Question

The U.S. Bureau of Labor Statistics reports that 11.3% of U.S. workers belong to unions (BLS website, January 2014). Suppose a sample of 400 U.S. workers is collected in 2014 to determine whether union efforts to organize have increased union membership.a. Formulate the hypotheses that can be used to determine whether union membership increased in 2014. H0: p Ha: p b. If the sample results show that 52 of the workers belonged to unions, what is the p-value for your hypothesis test (to 4 decimals)? c. At α = .05, what is your conclusion?

Answer 1

a) Null hypothesis:
`p \leq 0.113`

Alternative hypothesis:
`p > 0.113`

b)
`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

Replacing we got:

`z=\frac{0.13 -0.113}{\sqrt{(0.113(1-0.113))/(400)}}=1.074`

The p value for this case would be given by:

`p_v =P(z>1.074)=0.141`

c) For this case we see that the p value is higher than the significance level of 0.05 so then we have enough evidence to FAIL to reject the null hypothesis and we can't conclude that the true proportion workers belonged to unions is significantly higher than 11.3%

Explanation:

Information given

n=400 represent the random sample taken

X=52 represent the workers belonged to unions

`\hat p=(52)/(400)=0.13` estimated proportion of workers belonged to unions

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

`\alpha=0.05` represent the significance level

Confidence=95% or 0.95

z would represent the statistic

`p_v` represent the p value

Part a

We want to test if the true proportion of interest is higher than 0.113 so then the system of hypothesis are.:

Null hypothesis:
`p \leq 0.113`

Alternative hypothesis:
`p > 0.113`

Part b

The statistic is given by:

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

Replacing we got:

`z=\frac{0.13 -0.113}{\sqrt{(0.113(1-0.113))/(400)}}=1.074`

The p value for this case would be given by:

`p_v =P(z>1.074)=0.141`

Part c

For this case we see that the p value is higher than the significance level of 0.05 so then we have enough evidence to FAIL to reject the null hypothesis and we can't conclude that the true proportion workers belonged to unions is significantly higher than 11.3%