Respuesta :
Answer:
[tex]p_v =P(Z<-1.837)=0.033[/tex]
If we compare the p value with any significance level for example [tex]\alpha=0.05,0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the true proportion 1 is less than the true proportion 2, at 5% or 10% of significance .
Step-by-step explanation:
1) Data given and notation
[tex]X_{1}=40[/tex] represent the number of successes for 1
[tex]X_{2}=60[/tex] represent the number of successes for 2
[tex]n_{1}=135[/tex] sample of 1 selected
[tex]n_{2}=150[/tex] sample of 2 selected
[tex]\hat p_{1}=\frac{40}{135}=0.296[/tex] represent the sample proportion for 1
[tex]\hat p_{2}=\frac{60}{150}=0.40[/tex] represent the sample proportion 2
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 less than the proportion 2, the system of hypothesis would be:
Null hypothesis:[tex]p_{1} \geq p_{2}[/tex]
Alternative hypothesis:[tex]p_{1} < p_{2}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{\hat p_{1}-\hat 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{40+60}{135+150}=0.3509[/tex]
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.296-0.40}{\sqrt{0.3509(1-0.3509)(\frac{1}{135}+\frac{1}{150})}}=-1.837[/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 one left tailed test the p value would be:
[tex]p_v =P(Z<-1.837)=0.033[/tex]
If we compare the p value with any significance level for example [tex]\alpha=0.05,0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the true proportion 1 is less than the true proportion 2, at 5% or 10% of significance .
