Using goodness of fit for hypothesis testing of the best day for quality family time

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today, we're going to learn how to use goodness of fit for hypothesis testing of the best day for quality family time.  Here's our problem statement: A random sample of 773 subjects was asked to identify the day of the week that is best for quality family time.  Consider the claim that the days of the week are selected with a uniform distribution so that all days have the same chance of being selected.  The table below shows goodness-of-fit test results from the claim and data from the study.  Test that claim.

Part 1

OK, the first part of this problem is asking us to determine the null and alternative hypotheses.  For goodness of fit testing, that’s pretty much going to be the same thing every time.  The null hypothesis is going to be that everything is the same.  So in this case, all days of the week have an equal chance of being selected.  The alternative hypothesis will always be that at least one of those will be different. So at least one day of the week has a different chance of being selected.  Good job!

Part 2

Identify the test statistic.  Well, we work so many problems that by the time we get to Chapter 11, we’re pretty much in the habit of OK let’s get some data or some numbers, put them in StatCrunch, let StatCrunch chew some numbers, and spit out an answer.  But the answer that we’re looking for is already given to us here just below the problem statement.  It asks for the test statistic, and so here is our test statistic.  There’s a number: 3021.822.  So we just put that number here in the blank.  Excellent!

Part 3

This next part of the problem is exactly the same thing.  It asks us to identify the critical value.  The critical value is, again, listed up here in the results from some technology display that was already done by somebody.  So all we have to do is copy the number over.  Fantastic!

Part 4

And now the last part of the problem asks us to state the conclusion.  In this case, we’re going to compare the test statistic and the critical value.  Well, here’s the critical value, which marks the boundary region in the tail of our distribution that is the critical region or region of rejection.  Here’s our test statistic.  It’s well within the right tail of our distribution, and so therefore we are going to be inside the region of rejection.  Therefore we reject the null hypothesis.  Whenever we reject the null hypothesis, there is sufficient evidence.  And so, because the null hypothesis says everything is the same and we’re rejecting it, we are by default “accepting” the alternative, which says that at least one of the days is different, so it does not appear that all days have the same chance of being selected.  Fantastic!

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