Question

Suppose we take a random sample X1,…,X5 from a normal population with unknown variance σ2 and unknown mean μ. We test the hypotheses

H0:σ=2 vs. H1:σ<2
and we use a critical region of the form {S2<k} for some constant k.

(1) - Determine k so that the type I error probability α of the test is equal to 0.05.

(2) - For the value of k found in part (1), what is your conclusion in this test if you see the following data:

1.84,0.30,0.04,1.34,−0.95

Answer 1

H0:σ=2

H1:σ<2

n=5

This is chi-square ( ) left tail test.

(1)

test statistic, = (n-1 ) / = 4 . / 4 =

=0.05 = Type I error probabilirty

We reject the null hypothesis if test statistic,   < critical value= k = 1- , n-1= 0.95 , 4 = 0.71 = k

So, the value of k is 0.71 so that the type I error probability of the test is equal to 0.05.

(2)

If the samples are(Xi) 1.84,0.30,0.04,1.34,−0.95 then

=( (Xi2)-n.2 ) / (n-1) = 1.2136

So, = (n-1 ) / = = 1.2136 > = 0.71 = critical value = 0.95 , 4

So, we failed to reject null hypothesis . Data is coming from population with σ=2.

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