Question

Hypothesis Testing

Answer 1

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

Answer 2

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.

Answer 3

Hmm..so how would I set this up?

Answer 4

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

Answer 5

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 \(n=87\).

Answer 6

Ohhh..

Answer 7

So I have:

\(\sf \dfrac{0.4138 - 0.6}{\sqrt{\dfrac{0.6(1 - 0.6)}{87}}} \approx -3.55 \)

Answer 8

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.

Answer 9

Ah, I see..thanks!

Answer 10

Also note:

\[H_0 = null\ hypothesis\]
\[H_1 = Research\ Hypothesis\]

Answer 11

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

Answer 12

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.)