This question comes from an old final.A lawyer who knows a little bit about statistics decides to find out exactly how many sheets of designer toilet paper are in the rolls he usually buys at his local "Why-Pay-Less?" store. Although the package advertises that there are 1000 sheets per roll, he thinks the rolls run out too quickly and therefore must have less than 1000 sheets. He calls the manufacturer and learns that the rolls are normally distributed with a mean of 1000 sheets per roll and a standard deviation of 12 sheets.
He goes out and buys a package of 9 rolls. Treat this like a random sample. He counts each sheet in the 9 rolls and comes up with the following:
998, 999, 1001 ,1000, 921, 999, 1001, 998, 997
1. Does the lawyer have enough evidence to sue the manufacturer for false advertising? Use a 5% level of significance as your rule of thumb.
The Null Hypothesis is: the average = 1000
The Alternative Hypothesis is: the average is less than 1000
The test statistic looks like
Z = 990.44 - 1000 -------------------- = -2.39 about -2.40 SQRT(9) x 12 ----------- 9The area between Z=+2.40 and Z=-2.40 is 98.36 percent. So the area below Z=-2.40 is (100 - 98.36 / 2) = .82%. If the manufacturer's claim is true, then the lawyer only had a .82% chance of drawing a random sample of size 9 with an average as low as 990.44 or lower.This .82% chance is a lot lower than the 5% level that we're using as a "rule of thumb". I'd say he's got evidence.
2. Lawsuits are expensive and time-consuming and this lawyer is cautious. He hires you to analyze his data with this thought in mind: given what you have learned about means, medians, standard deviations, extreme observations, sample sizes, perhaps this data set has an unusual observation. If you think it does, go ahead and remove it from the sample and perform a second significance test. Would you advise him to pursue his lawsuit against the manufacturer?
If you think it does not, explain why not.
OK. The roll with 921 looks strange. 921 is about 79 sheets less than expected value for a single roll or its about a Z= 6 away (i.e. 6 standard errors for a sample of 1).
You have two routes here -- if you delete the 921 from the list and treat it like a sample of 8, the average is now 999.125. The new Z would be about -.20 or there is a 42% chance of getting an average of 999.125 for a sample of 8. You might advise him not to pursue a lawsuit.
On the other hand, you could recommend that he recheck the roll of 921 to make sure it is 921. Or that he take another sample. If he bought 9 more rolls and got another bad one like the 921...he really may have enough evidence to pursue a lawsuit. The company may be quite smart about cheating its customers.
Statistical Significance. We use the term "significant" in the statistical sense to mean that an observed difference is likely a real difference and not explainable in terms of sampling error. The level of likelihood used in our analysis is 95%. In other words, when we say a difference is significant we are saying that there is a 95% chance that it is a real difference and not the result of sampling error. This does not mean that the particular result is important in a managerial or any other sense.If a particular difference is large enough to be unlikely to have occurred due to chance or sampling error, then the difference is statistically significant.
Mathematical differences. By definition, if numbers are not exactly the same, they are different. This does not however suggest that the difference is either important or statistically significant.
Managerially important differences. If results or numbers are different to the extent that the difference would matter from a managerial perspective, we can argue that the difference is important. For example, the difference in consumer response to two different packages in a test market might be statistically significant but yet so small as to have little practical or managerial significance.
When you are in small sample situations and the population SD is unknown, the z-test must be modified. So we have something called a t-test.
It is like the z-scores you learned in Chapters 21 and 23 except the difference is in the calculation of the "SD of the box".
You are not allowed to just substitute the SD of the box with the sample SD when your samples are small (below 25).
And you can no longer use the normal curve on page A-105 but instead you must you the t-table on page A-106 to read off areas and get p-values.
The new SD is simply SD+ as described in Chapter 4.7. Or for some of you, it's the SD your calculator gives automatically.
_________________________________
SD+ = / number of measurements
/ -------------------------------- X SD
\/ number of measurements - one
An example.
Suppose from the population of UCLA male undergraduates, a random sample of size 16 is picked and each male's height is measured. Suppose for this random sample, the average height is 68 inches with a standard deviation of 2 inches. The Chancellor claims that UCLA men as tall as USC men (who average 69 inches in height). Is his claim correct based on the evidence from this sample?
The corrected SD = 2.0656.THere was a mistake in my notes in the previous lecture. Sorry, it's been corrected here.
2.0656 = SQRT(16/15) x 2Test is: 68 - 69 ----------------------- = -1.9465 = t SQRT(16) x 2.0656 ---------------- 16
with 15 degrees of freedom. This falls between 5% and 2.5% on the t table. If your rule of thumb was a 5% level of significance, we'd say that the chancellor's claim is not correct. We would reject the null hypothesis so the difference we are seeing is not due to chance. There is evidence that UCLA men are shorter.
If your rule of thumb was 1%, then we'd say would not reject the null so the difference we are seeing is due to chance and there is no evidence that UCLA men are shorter.
Generally, you are asked to either state a level of significance or are given a level of significance. Perhaps you can see why here...if we didn't like the results...we could say "oh, I don't use a 5% level, I use a 1%"
We will not be doing this chapter. Have a happy Thanksgiving.
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PageLast Update: 23 November 1998 by VXL