Known Standard Deviations
When we are comparing two means, we can say that the null hypothesis is…
Equation
$$ H_{0}: \mu_{1} = \mu_{2}$$
$$ H_{A}: \mu_{1} \neq \mu_{2} $$
$$ H_{A}: \mu_{1} \neq \mu_{2} $$
We could also have something like this:
Equation
$$ H_{0}: \mu_{1} \leq \mu_{2}$$
$$ H_{A}: \mu_{1} > \mu_{2} $$
$$ H_{A}: \mu_{1} > \mu_{2} $$
Or like this:
Equation
$$ H_{0}: \mu_{1} \geq \mu_{2}$$
$$ H_{A}: \mu_{1} < \mu_{2} $$
$$ H_{A}: \mu_{1} < \mu_{2} $$
The related t statistic to this is:
Equation
$$z = \frac{(\bar{x_{1}}-\bar{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^2}{n_{1}}+\frac{\sigma_{2}^2}{n_{2}}}} $$
In this case we assume the μ
1
– μ
2
= 0, but there may be times where we think the two are not equal and in fact there is a difference by an amount, so we want to test that it is the correct amount.
Challenge
Let’s say we known we have two samples, both with known standard devations and we want to test the null hypothesis that the two means are the same. One has a sample mean of 100, sample size 50, and standard deviation 5; the other has a sample mean 105, sample size 75 and standard deviation of 10. What is the associated z score and p value? What if we want to see if the second sample has a greater mean?