12.2: Two Proportions Z Test
- Page ID
- 105867
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Next, we will learn how to apply the Two Proportions \(Z\) Procedure to test a statistical claim about 2 population proportions. Consider the following example.
A group of researchers wants to know whether men are more likely than women to contract a novel coronavirus. They surveyed two random samples of 350 men and 340 women who were tested and found that 70 men and 36 women in the samples tested positive.
At \(1\%\) significance level, test the claim that men are more likely to have the novel coronavirus than women.
Note that the sample proportions are \(\hat{p}_1=\frac{70}{350}=0.2000\) and \(\hat{p}_2=\frac{36}{340}=0.1059\) for men and women respectively, also the pooled proportion is \(\hat{p}_p=\frac{70+36}{350+340}=0.1536\).
Now, let’s identify the statistical claim that needs to be tested:
“men are more likely to have the novel coronavirus than women”
It is not obvious, but the claim is about the population proportion of positive tested men being greater than the proportion of positive tested women, so we can symbolically express the claim as
\(p_1>p_2\) or \(p_1-p_2>0\)
Since the claim is about the 2 population proportions we will use the 2 Proportions \(Z\) Procedure. Let’s check if all necessary assumptions are satisfied:
- The samples are simple random and independent
- The number of positive and negative responses are both greater than 10 for both samples.
We will use the following template to perform the hypothesis testing:
In step 1, we will set up the hypothesis:
Since the claim is in the form of an inequality, therefore it must be set as an alternative hypothesis, therefore the null hypothesis is \(p_1-p_2=0\) and the test is right-tail, that is:
|
\(H_0: p_1-p_2=0\) \(H_a: p_1-p_2>0\) |
RT |
In step 2, we will identify the significance level:
The significance level can always be found in the text of the problem. In our case it is \(1\%\), thus:
\(\alpha=0.01\)
In step 3, we will find the test statistic using the formula:
\(z_0=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}_p\cdot(1-\hat{p}_p)}\cdot\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.2000-0.1059}{\sqrt{0.1536\cdot0.8464}\cdot\sqrt{\frac{1}{350}+\frac{1}{340}}}=3.427\)
In step 4, we will perform either the critical value approach or p-value approach to test the claim:
- In critical value approach, we construct the rejection region using the Z-curve:
RR: greater than \(z_{0.01}=2.326\)
- In p-value approach, we compute the p-value:
P-Value: \(P(Z>3.427)=0.0003\)
In step 5, we will draw the conclusion:
- In the critical value approach, we must check whether the test statistic is in the rejection region or not. Our test statistic is \(3.427\) and it is to the right of the critical value \(2.326\), thus it is in the rejection region.
- In the p-value approach, we must check whether the p-value is less than the significance level or not. Our p-value is \(0.0003\) and it is less than \(\alpha=0.01\).
Both tests suggest that we DO reject the null hypothesis in favor of the alternative.
In step 6, we will interpret the results:
Under \(1\%\) significance level we DO have sufficient evidence to suggest that men are more likely to have the novel coronavirus than women.
We discussed how to apply the Two Proportions \(Z\) Procedure to test a statistical claim about two population proportions.