NHST statistics for % of students late to class

[tomwilliam]

Homework Statement

The teachers claim that at least 20% of the students arrive late for class. The students say it is no more than 10%.
What is the null hypothesis (H_0) and alternative hypothesis (H_1) from the perspective of a) the teachers; b) the students?

Relevant Equations

none

This is a homework question in my daughter’s maths class. When I did stats I always had examples where there were two variables : an example being swallow wingspan and sex. The statement that males have larger wingspan would therefore be the H_1 and the H_0 would be that there is no impact of sex on wingspan.

In this example, I argue that the teachers’ claim is H_1 and the H_0 is that the data will not support that claim. My daughter says that the claim should be the status quo (H_0) and the H_1 is the alternative hypothesis that it’s not p greater than or equal to 0.2
Which of us is wrong?
Thanks!

 

[FactChecker]

The null hypothesis is the one that will be assumed unless the data indicates otherwise. I would chose the option with the least risk and disturbance as the null hypothesis. This problem statement does not it clear to me which should be the null hypothesis, but, with no other information, I would make the student's statement the null hypothesis.

 

[Mark44]

Homework Statement: The teachers claim that at least 20% of the students arrive late for class. The students say it is no more than 10%.
What is the null hypothesis (H_0) and alternative hypothesis (H_1) from the perspective of a) the teachers; b) the students?
Relevant Equations: none

This is a homework question in my daughter’s maths class. When I did stats I always had examples where there were two variables : an example being swallow wingspan and sex. The statement that males have larger wingspan would therefore be the H_1 and the H_0 would be that there is no impact of sex on wingspan.

In this example, I argue that the teachers’ claim is H_1 and the H_0 is that the data will not support that claim. My daughter says that the claim should be the status quo (H_0) and the H_1 is the alternative hypothesis that it’s not p greater than or equal to 0.2
Which of us is wrong?
Thanks!

Your daughter is almost correct. This is really two problems, each with its own pair of ##H_0## and ##H_a## hypotheses. For the teachers' claim, ##H_0: p = 0.2## and ##H_a: p > 0.2##.

 

[Mark44]

This problem statement does not it clear to me which should be the null hypothesis, but, with no other information, I would make the student's statement the null hypothesis.

My interpretation is that this is really two separate problems, each with its own pair of null and alternative hypotheses. Assuming that interpretation, the two parts are pretty straightforward.

 

[Hill]

As I understand it, the question is not "What is the null hypothesis (H_0) and alternative hypothesis (H_1)", but rather "What is the null hypothesis (H_0) and alternative hypothesis (H_1) from the perspective of a) the teachers; b) the students?" (my emphasis). The answer then is straightforward:
a) From the perspective of the teachers, H0 is "at least 20%".
b) From the perspective of the students, H0 is "no more than 10%".

 

[Mark44]

As I understand it, the question is not "What is the null hypothesis (H_0) and alternative hypothesis (H_1)", but rather "What is the null hypothesis (H_0) and alternative hypothesis (H_1) from the perspective of a) the teachers; b) the students?" (my emphasis).

This is my take as well.

a) From the perspective of the teachers, H0 is "at least 20%".
b) From the perspective of the students, H0 is "no more than 10%".

I disagree slightly here. The null hypotheses would be equalities so that for a) ##H_0: p = 0.2##, and for b) ##H_0 = 0.1##. The alternative hypotheses would be stated in terms of inequalities.

 

[Hill]

The null hypotheses would be equalities so that for a) H0:p=0.2, and for b) H0=0.1. The alternative hypotheses would be stated in terms of inequalities.

You are correct.

 

[FactChecker]

This is my take as well.

I disagree slightly here. The null hypotheses would be equalities so that for a) ##H_0: p = 0.2##, and for b) ##H_0 = 0.1##. The alternative hypotheses would be stated in terms of inequalities.

This presents a question. When testing the alternative hypotheses, do you use the parameters of the null hypothesis or of the alternative hypothesis? Is the goal to see if the alternative hypothesis, using its parameters, is possible/likely or to show that the null hypothesis, using its parameters, is not likely?

