Question

Test the hypothesis using the P value approach. Be sure the verify the requirements of the test.

H0: p=.77 versus H1: p is not equal to .77
n=500, x=380, α=.05

Answer 1

`p_v =2*P(z<-0.531)=0.595`

If we compare the p value and using the significance level given
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion is not significantly different from 0.77.

Explanation:

1) Data given and notation

n=500 represent the random sample taken

X=380 represent the number of people with some characteristic

`\hat p=(380)/(500)=0.76` estimated proportion of adults that said that it is morally wrong to not report all income on tax returns

`p_o=0.76` is the value that we want to test

`\alpha=0.05` represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

`p_v` represent the p value (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that the true proportion is 0.7 .:

Null hypothesis:
`p=0.77`

Alternative hypothesis:
`p \\eq 0.77`

When we conduct a proportion test we need to use the z statistic, and the is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

The One-Sample Proportion Test is used to assess whether a population proportion
`\hat p` is significantly different from a hypothesized value
`p_o`.

Check for the assumptions that he sample must satisfy in order to apply the test

a)The random sample needs to be representative: On this case the problem no mention about it but we can assume it.

b) The sample needs to be large enough

`np_o =500*0.77=385>10`

`n(1-p_o)=384*(1-0.77)=115>10`

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

`z=\frac{0.76-0.77}{\sqrt{(0.77(1-0.77))/(500)}}=-0.531`

4) Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level provided
`\alpha=0.05`. The next step would be calculate the p value for this test.

Since is a bilateral test the p value would be:

`p_v =2*P(z<-0.531)=0.595`

If we compare the p value and using the significance level given
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion is not significantly different from 0.77.