Respuesta :
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
[tex]\hat p[/tex] estimated proportion of people with union membership in the sample
[tex]p_o=0.113[/tex] is the value that we want to test
[tex]\alpha=0.025[/tex] represent the significance level (no given)
z would represent the statistic (variable of interest)
[tex]p_v[/tex] 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: [tex]p \leq 0.113[/tex]
Alternative Hypothesis: [tex]p >0.113[/tex]
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 [tex]\hat p[/tex] is significantly (different,higher or less) from a hypothesized value [tex]p_o[/tex]".
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
[tex]np_o =400*0.113=45.2>10[/tex]
[tex]n(1-p_o)=400*(1-0.113)=354.8>10[/tex]
3) Calculate the statistic
The statistic is calculated with the following formula:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}}[/tex]
On this case the value of [tex]p_o=0.113[/tex] 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:
[tex]p_v =P(z>2.2)=1-P(z<2.2)=0.014[/tex]
Using the significance level given [tex]\alpha=0.025[/tex] we see that [tex]p_v<\alpha[/tex] 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.
