BIOS14: Hypothesis testing（假设检验）using R

NOTES

1 The basic problem of hypothesis test

1.1 Example

year	weight of new born (mean)	size of data	SD
1989	3190 g	$\infty$	80 g
1990	3210 g	100	NA

It is obvious that the mean of 100 new born’s weight is 20g higher than that of 1989.

The point here is that what is shown by this 20 g diference. One explain is that it caused by randomness of sampling. Or it is true that baby born in 1990 is weighter than born in 1989.

To figure out this problem, we use ${\mu }_0$ indecate the mean of baby’s weight in 1989 and $\mu$ indecate that of 1990. We hypothesize that $\mu ={\mu }_0$, and use 100 samples in 1990 to test whether the hypothesis is true or not.

Here
Null hypothesis: $H_0:\mu =3190g$
If the null hypothesis is not stand, we need an alternative hypothesis:
Alternative hypothesis: $H_1:\mu \ne 3190g$

1.2 Two type error

$\alpha error$: $H_0$ is right but is rejected.
$\beta error$: $H_1$ is wrong but is accepted.

1.3 The procedure of hypothesis test

1. raise hypothesis:
   $H_0:\mu =3190g$
   $H_1:\mu \ne 3190g$
2. statistics ($\sigma$ is known, sample > 30):
   $$z=\frac{\bar{x}-{\mu }_0}{\sigma \sqrt{n}}=\frac{3210-3190}{80/\sqrt{100}}=2.5$$
3. $z_{\frac{\alpha }{2}}=\pm 1.96<2.5$, $z$ is in the critical region, so we reject $H_0$.