## iGreen one year ago Hypothesis Testing

1. iGreen

2. iGreen

I just need help finding the Test Statistic and P-value

3. anonymous

The study mentions proportions of patients responding to the treatment, so that's what your test should be concerned with. The t.s. for a proportion can be derived from the usual $$Z$$ t.s.: $Z=\frac{\bar{x}-\mu_0}{\sigma}=\frac{n\hat{p}-np_0}{\sqrt{np_0(1-p_0)}}=\frac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$ where $$p_0$$ is the proportion assumed under the null hypothesis.

4. iGreen

Hmm..so how would I set this up?

5. iGreen

Like, what do $$\sf \hat{p}, p_0,$$ and $$\sf n$$ stand for?

6. anonymous

Prior to the treatment, you have a mortality rate of $$60\%$$, so this is $$p_0$$. The study shows a mortality rate of $$\hat{p}=\dfrac{36}{87}\approx0.4138$$. Right away you also know the sample size is $$n=87$$.

7. iGreen

Ohhh..

8. iGreen

So I have: $$\sf \dfrac{0.4138 - 0.6}{\sqrt{\dfrac{0.6(1 - 0.6)}{87}}} \approx -3.55$$

9. anonymous

Right, now to find the $$p$$ value you can refer to a $$z$$ table, or if you're looking for better accuracy you might want to use a calculator to compute the area under the distribution curve.

10. iGreen

Ah, I see..thanks!

11. Australopithecus

Also note: $H_0 = null\ hypothesis$ $H_1 = Research\ Hypothesis$

12. Australopithecus

The hypothesis is since the new treatment reduces deaths The null hypothesis is the new treatment doesn't reduce deaths

13. anonymous

This computation shows that the critical value is $$|Z_{\alpha/2}|=|Z_{0.005}|\approx2.5758$$: http://www.wolframalpha.com/input/?i=Solve%5BIntegrate%5BPDF%5BNormalDistribution%5B0%2C1%5D%2Cx%5D%2C%7Bx%2C-t%2Ct%7D%5D%3D%3D.99%2Ct%5D In order to reject the null hypothesis, you need the test statistic $$Z$$ to satisfy $$|Z|>2.5758$$, which is clearly the case. This computation approximates the exact $$p$$ value: http://www.wolframalpha.com/input/?i=Integrate%5BPDF%5BNormalDistribution%5B0%2C1%5D%2Cx%5D%2C%7Bx%2C-Infinity%2C-3.5453%7D%5D (I've included this because the typical $$z$$ table doesn't provide the same level of precision.)