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

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

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Harper 27 mins 2021-10-13T01:18:26+00:00 1 Answer 0

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    2021-10-13T01:20:01+00:00

    Answer:

    z=3.536

    \hat p=0.204

    p_v =2*P(Z>3.54)\approx 0.0004  

    Step-by-step 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}=\frac{172}{677}=0.254 represent the proportion estimated for the sample 1

    p_{2}=\frac{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} \neq 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)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}   (1)  

    Where \hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{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)(\frac{1}{677}+\frac{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.  

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