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.

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Presentation transcript:

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 If we take a random sample of n=36, we use the C.L.T. to assume a normal sampling distribution.  = 3

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 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?  = 3  =.05 Z=-1.645

In other words, whenever the value of Z is less than , 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 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:

Calculate the Z-score, 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.