 

[Mark44]

When testing the alternative hypotheses, do you use the parameters of the null hypothesis or of the alternative hypothesis?

You use the assumed parameter in the null hypothesis.

 

[tomwilliam]

Thanks for all of your help. Indeed, this is two separate questions. Sorry if that wasn’t clear.
I do have a follow up about the equalities/inequalities.
We had worked on the basis of p > or equal to 0.2, given that it says « at least 20% ». Why is it = for the null hypothesis and inequality for the alternative ?

 

[tomwilliam]

Your daughter is almost correct. This is really two problems, each with its own pair of ##H_0## and ##H_a## hypotheses. For the teachers' claim, ##H_0: p = 0.2## and ##H_a: p > 0.2##.

Doesn’t that make me correct? The ##H_a: p > 0.2## corresponds to the teachers’ assertion being supported by the data, right ? Whereas H_0 would be that the teachers’ claim is not supported? (Although I’m not sure why it isn’t ##H_0: p < 0.2##
What am I missing? Thanks!

 

[Mark44]

Doesn’t that make me correct? The ##H_a: p > 0.2## corresponds to the teachers’ assertion being supported by the data, right ? Whereas H_0 would be that the teachers’ claim is not supported? (Although I’m not sure why it isn’t ##H_0: p < 0.2##
What am I missing? Thanks!

I misread your first post, somehow thinking that you were combining both parts of the problem into one hypothesis test. IOW, that the null hypothesis was the teachers' claim and that the alternate hypothesis was the students' claim. Sorry for my error here.

In this example, I argue that the teachers’ claim is H_1 and the H_0 is that the data will not support that claim. My daughter says that the claim should be the status quo (H_0) and the H_1 is the alternative hypothesis that it’s not p greater than or equal to 0.2

Rereading the OP more carefully, I think that your version of the two hypotheses is technically correct but not very helpful in how the two hypotheses will be defined. and that your daughter's version is wrong on the alternate hypothesis. The teachers are claiming that at least 20% of the students are late, so p > 0.2 would be the alternate hypothesis. As I recall from teaching a bunch of classes in statistics, the null hypothesis always consists of a point value for some parameter. In this case, p = 0.2.

 

[haruspex]

The teachers are claiming that at least 20% of the students are late, so p > 0.2 would be the alternate hypothesis. As I recall from teaching a bunch of classes in statistics, the null hypothesis always consists of a point value for some parameter. In this case, p = 0.2.

But what if the data shows so little tardiness that p = 0.2 is rejected?
I suggest the null hypothesis value of p should be that value <0.2 for which the likelihood of the data is maximised.

 

[Hill]

Wouldn't in this case just taking p=0.19 work, assuming that 1% is a desired precision?

 

[Mark44]

But what if the data shows so little tardiness that p = 0.2 is rejected?
I suggest the null hypothesis value of p should be that value <0.2 for which the likelihood of the data is maximised.

##H_0: p = 0.2## would be rejected only if the test statistic ##p^*## representing tardiness was too large, not that it was too small. In such a case, we would accept the null hypotheses, which is equivalent to rejecting the alternate hypothesis. In the textbook I'm using as a reference, "An Introduction to Mathematical Statistics and Its Application," by Richard J. Larsen and Morris L. Marx, every example of a hypothesis test gives the null hypothesis as some statistic being equal to a particular value.

In my experience of teaching statistics about 25 years ago, the null hypothesis was always as I described above while the alternate hypothesis was one of the forms <statistic ##\ne## <value> (two-tailed test), <statistic> < <value>, or <statistic> > <value>. The latter two forms implied that a one-tailed test would be used.

 

[Mark44]

But what if the data shows so little tardiness that p = 0.2 is rejected?
I suggest the null hypothesis value of p should be that value <0.2 for which the likelihood of the data is maximised.

Wouldn't in this case just taking p=0.19 work, assuming that 1% is a desired precision?

I don't think that things work like this. That is, that 0.20 - 0.01 means that p should be 0.19.