Chapter 9: Power I can make really good decisions. Chapter 9: Power Target Goal: I can make really good decisions. 9.1d h.w: pg 548: 23, 25

Hand draw Type I/II Curves

The probability of a Type II Error tells us the probability of accepting the null hypothesis when it is actually false. The complement of this would be the probability of not accepting (in other words rejecting) the null hypothesis when it is actually false. (good decision!)

Power To calculate the probability of rejecting the null hypothesis when it is actually false, compute Power = 1 – P(Type II Error), Or, (1 – b). This is called the power of a significance test.

A P-value describes what would happen supposing that the null hypothesis is true. Power describes what would happen supposing that a particular alternative hypothesis is true.

Ex: Exercise is good Can a six month exercise program increase the total body mineral content (TBBMC) of young women? A team of researchers is planning a study to examine this question.

Power = 1 – β = 0.80 Interpret results: This significance test correctly rejects the null hypothesis that exercise has no effect on TBBMC 80% of the time if the true effect of exercise is a 1% increase in TBBMC.

Power: We must use the shifted curve! Power: the probability of rejecting the null hypothesis when it is actually false. We must use the shifted curve!

How Does Power Change? Suppose H 0 : µ = 500 H a : μ > 500 What is the power of the test if µ = 520? Power = =.761 Rejection Region Suppose that  = $85 and n = 100. We would reject H 0 for x >  =.239 Power is the probability of correctly rejecting H 0. Notice that power is in the SAME curve as  Power = 1 – 

H 0 : μ = 500 H a : µ > 500 Find b and power. b =.03Power =.97 vs, Rejection Region Suppose that  = $85 and n = 100. We would reject H 0 for x > If we reject H 0, then  > 500. What if  = 530? Notice that, as the distance between the null hypothesized value for  and our alternative value for  increases,  decreases AND power increases

. H 0 :  = 500 H a :  > 500 Find  and power.  =.03power =.97 vs.  =.68power =.32 Rejection Region Suppose that  = $85 and n = 100. We would reject H 0 for x > If the null hypothesis is false, then  > 500. What if  = 510? Notice that, as the distance between the null hypothesized value for μ and our alternative value for μ decreases, β increases AND power decreases.

H 0 :  = 500 H a :  > 500 What happens if we use  =.01? Rejection Region Suppose that  = $85 and n = 100.  will increase and power will decrease. Rejection Region  Power 

What happens to , , & power when the sample size is increased? Reject H 0 Fail to Reject H 0 00  aa  The standard deviation will decrease making the curve taller and skinnier. The significance level (  ) remains the same – so the value where the rejection region begins must move. β decreases and power increases! Power

If power is too small, increase the Power 1.Increase α. 2.Consider an alternative that is further away from μ o. 3.Increase the sample size. 4.Decrease σ.

What does this look like? Increase α. Slides reference point to the left. Consider an alternative that is further away from μ o. Shift “new” curve to the right. Increase the sample size. Less spread and narrower curve. Decrease σ. Less spread and narrower curve.

Read Explore on your own applet on page 543.