Question

Assume that you plan to use a significance level of alpha α equals =0.05 to test the claim that p 1 p1 equals = p 2 p2. Use the given sample sizes and numbers of successes to find the pooled estimate p. Round your answer to the nearest thousandth. n 1 n1 equals =​677, n 2 n2 equals =3377 x 1 x1 equals =​172, x 2 x2 equals =654

Answer 1

z=3.536

`\hat p=0.204`

`p_v =2*P(Z>3.54)\approx 0.0004`

Explanation:

1) Data given and notation

`X_(1)=172` represent the number of people with the characteristic 1

`X_(2)=356` represent the number of people with the characteristic 2

`n_(1)=677` sample 1 selected

`n_(2)=3377` sample 2 selected

`p_(1)=(172)/(677)=0.254` represent the proportion estimated for the sample 1

`p_(2)=(654)/(3377)=0.194` represent the proportion estimated for the sample 2

`\hat p` represent the pooled estimate of p

z would represent the statistic (variable of interest)

`p_v` represent the value for the test (variable of interest)

`\alpha=0.05` significance level given

2) Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference between the two proportions, 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))=(172+654)/(677+3377)=0.204`

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

3) Calculate the statistic

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

`z=\frac{0.254-0.194}{\sqrt{0.204(1-0.204)((1)/(677)+(1)/(3377))}}=3.536`

4) Statistical decision

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

`p_v =2*P(Z>3.54)\approx 0.0004`

Comparing the p value with the significance level given
`\alpha=0.05` we see that
`p_v<\alpha` so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportion analyzed is significantly different between the two groups.