2Definitions Power THIS IS WHAT YOU WANT Type I Error Incorrectly rejecting a true null hypothesisType II ErrorFailing to reject the null hypothesis when it should be rejectedPowerRejecting the null hypothesis when Ha is trueTHIS IS WHAT YOU WANT
3Helpful Chart for remembering definitions Drawing this chart before doing any power presentation will be very helpful!!!Truth About The PopulationDecision From SampleHo: is true Ha: is trueType I ErrorPOWER1-Type II ErrorCorrectReject Ho:Fail to Reject Ho:
4Let’s start at the beginning... Ho = .05The 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 PowerThe most frequently used level is .05.In this presentation, will always equal .05.
5Review 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...… .05Type Two Error is found when…we fail to reject Ho and Ha is true
6The 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.
7Let’s put the two curves together… HoHa=.05Notice 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.
8Another 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
9Another 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
10You 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 HoReject Ho
11PowerThe power of this test is the shaded region. Once again, power is found when we reject Ho and Ha is correct.Fail to reject HoReject Ho=10=0To 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.
12Type I ErrorType one error is the shaded region and is found when we fail to accept Ho but Ho is true.Fail to accept HoReject Ho=10=0The 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, , )
13Type II ErrorFail to reject HoReject Ho=10=0Type 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.
14An Example:If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1Find the power of the test using =.05 and =.316=0 =.05=.3161.) Draw the Ho curve2.) Label the appropriate level on the Ho curve. Also label the mean and standard deviation.
15An Example Continued:If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1Find the power of the test using =.05 and =.316= = =1.1=.316Ho: Ha:3.) Draw the Ha curve4.) In this problem we want to find Power, so shade the desired region
16An Example Continued: Conclusion: If the null hypothesis is Ho: =0 and the alternative hypothesis is Ha: =1.1Find the power of the test using =.05 and =.3165.) Find the z-score of the level.Ho: Ha:invNorm(.95,0,.316) = .526.) Use normalcdf to find the probability of getting Power.Normalcdf(.52,e99,1.1,.316)=.966=.316Conclusion:The probability of correctly rejecting Ho and accepting Ha is 96.6%= = =1.1z=.52
17A couple problems to try on your own… From Moore’s Basic Practice of Statistics1.)You have a SRS of size n=9 from a normal distribution with =1. You wish to testHo: = 0 and Ha: >0You 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 = 0Find 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< , 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.