Question

Let n1equals50​, Upper X 1equals30​, n2equals50​, and Upper X 2equals10. Complete parts​ (a) and​ (b) below. a. At the 0.05 level of​ significance, is there evidence of a significant difference between the two population​ proportions? Determine the null and alternative hypotheses. Choose the correct answer below. A. Upper H 0 : pi 1 equals pi 2 Upper H 1 : pi 1 not equals pi 2 Your answer is correct.B. Upper H 0 : pi 1 greater than or equals pi 2 Upper H 1 : pi 1 less than pi 2 C. Upper H 0 : pi 1 less than or equals pi 2 Upper H 1 : pi 1 greater than pi 2 D. Upper H 0 : pi 1 not equals pi 2 Upper H 1 : pi 1 equals pi 2 Calculate the test​ statistic, Upper Z Subscript STAT​, based on the difference p1minusp2.

Answer 1

Null hypothesis:
`p_(1) = p_(2)`

Alternative hypothesis:
`p_(1) \\eq p_(2)`

`z=\frac{0.6-0.2}{\sqrt{0.4(1-0.4)((1)/(50)+(1)/(50))}}=4.082`

`p_v =2*P(Z>4.082)=4.46x10^(-5)`

So the p value is a very low value and using any significance level for example
`\alpha=0.05` always
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion 1 is significantly different from proportion 2.

Explanation:

1) Data given and notation

`X_(1)=30` represent the number of people with a characteristic in 1

`X_(2)=10` represent the number of people with a characteristic in 2

`n_(1)=50` sample of 1 selected

`n_(2)=50` sample of 2 selected

`p_(1)=(30)/(50)=0.6` represent the proportion of people with a characteristic in 1

`p_(2)=(10)/(50)=0.2` represent the proportion of people with a characteristic in 2

z would represent the statistic (variable of interest)

`p_v` 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:
`p_(1) = p_(2)`

Alternative hypothesis:
`p_(1) \\eq p_(2)`

We need to apply a z test to compare proportions, and the statistic is given by:

`z=\frac{p_(1)-p_(2)}{\sqrt{\hat p (1-\hat p)((1)/(n_(1))+(1)/(n_(2)))}}` (1)

Where
`\hat p=(X_(1)+X_(2))/(n_(1)+n_(2))=(30+10)/(50+50)=0.4`

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:

`z=\frac{0.6-0.2}{\sqrt{0.4(1-0.4)((1)/(50)+(1)/(50))}}=4.082`

4) Statistical decision

For this case we don't have a significance level provided
`\alpha`, but we can calculate the p value for this test.

Since is a two sided test the p value would be:

`p_v =2*P(Z>4.082)=4.46x10^(-5)`

So the p value is a very low value and using any significance level for example
`\alpha=0.05` always
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion 1 is significantly different from proportion 2.