STAT 211 Topic 7

Lecture 14 Notes

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Hypothesis Testing

1. Form hypotheses (null and alternative)
2. Find test statistic (value from sample data on which decision is based
   • z-test (if it follows Z distribution)
   • t-test (if it follows T distribution)
   • χ²-test (if it follows χ² distribution)
3. Find rejection region
4. Reach conclusion

Significance Level

$\alpha$ represents "significance level" and is usually predetermined. It is the probability that we make a type 1 error (see below).

2 types of errors:

1. Rejecting the null hypothesis when the null hypothesis is true
2. Fail to reject the null hypothesis although the null hypothesis is false

Case 1: Normal population with known σ

Null hypothesis:

$H_0:\mu ={\mu }_0$

Test Statistic:

$z=\frac{\overline{x}-{\mu }_0}{\sigma /\sqrt{n}}$

Rejection Region:

$\mu >{\mu }_0;z\ge z_{\alpha }$

$\mu <{\mu }_0;z\ge z_{\alpha }$

$\mu \ne {\mu }_0;z\ge z_{\alpha /2}\text{or}z\le -z_{\alpha /2}$

Case 2: Large Sample

Similar to Case 1, but we use our sample standard deviation instead of σ

Null hypothesis:

$H_0:\mu ={\mu }_0$

Test Statistic:

$z=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

Rejection Region:

$\mu >{\mu }_0;z\ge z_{\alpha }$

$\mu <{\mu }_0;z\ge z_{\alpha }$

$\mu \ne {\mu }_0;z\ge z_{\alpha /2}\text{or}z\le -z_{\alpha /2}$

Case 3: Normal population with unknown σ and small sample

Similar to Case 2, only we use t-test instead of z-test.

Null hypothesis:

$H_0:\mu ={\mu }_0$

Test Statistic:

$t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

Rejection Region:

$\mu >{\mu }_0;t\ge t_{\alpha }$

$\mu <{\mu }_0;t\ge t_{\alpha }$

$\mu \ne {\mu }_0;t\ge t_{\alpha /2}\text{or}t\le -t_{\alpha /2}$

Example

Suppose we have the following information

• $\sigma =10$ (known standard deviation)
• $\mu =170$ (our claim about a normal distribution)
• $n=30$ (sample size)
• $\overline{X}=180$ (sample average)
• $s=12$ (sample standard deviation)

Null hypothesis

$H_0:\mu =170$

Alternative Hypothesis

$H_a:\mu \ne 170$

Test Statistic

$z=\frac{180-170}{10/\sqrt{30}}=\sqrt{30}\approx 5.48$

If our significance level $\alpha =1\mathrm{\%}$

We reject the null hypothesis if $z>z_{0.01}$

$5.48>2.58$, so we reject the null hypothesis.

One-sided vs. Two-sided

If the null hypothesis is that the average is a certain number ($\mu =17$):

a one-sided test

would be if our alternative hypothesis is either $\mu >17$ or $\mu <17$.

a two-sided test

includes both cases of the above alternatives in a single alternative hypothesis: $\mu \ne 17$.

Lecture 15

Lecture 15 Notes

Testing Population Proportion

• We must have a large sample
• $H_0:p=p_0$ (the proportion is claimed to be some number $p_0$)
• $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

  Note: be careful to use $p_0$ in the denominator, not $\hat{p}$!

Rejection Region:

$p>p_0;z\ge z_{\alpha }$

$p<p_0;z\ge z_{\alpha }$

$p\ne p_0;z\ge z_{\alpha /2}\text{or}z\le -z_{\alpha /2}$

Closer look at Hypothesis Testing

If the null hypothesis is true, the significance level $\alpha$ is the probability that the null hypothesis will be rejected:

If $H_0$ is true:

• $P(z\ge z_{\alpha })=\alpha$
• $P(z\le -z_{\alpha })=\alpha$
• $P(z\ge z_{\alpha /2}\text{or}z\le -z_{\alpha /2})=\alpha$

Errors

Type 1

Rejecting the null hypothesis when the null hypothesis is true

We let the probability that the null hypothesis is true and the test statistic falls into the rejection region $P(\text{Type 1 error})=\alpha$

Type 2

Fail to reject the null hypothesis although the null hypothesis is false

we don't care about this, but the book discusses a way to calculate this probability

P-Values

Suppose that we reject the null hypothesis at 5% significance level, but not at 1%.

Our statistic $z$ changes from sample to sample, but if the null hypothesis is true, $z\sim N(0,1)$ and our statistic should be somewhere around 0. If our $z=5$ for example, then we know that something is wrong with our null hypothesis.

P-value is the probability that another test statistic $Z$ is further away from mean 0 than our $z$ above. If this value very small (i.e. smaller than the significance level $\alpha$), then we reject the null hypothesis.

• $H_0:\mu ={\mu }_0$
• $H_a:\{\begin{array}{ll}\mu >{\mu }_0& P(Z>z)=1-\mathrm{\Phi }(z)\\ \mu <{\mu }_0& P(Z<z)=\mathrm{\Phi }(z)\\ \mu \ne {\mu }_0& P(Z>|z|\text{or}Z<-|z|)=2\times \min (P(Z>z),P(Z<z))\end{array}$

What about t-test?

Substitute $t$ for $z$

• $H_0:\mu ={\mu }_0$
• $H_a:\{\begin{array}{ll}\mu >{\mu }_0& P(T_{n-1}>t)\\ \mu <{\mu }_0& P(T_{n-1}<t)\\ \mu \ne {\mu }_0& P(T_{n-1}>|t|\text{or}{T}_{n-1}<-|t|)=2\times \min (P(T_{n-1}>t),P(T_{n-1}<t))\end{array}$