iGreen
  • iGreen
Hypothesis Testing
Mathematics
schrodinger
  • schrodinger
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iGreen
  • iGreen
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iGreen
  • iGreen
I just need help finding the Test Statistic and P-value
anonymous
  • 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.

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iGreen
  • iGreen
Hmm..so how would I set this up?
iGreen
  • iGreen
Like, what do \(\sf \hat{p}, p_0,\) and \(\sf n\) stand for?
anonymous
  • 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 \(n=87\).
iGreen
  • iGreen
Ohhh..
iGreen
  • iGreen
So I have: \(\sf \dfrac{0.4138 - 0.6}{\sqrt{\dfrac{0.6(1 - 0.6)}{87}}} \approx -3.55 \)
anonymous
  • 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.
iGreen
  • iGreen
Ah, I see..thanks!
Australopithecus
  • Australopithecus
Also note: \[H_0 = null\ hypothesis\] \[H_1 = Research\ Hypothesis\]
Australopithecus
  • Australopithecus
The hypothesis is since the new treatment reduces deaths The null hypothesis is the new treatment doesn't reduce deaths
anonymous
  • 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.)

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