Statistical Hypothesis

larz · #1 $Old$ June 20th, 2009, 07:01 PM

A bowl contains 2 red balls, 2 white balls, and fifth ball that is either red or white. Let p denote the probability of drawing a red ball from the bowl. We shall test the simple null hypothesis H0: p=3/5 against the simple alternative hypothesis H1: p=2/5. Draw 4 balls at random from the bowl, one at a time and WITH replacement. Let X equal the number of red balls drawn.

Define the critical region C for this test is terms of X.
For the critical region C, find the values of $\alpha$ and $\beta$ .

Thank you for helping

matheagle · #2 $Old$ June 20th, 2009, 07:39 PM

This a binomial test and I can give you an $\alpha$ if you give me a decision rule/critical region.
Similarly if you give me an $\alpha$ I can then give you the critical region.
Clearly we reject the null hypothesis and accept the alternative for small X,
since p under the alternative is smaller than p under the null.

If we make the critical region X=0,

then $\alpha=P(X=0)={4 \choose 0} (3/5)^0(2/5)^4$

and $\beta=P(X>0)$ when p=2/5.

If we make the critical region X=0 or X=1, then

$\alpha=P(X=0)+P(X=1)={4 \choose 0} (3/5)^0(2/5)^4+{4 \choose 1} (3/5)^1(2/5)^3$

and $\beta=P(X>1)$ when p=2/5.

larz · #3 $Old$ June 21st, 2009, 08:20 AM

That is everything I have for this question. I came across some formulas to use the two probabilities to get the critical region by testing the hypothesis, but am not sure which to use. Would the probability of H0 be considered p0 and the probability of H1 be just p? That's what doesn't seem right. I'm looking for a formula for p0 and p1. Thanks

matheagle · #4 $Old$ June 21st, 2009, 09:26 AM

p0 is p under $H_0$ and that's just 3/5 in this problem.
The 3/5 represents a third red ball in the urn.

larz · #5 $Old$ June 27th, 2009, 02:50 PM

Don't I have to use an actual equation to find the critical region C? I am not sure if I can use the one from the chart for $H_{1}: \mu<\mu_{0}$ ( $\bar{x} \ge \mu_{0}+z_{\alpha}\sigma/\sqrt{n})$ since H1 is an actual value (p=2/5).

Do you think this question wants me to choose how many red balls so I have an actual X? Thanks

If I make my critical region $X \le 1$ , how can I figure out the $\alpha$ ? Thank you

matheagle · #6 $Old$ June 28th, 2009, 07:40 PM

You cannot use a normal approximation here.
The sample is small and the underlying distribution is binomial.
So drop the z's.

Quote:

Originally Posted by larz $View Post$

If I make my critical region $X \le 1$ , how can I figure out the $\alpha$ ? Thank you

I DID that!
X=0 or X=1 IS $X \le 1$ .
You need to learn what alpha is.
It's the probability of rejecting the null hypothesis assuming that the null is correct.
It's the probability of that error.