11.3: One Proportion Z Test
- Page ID
- 105862
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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 One Proportion \(Z\) Procedure to test a statistical claim about a population proportion. Consider the following example.
The ABC News/Ipsos poll was conducted by Ipsos Public Affairs‘ Knowledge Panel® on March 18-19, 2020, in English and Spanish, among a random national sample of 512 adults. In the poll, 282 Americans approve of the president's management of the crisis. At \(10\%\) significance level, test the claim that the majority of the Americans approve the president's management of the crisis.
Note that the sample proportion is \(\hat{p}_i=\frac{282}{512}=0.5508\) or \(55.08\%\).
Now, let’s identify the statistical claim that needs to be tested:
“the majority of the Americans approve”
It is not obvious, but the claim is about the population proportion being greater than \(50\%\), so we can symbolically express the claim as
\(p>0.50\)
Since the claim is about the population proportion, we will use the One Proportion \(Z\) Procedure. Let’s check if all necessary assumptions are satisfied:
- The sample is simple random
- The number of positive and negative responses are both greater than 10.
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 \(p=0.5\) and the test is right-tail, that is:
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\(H_0: p=0.5\) \(H_a: p>0.5\)
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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 \(10%\), thus:
\(\alpha=0.10\)
In step 3, we will find the test statistic using the formula:
\(z_0=\frac{\hat{p}_i-p_0}{\sqrt{p_0(1-p_0)/n}}=\frac{0.5508-0.5}{\sqrt{0.5(1-0.5)/512}}=2.30\)
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:
RR: greater than \(z_{0.1}=1.28\)
- In p-value approach, we compute the p-value:
P-Value: \(P(Z>2.30)=0.011\)
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 \(2.30\) and it is to the right of the critical value \(1.28\), 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 \(0.011\) and it is less than \(\alpha\).
Both tests suggest that we DO reject the null hypothesis in favor of the alternative.
In step 6, we will interpret the results:
Under \(10\%\) significance level we have sufficient evidence to suggest that the majority of the Americans approve the president’s management of the crisis.
We discussed how to apply the One Proportion \(Z\) Procedure to test a statistical claim about a population proportion.