How to solve the power test without software?

Question

1. 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

This question is asking you to test the claim that the mean reaction time in seconds is less than 1.6 seconds using a confidence interval and then to find the power of the test assuming that the true population mean is 1.45 seconds.

1.     We are constructing a one sided confidence interval to find the upper bound.

If the upper bound is less than 1.6, we can reject Ho.

Ho: µ=1.6

Ha: µ<1.6

Components of Confidence Interval

n=10

xbar = 1.41

s=0.3071

t critical = invT(0.99,9) = 2.8214

The entire significance level is in one tail since it is a one sided test.

Upper bound = xbar + tcritical*(s/√n)

                             = 1.41 + 2.8214*(0.3071/√10)

                             = 1.684  Upper Bound of CI

Since 1.6 seconds is included in the 99% confidence interval (∞,1.684) there is not enough evidence to conclude that the mean reaction time is less than 1.6 seconds.

2.     Find the rejection region assuming Ho true

1.6 – 2.8214 *(0.3071/√10) = 1.326

Reject Ho if xbar < 1.326

Find the power by calculating P(Reject Ho | Ha true) = P(xbar < 1.326 | µ=1.45)

=P(t<(1.326 – 1.45)/(0.3071/√10)) = P(t < -1.2769)

=tcdf(-10000, -1.2769,9)

=0.1168

The probability that we reject Ho correctly given µ=1.45

Is 0.1168

Let me know if you have any questions about this. It's a complicated concept.

Dian