Question

In a recent poll 1087 randomly selected adults were asked whether they approve of labor​ unions; 61​% said yes. In​ 1936, about 63​% of adults approved of labor unions. At the 1​% significance​ level, do the data provide sufficient evidence to conclude that the percentage of adults who approve of labor unions has decreased since​ 1936? Use the​ one-proportion z-test to perform the appropriate hypothesis​ test, after checking the conditions for the procedure.

What is the test statistic?

Answer 1

Let p be the true current proportion of adults who approve of labor unions.

We want to test if this proportion of adults who approve of labor unions has decreased since​ 1936. That means, we want to test if p<0.63

The hypotheses are

$\begin{align*} &H_0:p=0.63\leftarrow\text{the current proportion of adults who approve of labor unions is 63\% }\\ &H_a:p<0.63\leftarrow\text{the current proportion of adults who approve of labor unions is less than 63\% }\\ \end{align*}$

The level of significance to test the hypotheses is $\begin{align*} \alpha=0.01 \end{align*}$

From the sample we know the following information

n=1087 is the sample size

$\begin{align*} \hat{p}=0.61 \end{align*}$ is the sample proportion of adults who approve of labor unions at present

$\begin{align*} p_{H_0}=0.63 \end{align*}$ is the hypothesized value of current proportion of adults who approve of labor unions

The standard error of proportion is

$\begin{align*} \sigma_{\hat{p}}=\sqrt\frac{p_{H_0}(1-p_{H_0})}{n}=\sqrt\frac{0.63(1-0.63)}{1087}=0.0146 \end{align*}$

The values of

$\begin{align*} np_{H_0}=1087\times 0.63=684.81\\ n(1-p_{H_0})=1087\times(1- 0.63)=402.19 \\ \end{align*}$

are both greater than 5. Hence we can use normal distribution as the sampling distribution of proportion.

That is, we can use one-proportion z-test to perform the appropriate hypothesis​ test.

The test statistics is

$\begin{align*} z=\frac{\hat{p}-p_{H_0}}{\sigma_{\hat{p}}}=\frac{0.61-0.63}{0.0146 }=-1.37 \end{align*}$

this is a 1 tailed (left tail) test (The alternative hypothesis has "<")

The left tail critical value for $\begin{align*} \alpha=0.01 \end{align*}$ is

$\begin{align*} P(Z<-z_{\alpha})=\alpha=0.01\\\implies P(Z>z_{\alpha})=0.01\\\implies P(Z<z_{\alpha})=0.99 \end{align*}$

Using the standard normal tables, we can get for z=2.33, P(Z<2.33) = 0.99

The left tail critical value is -2.33

We will reject the null hypothesis, if the test statistics is less than the critical value.

Here, the test statistics is -1.37 and it is not less than the critical value -2.33. Hence we do not reject the null hypothesis.

We conclude that there is no sufficient evidence to support the claim that the percentage of adults who approve of labor unions has decreased since​ 1936.