Question

Suppose a national survey conducted among a simple random sample of 1528 American adults, 800 indicate that they think the Civil War is still relevant to American politics and political life.

1. What are the correct hypotheses for conducting a hypothesis test to determine if these data provide strong evidence that the majority of the Americans think the Civil War is still relevant?
2. Calculate the test statistic for this hypothesis test.
3. Calculate the p-value for this hypothesis test.

Answer 1

1) Null hypothesis:
`p \leq 0.5`

Alternative hypothesis:
`p > 0.5`

2)
`z=\frac{0.524 -0.5}{\sqrt{(0.5(1-0.5))/(1528)}}=1.876`

3) Since is a right tailed test the p value would be:

`p_v =P(z>1.876)=0.0303`

Explanation:

Data given and notation

n=1528 represent the random sample taken

X=800 represent the adults that said indicate that they think the Civil War is still relevant to American politics and political life

`\hat p=(800)/(1528)=0.524` estimated proportion of adults indicate that they think the Civil War is still relevant to American politics and political life

`p_o=0.5` is the value that we want to test

`\alpha` represent the significance level

z would represent the statistic (variable of interest)

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

1) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that true proportion is higher than 0.5.:

Null hypothesis:
`p \leq 0.5`

Alternative hypothesis:
`p > 0.5`

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`.

2) Calculate the statistic

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

`z=\frac{0.524 -0.5}{\sqrt{(0.5(1-0.5))/(1528)}}=1.876`

3) 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 next step would be calculate the p value for this test.

Since is a right tailed test the p value would be:

`p_v =P(z>1.876)=0.0303`