tailed test, and you want the lower confidence bound:

Confidence Interval (CI): X̄ - Zα * SEM

Where Zα is the critical value from the standard normal distribution corresponding to the significance level of 0.01. Zα ≈ -2.33 for a one-tailed test.

CI = 1.35 - (-2.33 * 0.0919) ≈ 1.561 seconds

So, the lower bound of the confidence interval is approximately 1.561 seconds.

(2) Choose an appropriate OC curve to determine the approximate power of the test:



To determine the power of the test, you need to specify the following:

1. The true population mean under the alternative hypothesis (μa). In this case, μa is 1.45 seconds.
2. The significance level (α), which is 0.01.

Now, you can use statistical software or tables to calculate the power of the test based on these values. The power of the test represents the probability of correctly rejecting the null hypothesis when the true mean reaction time is 1.45 seconds.

The power calculation involves finding the area under the sampling distribution curve for the alternative hypothesis, to the left of the critical value corresponding to the significance level α = 0.01. The power can be calculated using the normal distribution or statistical software.

It's important to note that the power of the test depends on the sample size and the effect size (the difference between the true population mean under the alternative hypothesis and the null hypothesis). In this case, the effect size is 1.6 - 1.45 = 0.15 seconds.

The power of the test will tell you the likelihood of detecting a true decrease in the mean reaction time when it exists, given your sample size and significance level.