For 17 girls diagnosed with anorexia, weight change after family therapy was as follows:
11,11, 6, 9, 14, -3, 0, 7, 22, -5 , -4, 13, 13, 9, 4 , 6, 11
Partial results are shown below. Fill in the missing results:
And we had gotten the table completed as far as this. We also along the way found out that the mean was 7.29
#1 CHILL !
I mean this most seriously. This really is the first step.
What is it you are asked to do in the problem? All that is left is to find the t-value. Here is where several of the students went wrong. So many of them went wrong I would have thought they had cheated, but they were sitting all around the room. Barring some secret hand signals, that was not possible.
Many of the students obtained a value of around 2.12, which is very much NOT correct. I was confused and then I realized that while *I* knew that the problem was asking for the obtained t-value, what the students had computed was the critical t-value with 16 degrees of freedom. The problem did not specify and the textbook author, like me, just assumed that you would know that the value shown on a print-out was the obtained t-value, not the critical t-value.
Well, sure you would know that if like me, and no doubt like the author of the textbook, you had been looking at printouts from statistical programs for the past 30 years. These students could not be expected to know that, so, I ended up giving them full credit if that is what the answered.
What you should know now
- The t-value referenced in the print-out is the OBTAINED t, not the critical t-value for that number of degrees of freedom.
- The formula for obtaining t is (obtained mean – hypothesized mean)/ standard error
- Your hypothesized mean is 0
- Your obtained mean is 7.29
- The standard error is the standard deviation divide by the square root of N
- The critical value for t for 16 degrees of freedom when p < .05 is 2.12
- The lower confidence limit is the mean MINUS the CRITICAL t times the standard error
- The lower confidence limit is 3.6
- The difference between the mean and the lower confidence limit is 3.69
- The standard deviation is the square root of the sum of squared deviations from the mean divided by n -1
#3 SELECT A STRATEGY
There are a number of ways to find the t-value. All involve subtracting the hypothesized mean from the obtained mean and dividing by the standard error. Some ways are harder than others. You could compute the standard deviation and divide by the square root of N but that is a lot of work. Here is what I think is the easiest way
- Divide 3.69 by 2.12 — that will give us the standard error
- Subtract 0 from 7.29
- Divide 7.29 by the standard error
In this case, it was this step and the previous one where people ran into trouble. What is interesting is that they did not realize what they DIDN’T understand. That is, they didn’t understand that the t-value they were expected to produce was the obtained t-value, not the critical t-value.
You could (and many people did) compute the standard deviation, then divide it by the square root of N to get the standard error and it would give the correct answer, but it just seems more work than dividing 3.69 by 2.12.
#4 DO IT
Carry out your strategy.
- 3.69/ 2.12 —- The standard error is 1.74
- 7.29 -0 = 7.29
- Divide 7.29 by 1.74 = 4.19
#5 TEST IT
Do a reality check. No one in the class asked which t-value it should be and it never occurred to me that people would not automatically know that it was the obtained t-value that is of interest. I mean, seriously, what’s the purpose of doing a study to find a critical value of t that was established a hundred years ago? I’m not surprised though, that people who are not experienced statisticians don’t immediately think of that. Probably a lot of what statisticians do doesn’t seem very obvious so maybe it’s just another of those weird things.
So, I guess it is up to me on Thursday to explain to the class that you have a critical value for a test statistic and an obtained value.
A lot of #5 comes from experience. For example, immediately, when I saw t-values of around 2 that the students had obtained, I thought that can’t be right, because even with 17 people, 7 pounds is pretty far from 0, it seemed like it ought to be significant.
So …. this brings me to number 6
The more problems you do, the better you get at solving them. People often get the impression that people who are good at math have some kind of special math brain. It’s not true. If you are telling yourself that you are just not good at math, cut it out right now before I come over there and smack you.
I married a rocket scientist – literally – someone whose idea of the way to a woman’s heart was to write a program to generate fractals and email her a pink fractal for Valentine’s Day. It worked, too. And yet — I can guarantee you that he, and I, both ran into the same obstacles in learning mathematics that anyone else does. The only difference between us and our friends who quit school and ended up working at Wal-Mart is that we spent hours and hours and hours learning programming, statistics(well, I learned the statistics), Calculus, Physics (well, he learned the physics).
Last week, more than one student said to me, with some frustration.
“Dr. De Mars, I studied for HOURS for this class.”