# Thread: test statistic and probaility

1. ## test statistic and probaility

Assume that
X1, . . . ,X10 ∼ Normal(μ1, σ1^2) and that Y1, . . . , Y20 ∼ Normal(μ2, σ2^2).

None of the population parameters are known. Let = μ1 − μ2.

To test H0 : ≥ 0 versus H1 : < 0 at significance level α = 0.05, we observe samples ~x and ~y.

(a) What test should be used in this situation?

If we observe ~x and ~y that result in¯x = −0.82, s1 = 4.09, ¯y = 1.39, and s2 = 1.22, then what is the value of the test
statistic?

(b) If we observe ~x and ~y that result in s1 = 4.09, s2 = 1.22,, and a test statistic
value of 1.76, then which of the following R expressions best approximates the
significance probability?

i. 2*pnorm(-1.76)

ii. pt(-1.76,df=28)

iii. pt(1.76,df=10)

iv. pt(-1.76,df=10)

v. 2*pt(1.76,df=28)

(c) True of False: if we observe ~x and ~y that result in a significance probability of p = 0.96, then we should reject the null hypothesis.

i try to solve this problem ......by using R program

(a)We should use Welch’s approximate T-test in this situation.

> v=(((((4.09)^2)/10)+(((1.22)^2)/20))^2)/((((4.09/10)^2)/9)+(((1.22/20)^2)/19))
> v
[1] 162.5339
> qt=qt(.975,v)
> qt
[1] 1.974667
> tw=((-.82-1.39)-0)/(sqrt(((4.09^2)/10)+((1.22^2)/20)))
> tw
[1] -1.671927
ᅵtwᅵ=1.671926<qt=1.974667

We decline to reject at significance level α=0.05

(b)
(ii) pt(-1.76,df=28)

(c)

i think i have problem in (a), and (c) please let me know how to solve this problem

2. in part (a) you need to know if you can assume that the two population variances are equal or not
It seems that you cannot.
go to http://en.wikipedia.org/wiki/Student's_t-test
and see where it says Unequal sample sizes, unequal variance
the unequal variances is the point, not the sample sizes