Respuesta :
Answer:
[tex]z=\frac{0.42-0.45}{\sqrt{\frac{0.42(1-0.42)}{100}+\frac{0.45(1-0.45)}{100}}}=-0.428[/tex]
[tex]p_v =2*P(Z<-0.428)=0.669[/tex]
And we can use the following R code to find it: "2*pnorm(-0.428)"
The p value is a very high value and using any significance given [tex]\alpha=0.05[/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 two proportions are not significantly different.
Step-by-step explanation:
1) Data given and notation
[tex]X_1 =42[/tex] people with some characteristic desired on sample 1
[tex]X_2 =45[/tex] people with some characteristic desired on sample 2
[tex]n_{1}=100[/tex] sample selected 1
[tex]n_{2}=100[/tex] sample selected 2
[tex]p_{1}=\frac{42}{100}=0.42[/tex] represent the proportion estimated for 1
[tex]p_{2}=\frac{45}{100}=0.45[/tex] represent the proportion estimated for 2
[tex]\alpha=0.05[/tex] represent the significance level
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 for the two samples are different , 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{\frac{p_1 (1-p_1)}{n_{1}}+\frac{p_2 (1-p_2)}{n_{2}}}}[/tex] (1)
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.42-0.45}{\sqrt{\frac{0.42(1-0.42)}{100}+\frac{0.45(1-0.45)}{100}}}=-0.428[/tex]
4) Statistical decision
We can calculate the p value for this test.
Since is a two tailed test the p value would be:
[tex]p_v =2*P(Z<-0.428)=0.669[/tex]
And we can use the following R code to find it: "2*pnorm(-0.428)"
The p value is a very high value and using any significance given [tex]\alpha=0.05[/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 two proportions are not significantly different.
