Question

We are conducting a hypothesis test to determine if fewer than 80% of ST 311 Hybrid students complete all modules before class. Our hypotheses are H0 : p = 0.8 and HA : p < 0.8. We looked at 110 randomly selected students and found that 97 of these students had completed the modules before class. What is the appropriate conclusion for this test?

Answer 1

`z=\frac{0.881 -0.8}{\sqrt{(0.8(1-0.8))/(110)}}=2.124`

Null hypothesis:
`p\geq 0.8`

Alternative hypothesis:
`p < 0.8`

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

`p_v =P(Z<2.124)=0.983`

So the p value obtained was a very high value and using the significance level assumed
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.

Be Careful with the system of hypothesis!

If we conduct the test with the following hypothesis:

Null hypothesis:
`p\leq 0.8`

Alternative hypothesis:
`p > 0.8`

`p_v =P(Z>2.124)=0.013`

So the p value obtained was a very low value and using the significance level assumed
`\alpha=0.05` we have
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis.

Explanation:

1) Data given and notation

n=110 represent the random sample taken

X=97 represent the students who completed the modules before class

`\hat p=(97)/(110)=0.882` estimated proportion of students who completed the modules before class

`p_o=0.8` is the value that we want to test

`\alpha` represent the significance level

z would represent the statistic (variable of interest)

Alternative hypothesis:
`p < 0.8`

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{0.881 -0.8}{\sqrt{(0.8(1-0.8))/(110)}}=2.124`

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 assumed
`\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<2.124)=0.983`

So the p value obtained was a very high value and using the significance level assumed
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.

Be Careful with the system of hypothesis!

If we conduct the test with the following hypothesis:

Null hypothesis:
`p\leq 0.8`

Alternative hypothesis:
`p > 0.8`

`p_v =P(Z>2.124)=0.013`

So the p value obtained was a very low value and using the significance level assumed
`\alpha=0.05` we have
`p_v<\alpha` so we can conclude that we have enough evidence to reject the null hypothesis.