Question

70% of the U.S. population recycles. According to a green survey of a random sample of 250 college students, 204 said that they recycled. At alpha = 0.01, is there sufficient evidence to conclude that the proportion of college students who recycle is greater than 70%?

Answer 1

`z=\frac{0.816 -0.7}{\sqrt{(0.7(1-0.7))/(250)}}=4.002`

`p_v =P(z>4.002)=0.0000314`

So the p value obtained was a very low value and using the significance level given
`\alpha=0.01` we have
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the proportion of people said that they recycled is significantly higher than 0.7 or 70%

Explanation:

Data given and notation

n=250 represent the random sample taken

X=204 represent the people said that they recycled

`\hat p=(204)/(250)=0.816` estimated proportion of people said that they recycled

`p_o=0.7` is the value that we want to test

`\alpha=0.01` represent the significance level

Confidence=99% or 0.99

z would represent the statistic (variable of interest)

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

Concepts and formulas to use

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

Null hypothesis:
`p\leq 0.7`

Alternative hypothesis:
`p > 0.7`

When we conduct a proportion test we need to use the z statisitc, 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`.

Calculate the statistic

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

`z=\frac{0.816 -0.7}{\sqrt{(0.7(1-0.7))/(250)}}=4.002`

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=0.01`. 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>4.002)=0.0000314`

So the p value obtained was a very low value and using the significance level given
`\alpha=0.01` we have
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the proportion of people said that they recycled is significantly higher than 0.7 or 70%