Question

A university has found over the years that out of all the students who are offered admission, the proportion who accept is .70. After a new director of admissions is hired, the university wants to check if the proportion of students accepting has increased significantly. Suppose they offer admission to 1200 students and 888 accept. Is this evidence at the α = .05 level that there has been a significant increase in proportion of students accepting admission?

Answer 1

`z=\frac{0.74 -0.7}{\sqrt{(0.7(1-0.7))/(1200)}}=3.02`

`p_v =P(z>3.02)=0.0013`

So the p value obtained was a very low 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 reject the null hypothesis, and we can said that at 5% of significance the proportion of students that accept is significanlty higher than 0.7

Explanation:

Data given and notation

n=1200 represent the random sample taken

X=888 represent the students that accept

`\hat p=(888)/(1200)=0.74` estimated proportion of students that accept

`p_o=0.7` 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)

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 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`.

Calculate the statistic

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

`z=\frac{0.74 -0.7}{\sqrt{(0.7(1-0.7))/(1200)}}=3.02`

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 right tailed test the p value would be:

`p_v =P(z>3.02)=0.0013`

So the p value obtained was a very low 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 reject the null hypothesis, and we can said that at 5% of significance the proportion of students that accept is significanlty higher than 0.7