Respuesta :
Answer:
Null hypothesis:[tex]p_{1} = p_{2}[/tex]
Alternative hypothesis:[tex]p_{1} \neq p_{2}[/tex]
[tex]z=\frac{0.41-0.43}{\sqrt{0.42(1-0.42)(\frac{1}{1300}+\frac{1}{1300})}}=-1.033[/tex]
[tex]p_v =2*P(Z<-1.033)=0.302[/tex]
So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion 1 is not significantly different from the proportion 2.
Step-by-step explanation:
1) Data given and notation
n = 1300 sample size selected
[tex]p_{1}=0.41[/tex] represent the proportion of adults approve of President1.
[tex]p_{2}=0.42[/tex] represent the proportion of adults approve of President2.
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the proportion 1 is different from proportion 2 , the system of hypothesis would be:
Null hypothesis:[tex]p_{1} = p_{2}[/tex]
Alternative hypothesis:[tex]p_{1} \neq p_{2}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{0.41+0.43}{2}=0.42[/tex]
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.41-0.43}{\sqrt{0.42(1-0.42)(\frac{1}{1300}+\frac{1}{1300})}}=-1.033[/tex]
4) Statistical decision
For this case we don't have a significance level provided [tex]\alpha[/tex], but we can calculate the p value for this test.
Since is a two sided test the p value would be:
[tex]p_v =2*P(Z<-1.033)=0.302[/tex]
So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion 1 is not significantly different from the proportion 2.
