A Little t-Test Puzzle

Dear all hard-working, puzzle-loving students,

Yesterday in class, everyone put a relative's IQ score on his/her index card. With Suzanne's help, I then entered those IQ scores into the computer. Next, I asked the computer to compute a 95% confidence interval.

As you will recall, I used the confidence interval to test a null hypothesis that said that the population mean (of IQ scores) was equal to 100. That null statement did not coincide with my hunch, for I truly believed that the sample of IQ scores that we were dealing with came from a population--an ABSTRACT population, of course--where the mean was greater than 100. (This last statement was my "research hypothesis.")

Once Suzanne looked at the computer screen and told us what the numerical values were for the end points of the confidence interval, I was able to reject the null hypothesis. I was allowed to do that because the number specified in my null hypothesis (100) was NOT inside the confidence interval that extended from about 108 to 123.

This morning, I took the index cards, entered the same IQ data into the computer, and this time asked the computer to compute a CALCULATED VALUE (instead of a confidence interval). Very quickly, my computer complied with my request. The calculated value (labeled "t") turned out equal to 4.175. When I compared this calculated value against the proper critical value (that I had to look up in the back of a statistics book), I was allowed to reject the null hypothesis. (This came as no surprise to me, for the same data will ALWAYS produce the same reject/fail-to-reject decision regardless of whether the null hypothesis is evaluated via a calculated value or via a confidence interval.)

So, here's a summary of the hypothesis testing that I did this morning using the IQ data from yesterday's class session:

  1. Null hypothesis: m = 100
  2. Alternative hypothesis: m = something other than 100
  3. Level of significance: .05
  4. Summary of sample data: mean = 116, calculated t = 4.175
  5. Critical value and p: CV = ??? ; p = ???
  6. Decision about Ho: Reject the null hypothesis

Here, now, is your puzzle question. You'll note that in step 5 I have not told you the numerical value of the critical value that I located in the back of the statistics book, nor have I told you the numerical value of the "p" number that was displayed on my computer screen after the computer analyzed the IQ data. Your puzzle task is to say something about the CV and the p numbers. No matter how hard you might think about this little puzzle, you won't be able to come up with the exact values of these two numbers. That's not what you're supposed to do! Instead, you're supposed to fill in the blanks of these two statements:

  1. The CV number must have been _____ (bigger/smaller) than ___.
  2. The p number must have been ____(bigger/smaller) than ___.

The first blank in each statement should be filled with a WORD, while the second blank in each statement should be filled with a NUMBER. If you'll send me a reply in which your message is simply a word (bigger/smaller) and a number after the letter "a," and a word (bigger/smaller) and a number after the letter "b," I'll send you back a short message that will simply say "Right!" or "Try again!"

Good luck to all you puzzle solvers!!!!!!!!!!!!

Sky Huck

Copyright © 2012

Schuyler W. Huck
All rights reserved.

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