Considerate the follow data about voting intention for $1000$ couples. The variables $X_{1} ,X_{2} $ represent the women and men respectively, and $0$ represent voting for the coalition party and $1$ for the opssition party.
| $X_{1}= 0$ | $X_{1}=1$ | |
|---|---|---|
| $X_{2}=0$ | $245$ | $170$ |
| $X_{2}=1$ | $218$ | $367$ |
- Calculate the oddsratio $\hat{R}$
- If the size of the sample grows to $\infty$, the distribution of $\log(\hat{R})$ converges to a normal with expected value $\log(R)$ (I think my teacher wanted to write $\hat{R}$), the true log-oddratio of the underlying distribution and variances, $$\frac{1}{n_{0,0}}+ \frac{1}{n_{0,1}}+\frac{1}{n_{1,0}}+\frac{1}{n_{1,1}}$$
where $n_{i,j}$ is the observations with $X_{1}=i$ and $X_{2}=j$.
Do you support the hypotesis that men and women vote independently each other with significance level $\alpha = 0.05$?
For 1 we have that $\hat{R}=2.4226$.
For 2, under $H_{0}$ (i.e women and men vote independently each other) we have that the oddratio is $1$,son $\log(R)=0$ and the distribution of $\log(\hat{R})\rightarrow N(0, \frac{1}{n_{0,0}}+ \frac{1}{n_{0,1}}+\frac{1}{n_{1,0}}+\frac{1}{n_{1,1}})$.
But, because $\hat{R}=1$ then $n_{0,0}=n_{0,1}$ and $n_{1,0}=n_{1,1}$, so
$$\log(\hat{R})\rightarrow N\left(0, \frac{2}{n_{0,0}}+\frac{2}{n_{1,1}}\right)$$
I don't know what else to do
I don't understand what you are doing in the 2nd part of the question. $R$ is the (unknown) true odds ratio (the parameter of interest), and you have calculated its estimate (the observed value of $\hat R$) in the first part.
For a large sample test, your test statistic for testing $H_0: R=1 \iff\ln (R)=0$ is
$$T=\frac{\ln (\hat R)}{\sqrt{\frac{1}{n_{0,0}}+ \frac{1}{n_{0,1}}+\frac{1}{n_{1,0}}+\frac{1}{n_{1,1}}}} \stackrel{d}\longrightarrow N(0,1) \quad,\text{ under }H_0$$
Based on given data, I get
$$\hat R=\frac{n_{0,0}\times n_{1,1}}{n_{0,1}\times n_{1,0}}=\frac{245\times 367}{218\times 170}\approx 2.42620 \implies \ln (\hat R)\approx 0.88633$$
And
$$\frac{1}{n_{0,0}}+ \frac{1}{n_{0,1}}+\frac{1}{n_{1,0}}+\frac{1}{n_{1,1}}=\frac1{245}+\frac1{218}+\frac1{170}+\frac1{367} \approx 0.01727$$
So find the observed value of $T$ and draw your conclusion, either using p-value or standard normal tables. The alternative hypothesis is presumably $H_1: \ln R\ne 0$.