Understanding Power By Jessica Jorge

Definitions Power THIS IS WHAT YOU WANT Type I Error Incorrectly rejecting a true null hypothesis Type II Error Failing to reject the null hypothesis when it should be rejected Power Rejecting the null hypothesis when Ha is true THIS IS WHAT YOU WANT

Helpful Chart for remembering definitions Drawing this chart before doing any power presentation will be very helpful!!! Truth About The Population Decision From Sample Ho: is true Ha: is true Type I Error  POWER 1- Type II Error  Correct Reject Ho: Fail to Reject Ho:

Let’s start at the beginning... Ho   = .05 The  level is the pre-determined place where Ho is rejected. That is, if a observation falls above the  level, one would consider it significant, and would therefore reject Ho. The level is very important for understanding Power The most frequently used  level is .05. In this presentation,  will always equal .05.

Review Questions  Type One Error is denoted by the symbol…. Power is found when… the null hypothesis is rejected and the alternative hypothesis is true. THIS IS WHAT YOU WANT!! The most frequently used  level is... … .05 Type Two Error is found when… we fail to reject Ho and Ha is true

The Ha curve is the second curve used when looking at power. Similarly, the Ha curve is a mound shaped symmetrical curve, but will have a different mean than the Ho curve and sometimes a different standard deviation. Note: Unless told otherwise, assume  is the same for the Ho and Ha curve. Also note that the Ha curve does not have an  level marked. ONLY the Ho curve will have an  level. 

Let’s put the two curves together… Ho Ha =.05 Notice the  level is marked clearly with a vertical line. Drawing this line clearly is helpful to visualize what is happening. Always place the Ha curve in accordance to the Ho curve.

Another little quiz…. What curve is this? Ho If the mean of the Ho curve is =10, and the mean of the Ha curve is =0, where would the Ha curve be placed? =10

Another little quiz…. What curve is this? Ho Ho =0 Ha If the mean of the Ho curve is =10, and the mean of the Ha curve is =0, where would the Ha curve be placed? =10

You will now see the importance of the  level As previously mentioned, the  level plays an important role in Power. In this illustration we have added where to reject and accept Ho according to the  level. If an observation falls to the left of the  level, we fail to reject Ho (accept Ho), and if it falls to the right, we reject. Fail to reject Ho Reject Ho

Power The power of this test is the shaded region. Once again, power is found when we reject Ho and Ha is correct. Fail to reject Ho Reject Ho =10 =0 To find the probability of getting power, one can find the z-score of the  level. Because power says that Ha is true, use the mean and standard deviation of the Ha curve to find the z-score. A quick way to find the z-score is to use the invNorm function on the TI-83. Ex. invNorm(.05, ,,) Once the z-score is calculated, simply use normalcdf to calculate the probability of finding power.

Type I Error Type one error is the shaded region and is found when we fail to accept Ho but Ho is true. Fail to accept Ho Reject Ho =10 =0 The probability of getting Type I Error will always be the  level. To find the probability of getting Type I Error, first find the z-score of the  level. You can use the invNorm function of your calculator to do this. Next, you can use the z-table to find the probability of Type II Error. An easier and more accurate way is to use the normalcdf function on you calculator. Ex. Normalcdf(lower bound, upper bound, , )

Type II Error Fail to reject Ho Reject Ho =10 =0 Type II Error is found when we fail to reject Ho based on the sample, but Ha is true. To find the probability of getting Type II Error, find the z-score of the  level. (Use invNorm, the same function used in calculating Power and Type I error.) Once the z-score is found, simply use either the z-table to find the p-value (same as the probability) OR use normalcdf. If you use normalcdf, be sure to use the mean and standard deviation of the Ha curve.

An Example: If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1 Find the power of the test using =.05 and =.316 =0 =.05 =.316 1.) Draw the Ho curve 2.) Label the appropriate  level on the Ho curve. Also label the mean and standard deviation.

An Example Continued: If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1 Find the power of the test using =.05 and =.316 =0 =.05 =1.1 =.316 Ho: Ha: 3.) Draw the Ha curve 4.) In this problem we want to find Power, so shade the desired region

An Example Continued: Conclusion: If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1 Find the power of the test using =.05 and =.316 5.) Find the z-score of the  level. Ho: Ha: invNorm(.95,0,.316) = .52 6.) Use normalcdf to find the probability of getting Power. Normalcdf(.52,e99,1.1,.316)=.966 =.316 Conclusion: The probability of correctly rejecting Ho and accepting Ha is 96.6% =0 =.05 =1.1 z=.52

A couple problems to try on your own… From Moore’s Basic Practice of Statistics 1.)You have a SRS of size n=9 from a normal distribution with =1. You wish to test Ho:  = 0 and Ha:  >0 You decide to reject Ho if x-bar > 0 and to accept Ho otherwise. Find the probability of Type I error. That is, find the probability that the test rejects Ho when in fact  = 0 Find the probability of Type II error when  = .3. Find the probability of Type II error when  = 1. The hypothesis are:  = 300 and Ha:  < 300. The sample size is n=6, and the population is assumed to have a normal distribution with =3. A 5% significance test rejects Ho if z< -1.645, where the test statistic z is: Z= x-bar – 300/(3/sqr6) Find the power of this test against the alternative hypothesis  = 299. Find the power against the alternative  = 295. Is the power against  = 290 higher or lower than the value you found in b? (Don’t actually calculate that power.) Explain your answer.