They want you to compute under the SIMPLE hypothesis of
1a) The strength of concrete depends to some extent on the method used for drying it. Two different drying methods were tested independently on specimens. The strength using each of the methods follow a normal distribution with mean μ_x and μ_y respectively and the same variance. The results are:
method 1: n1=7, x bar=3250, s1=210
method 2: n2=10, y bar=3240, s2=190
Do the methods appear (use alpha=0.05) to produce concrete with different mean strength ?
For this one, the test is H_o: μ_x=μ_y v.s. H_a: μ_x≠μ_y. I computed a p-value of >0.2, so the p-value is greater than alpha(which is 0.05), and so we fail to reject H_o and the answer is "no".
Now I am stuck with part b...
1b) Suppose σ_x=210, σ_y=190 (n1, n2, x bar, y bar same as part a). Find the probability of deciding that the methods are not different when the true difference in means is 2.
I am having trouble understanding what the question is asking for. Is it asking for P(type II error)? If so, how can I find it in this case?
Any help is greatly appreciated!
P.S. By the way, this topic is also discussed in SOS mathematics cyberboard forum.
Now I am a bit confused...
(i) P(fail to reject H_o | H_o is false) = β
(ii) P(fail to reject H_o | μ_x-μ_y=2)
Are the two probabilities above equal? In (i), we are given (conditional on) that H_o is false, so μ_x≠μ_y. In (ii), we are given that μ_x-μ_y=2.
But μ_x≠μ_y and μ_x-μ_y=2 aren't equivalent (μ_x≠μ_y does not imply that μ_x-μ_y=2, it could be that μ_x-μ_y=3, μ_x-μ_y=4, etc.), so I think (i) and (ii) are not equal. And it looks like the question is asking for (ii), right?
μ_x-μ_y=2 guarantees μ_x≠μ_y, but the converse is false, hence not if and only if...
As I wrote yesterday. There is a different beta for every different possible outcome under the alternative hypothesis. And one minus beta is the power FUNCTION. There are a lot of beta's. YOU were asked to obtain just one, when the difference of the two populations means were equal to 2.
I've been seeing two versions of the definition for β...
Definition 1: β = P(fail to reject H_o | H_o is false)
Definition 2: β = P(fail to reject H_o | H_a is true)
Are these two definitions consistent?
So in hypothesis testing, saying H_o is false is EXACTLY THE SAME thing as saying that H_a is true, am I right?
yes, we end up deciding on one or the other
But most statisticians either say we accept
or we fail to accept .
Some won't say we accept , because this really is like a proof by contradiction.
We assume to be correct, only to hope that it isn't and thus deciding on .