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Chapter 9 Continued... III. One-Tailed Tests (large sample) • Hilltop Coffee states that each can has at least 3 pounds of coffee. • The Fed. Trade Commission randomly tests corporate claims. • If Hilltop’s claim is correct, 3. Ho: 3 Ha: < 3 If we reject Ho, Hilltop is violating their claim. A. Sampling Distribution x =3 n x If we take a random sample of n=36, we use the C.L.T. to assume a normal sampling distribution. B. How low is too low? • Suppose we measured out each coffee can and calculated a sample mean weight of 2.99 pounds. • Do you think this is enough evidence to reject Ho and conclude that Hilltop is underfilling their cans? Probably not. • If we did reject Ho, we might make a type I error. • What if x-bar was 1.99 pounds? Maybe this is too low and we should reject Ho. C. The role of z-scores • Remember a z-score tells us how many standard deviations a sample mean falls from the expected value, or population mean. • So would a sample mean that was 1 standard deviation below =3 be far enough below to reject Ho? • We need to consider the probability involved in calculating such a sample mean. D. The Rejection Range = .05 Z=-1.645 =3 x If we get a sample and calculate Z=1.645 below the mean, only a probability of .05 remains in the lower tail of the sampling distribution. Maybe this is low enough? • In other words, whenever the value of Z is less than -1.645, the probability of making a type I error would be .05. • Thus, we would reject Ho if Z<-1.645, if we believed that .05 was an acceptable degree of risk. • If we wanted to lower that to .01, our rejection range would lie below Z=-2.33. E. Methodology 1. Specify a maximum allowable probability of a type I error (). This is the probability of rejecting Ho when it is true. 2. Find Z that corresponds to . This is the critical Z score. If =.01, then Z corresponds to the area under the curve of .4900. Z=.01=2.33 Thus, reject Ho if Z<-2.33. Methodology continued 3. Take a sample, calculate the mean and standard error. 4. Calculate Z and compare to the critical Z. 5. If your Z is greater (in absolute value) than the critical Z, reject Ho. Example Ho: 3 lbs. Ha: < 3 lbs. A sample of 36 cans is taken and the sample mean is 2.92 lbs. Previous studies have found that historically cans are filled with a standard deviation of .18 lbs. The standard error of the sampling distribution is: .18 x .03 6 n Calculate the Z-score, Z x 2.92 3 2.67 x .03 Since this Z-score is greater (in absolute value) than the critical value of 2.33, we reject Ho and conclude that they are underfilling their coffee cans.