1. Chapter 9 Part C

2. III. One-Tailed Tests B. P-values Using p-values is another approach to conducting a hypothesis test, yielding the same result. In general: a p-value is the probability of obtaining a sample result that is at least as unlikely as what is observed. For example, suppose you calculate a z-score of negative 2.80.

3. A diagram The standard normal probability table tells us that there is a.0026 probability of observing a sample mean less than or equal to what was observed. Z 0 Z=-2.80 P-value is P=.0026.4974

4. Z 0 Z=-2.80 P-value is P=.0026 How do you use it? If you’re testing the hypothesis at the 99% level, then  =.01. If p-value< , reject Ho. Z=-2.33  =.01

5. Result of the test. So in this example,.0026<.01, so we reject Ho. Had we been using our previous methodology of comparing the test statistic to the critical value, we would get the same decision. Since Z=-2.80 is greater (absolute value) than Z  =.01 =-2.33, we would reject Ho.

6. An Example Ho:   10 Ha:  < 10 n=36, =8, s=5,  =.01, Z  =.01 =-2.33 Test statistic, Z=(-2)/(5/6)=-2.4 p-value for a Z=-2.4 is.0082 and since this is less than  =.01, reject the null Ho. MAKE SURE YOU CAN DO THIS!

7. IV. Two-Tailed Tests about  : (large sample) The big important gold star difference when you do a two-tailed test is that the rejection range is split equally in each tail of the sampling distribution. If you get a test statistic that is either too low, or too high, you will reject Ho.

8. A. An Example A lathe is set to cut bars of steel into perfect lengths of 6 centimeters. If the bars are anything but 6 cm, our customers will incur substantial costs in resizing the bars and will find other suppliers. Ho:  = 6 cm Ha:   6 cm

9. The rejection range If you are testing this hypothesis at the  =.05 level of significance, then.025 (  /2) is in each tail.  = 6.4750.025

10. Critical Z and Test Statistic Find the z-score that is associated with.4750, and you’ll find Z  =.025 =±1.96. In a sample of 121 bars, you find a sample mean of 6.08 cm with a standard deviation of.44 cm. With a test statistic of 2.0, you would reject Ho and conclude, with 95% confidence, that the bars are not exactly 6 cm.

11. B. P-values for 2-tailed tests Because the significance level (  ) is split between each tail of the sampling distribution, so is the p- value. The p-value is reported as 2 times the area in one tail. If 2*(area in tail) < , reject Ho.

12. Example In the previous example, Z=2.0. The area in the tail, beyond Z=2.0 is (.5-4772)=.0228 The p-value = 2(.0228) =.0456, and since this is less than  =.05, reject Ho. Z 0 p=.0228  /2=.025

13. C. Confidence Intervals Recall, that with a (1-  ) confidence coefficient, a confidence interval is constructed as: If you have a 2-tailed hypothesis test like: Ho:  =  0 Ha:    0

14. If you take a sample, and your confidence interval includes the hypothesized value of  0, then you cannot reject the null. If your  0 is outside of the confidence interval, reject the null.

15. Back to our example Our sample mean was 6.08 cm, with a standard error of (.44/11)=.04. So our 95% confidence interval would look like: or a range between 6.0016 and 6.1584. Since 6 cm is NOT in the interval, we reject Ho.