how do I do a chi square test when I am not give the alpha?

Question

A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Calculate the chi-square test statistic χ2 to test the claim that the distribution is uniform. Brand 1 2 3 4 5 Customers 55 30 18 62 32

Answer 1

The default would depend on which branch of science you are usually working in.  If they don't tell you otherwise, it's usually okay to assume alpha of .05.  When you answer a question that omits that specification, you can STATE the alpha you used.

 

However, it isn't necessary to know the alpha in order to calculate the TEST statistic.  You only need to know the alpha in order to find the critical value that the test statistic will be compared against.

 

I notice that there are only 197 observations, but we'll assume that's okay and we got 3 "no preferences".  

 

You're going to compare each result above to the expected result (200/5 = 40), square the difference, and divide by 40.  Add those 5 numbers up.  That's your test statistic.  (It will be between 33 and 34, varying a bit depending on whether you used 197 or 200.)

 

Then, the next step would be to look up the degree of significance based on the degrees of freedom.  With 5 categories, if the customer didn't like one of the first 4, then he MUST like the 5th, so there are 4 degrees of freedom. For alpha of .05, the critical value will be around 9.49, which is WAY under the test statistic, so our customers do have real preferences. 

 

 

 

Answer 2

Just calculate the chi-square statistic as usual and report the p-value.  Most of the time it will be pretty clear whether you can reject the claim for any reasonable value of alpha (e.g. alpha = 0.01 or 0.05 or 0.10).  If your p-value is less than 0.01 you can safely reject the claim.  If your p-value is greater than 0.10 you can not reasonably reject the claim.  If your p-value lies between 0.01 and 0.10 you can make a statement like "We can reject the claim at alpha = 0.10 but not at alpha = 0.05" (or whatever numbers make sense given the p-value).  

 

Just by eyeballing the data it seems like it is not uniform, but you should go ahead and calculate the statistic anyway.  By the way, your data only adds up to 197 samples, so maybe 3 customers didn't return the survey (or maybe they had no preference).