ANCOVA: overview
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ANCOVA  $z$ test for a single proportion 


Independent variables  Independent variable  
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)  None  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
THIS TABLE IS YET TO BE COMPLETED  Null hypothesis  
  $\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to H0  
n.a.  Alternative hypothesis  
  Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  
n.a.  Assumptions  
 
 
n.a.  Test statistic  
  $z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
$p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size  
n.a.  Sampling distribution of $z$ if H0 were true  
  Approximately standard normal  
n.a.  Significant?  
  Two sided:
 
n.a.  Approximate $C\%$ confidence interval for $\pi$  
  Regular (large sample):
 
n.a.  Equivalent to  
  When testing two sided: goodness of fit test, with categorical variable with 2 levels  
n.a.  Example context  
  Is the proportion smokers amongst office workers different from $\pi_0 = .2$? Use the normal approximation for the sampling distribution of the test statistic.  
Pratice questions  Pratice questions  