Question

The U.S. Bureau of Labor Statistics reports that 11.3% of U.S. workers belong to unions. Suppose a sample of 400 U.S. workers is collected in 2014 to determine whether union efforts to organize have increased union membership at 0.025 level of significance. The sample results in a test statistic (z) of 2.2.
We conclude that union membership increased in 2014. (Enter 1 if the conclusion is correct. Enter 0 otherwise.)

Answer 1

Answer:

1. The conclusion is statistically correct at the significance level given.

Step-by-step explanation:

1) Data given and notation n  

n=400 represent the random sample taken  

X represent the people with union membership in the sample

$\hat p$ estimated proportion of people with union membership in the sample

$p_o=0.113$ is the value that we want to test  

$\alpha=0.025$ represent the significance level (no given)  

z would represent the statistic (variable of interest)  

$p_v$ represent the p value (variable of interest)  

p= population proportion of people with union membership

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the proportion of people with union membership exceeds 11.3%. :  

Null Hypothesis: $p \leq 0.113$

Alternative Hypothesis: $p >0.113$

We assume that the proportion follows a normal distribution.  

This is a one tail upper test for the proportion of  union membership.

The One-Sample Proportion Test is "used to assess whether a population proportion $\hat p$ is significantly (different,higher or less) from a hypothesized value $p_o$".

Check for the assumptions that he sample must satisfy in order to apply the test

a)The random sample needs to be representative: On this case the problem no mention about it but we can assume it.

b) The sample needs to be large enough

$np_o =400*0.113=45.2>10$

$n(1-p_o)=400*(1-0.113)=354.8>10$

3) Calculate the statistic  

The statistic is calculated with the following formula:

$z=\frac{\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}}$

On this case the value of $p_o=0.113$ is the value that we are testing and n = 400.

Since we have already the statistic calculated z=2.2, we just need to calculate the p value in order to check if we can reject or not the null hypothesis.

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

Based on the alternative hypothesis the p value would be given by:

$p_v =P(z>2.2)=1-P(z$

Using the significance level given $\alpha=0.025$ we see that $p_v$ so we have enough evidence at this significance level to reject the null hypothesis. And on this case makes sense the claim that the union membership increased in 2014.