Question

A group of retired admirals, generals, and other senior military leaders, recently published a report, "Too Fat to Fight". The report said that 75 percent of young Americans between the ages of 17 to 24 do not qualify for the military because of failure to graduate [from high school], criminal records or physical problems. The study cited Department of Defense and health data. A research group does not agree with this statistic. In a random sample of 180 young Americans 125 did not qualify for the military. Using a 5% significance level, is there sufficient evidence to conclude that less than 75% of Americans between the ages of 17 to 24 do not qualify for the military?

Answer 1

`z=\frac{0.694 -0.75}{\sqrt{(0.75(1-0.75))/(180)}}=-1.735`

`p_v =P(z<-1.735)=0.041`

If we compare the p value obtained and 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 americans between 17 to 24 that not qualify for the military is significantly less than 0.75 or 75% .

Explanation:

1) Data given and notation

n=180 represent the random sample taken

X=125 represent the number of americans between 17 to 24 that not qualify for the military

`\hat p=(125)/(180)=0.694` estimated proportion of americans between 17 to 24 that not qualify for the military

`p_o=0.75` 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 less than 75% of Americans between the ages of 17 to 24 do not qualify for the military :

Null hypothesis:
`p\geq 0.75`

Alternative hypothesis:
`p < 0.75`

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`.

3) Calculate the statistic

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

`z=\frac{0.694 -0.75}{\sqrt{(0.75(1-0.75))/(180)}}=-1.735`

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 left tailed test the p value would be:

`p_v =P(z<-1.735)=0.041`

If we compare the p value obtained and 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 americans between 17 to 24 that not qualify for the military is significantly less than 0.75 or 75% .