Stockbrokers and lawyers hypothesis testing

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Dreary

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Here is an interesting problem involvig hypothesis testing:

A national survey of lawyers found that lawyers drank an average of 6.8 cups of coffee each week. A random sample of 36 stockbrokers found that the stockbrokers drank an average of 6.2 cups of coffee each week with a standard deviation of 0.5. At the 5% significance level, which of the following is most accurate?

a. Stockbrokers drink the same amount of coffee per week as lawyers
b. Stockbrokers definitely drink less coffee per week than lawyers.
c. The analyst cannot conclude that stockbrokers drink the same amount of coffee per week as lawyers.
d. Stockbrokers drink more coffee per week than lawyers.

My paraphrasing of the reported answer is as follows:

To answer this you need to state your null hypothesis as u=6.8, that is assume stockbrokers drink the same amount of coffee as lawyers. Then do a two-tailed z test to see that the value 6.2 for stockbrokers falls outside the range, and you reject the null (that stockbrokers drink the same amount of coffee as lawyers).

Options a, b, and d do not work because our conclusion is that stockbrokers do not drink the same amount of coffee as lawyers. So the answer is C.

However, if we set the null hypothesis to u >= 6.8 (i.e., stockbrokers drink same or more coffee than lawyers), then we should reject the null if we find that 6.2 falls outside this range, and accept the alternative (that stockbrokers drink less coffee than lawyers). Wouldn't then answer "b" be the correct answer? So it seems that stating the null makes a difference!

Dreary
 
Just eyeballing this: the difference (6.8 - 6.2) is less than two st dev's, so we know that we can't confidently say (at 95%) the means are different. Any choice indicating a proven difference (b, d) are immediately stricken.

> Then do a two-tailed z test to see that the value 6.2 for stockbrokers falls outside the range

Is that true? It's been a while since I've done these, but I thought 95% was about 2 st dev's (here, 1.0) difference.

So I'd say a. is the answer.
 
^ Gotta check that out Darien..

It's not std. dev.'s that matter here; it's standard error. The standard error of the mean is 0.5/Sqrt(36) = 0.083. So our test statistic here is (6.2 -6.8)/0.083 = -7.23 which is distributed normally under the null because the sample size is > 30.

But now there is a quandary. You reject any H0 at 0.05, so the answer seems to be b) but that word "definitely" is an issue. Any one of those lawyers would say "could the difference be from sampling error?" to which I would say "well with probability 10^-16 or whatever" which may or may not be definitely.

C) is sort-of always true because you just cant do a hypothesis test to prove two things are the same and, in fact, with probability 1 they aren't the same. So C) is correct but substanceless.

a) and d) are way out.
 
What if I set the null hypothesis as Ho: u <= 6.8, and Ha: u > 6.8?

Then, Z would be as before (-7.23), which is less than the critical value. So I should reject the null, and conclude that "d" is the correct answer? What am I missing?
Dreary
 
Your rejection region in that case is Z > 1.65 and since -7.23 is not greater than 1.65 you would fail to reject the null.
 
Thanks Joey, that's what i was missing.
Dreary
 
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