Answer:
Z= (X1`-X2`)- (u1-u2)/[tex]\sqrt{ \frac{ s_1^2}{n_1^2} +\frac{ s_2^2}{n_2}[/tex] (here s1=σ1 and s2= σ2)
Step-by-step explanation:
Let X1` be the mean of the first random sample of size n1 from a normal population with a mean u1 and known standard deviation σ1.Let X2` be the mean of the second random sample of size n2 from another normal population with a mean u2 and known standard deviation σ2. Then the sampling distribution of the difference X1`-X2` is normally distributed with a mean of u1-u2 and a standard deviation of
[tex]\sqrt{ \frac{ s_1^2}{n_1^2} +\frac{ s_2^2}{n_2}[/tex] . (here s1=σ1 and s2= σ2) In other words the variable
Z= (X1`-X2`)- (u1-u2)/[tex]\sqrt{ \frac{ s_1^2}{n_1^2} +\frac{ s_2^2}{n_2}[/tex] (here s1=σ1 and s2= σ2)
is exactly standard normal no matter how small sample sizes are . Hence it is used as a test statistic for testing hypotheses about the difference between two population means.
The procedure is stated below.
1) Formulate the null and alternative hypotheses.
2) Decide on significance level ∝
3) Use the test statistic Z= (X1`-X2`)- (u1-u2)/[tex]\sqrt{ \frac{ s_1^2}{n_1^2} +\frac{ s_2^2}{n_2}[/tex] (here s1=σ1 and s2= σ2)
4) Find the rejection region
5)Compute the value of Z from the sample data
6) Rehect H0 if Z falls in the critical region, accept H0 , otherwise.
