# F-test / t-test

1. Jul 24, 2011

### Maybe_Memorie

1. The problem statement, all variables and given/known data

A study of weight gain in male rats compared two diets, A and B,
which were formulated in terms of level and source of protein.
A total of twenty two rats were randomly assigned to the diets, eleven to each diet.
Summary statistics for the weight gains (grams) by the end of the study are shown.

A B
Mean 99.5 78.7
St. Dev 10.9 16.5
n 11 11

(a) An F-test was carried out to compare the sample standard deviations.
Why might this have been done? Carry out the F-test: specify the null hypothesis,
the significance level and critical value you use in the test and intrepret the
test result.

(b) Carry out a t-test to compare the sample means. State ecplicitly the null hypothesis,
the degrees of freedom for the reference distribution, and the critical values for your test.
Interpret the results of the test in practical terms.

(c) It has been suggested that rather than carry out a test, it would be better to calculate
a confidence interval. Explain why this is either correct or not correct. Calculate and interpret an appropriate confidence interval.

(d) It was asserted that had the raw data been available, a paired t-test would be the appropriate test for analyzing the study results. Discuss.

(e) Before a study such as this one can be carried out, the researchers need to decide on a sample size for the study. Discuss the issues that need to be considered in arriving at a suitable sample size for
this type of study.

2. Relevant equations

3. The attempt at a solution

(a) I used F = (s1)^2/(s2)^2
This gave (16.5)^2/(10.9)^2 = 2.29

The null hypotheses, Ho: s1=s2

Is this correct?

I don't know how to get the significance level or critical value.

(b) I used t = (x1 - x2)/root[(s1^2 - s2^2)/n)
which gave me 3.49

Ho: u1 = u2

11-1=10 degrees of freedom

Is it okay to do a 95% confidence interval? That's what all the examples do, but I don't know why...

(c) Using (x1 - x2)+-t*(root[(s1^2 - s2^2)/n)

I need to get t* from the previous part.

For (d) and (e) any advice would be greatly appreciated as I'm lost.

2. Jul 24, 2011

### I like Serena

Hi Maybe_Memorie!

First I think you need some background information.

To do any statistical test you need to define a null hypothesis Ho, an alternative hypothesis Ha, and a significance level alpha.
An important choice is whether you select a Ha to be 1-sided or 2-sided.
The significance level alpha is usually chosen as 5%, meaning the chance that you draw the wrong conclusion should be less than 5%.

This is a typical experiment where two populations are compared.
To do that we'll do an "Independent two-sample t-test".
For this it matters whether the population variances are the same or not.
If they are the same, they must be "pooled".
So you would either do the "Independent two-sample t-test with equal variances" or the "Independent two-sample t-test with unequal variances".
Do you have the appropriate formulas available?
The formula you used for (b) is not quite the right formula.

To find out if the variances can be assumed equal the F-test is done.
To do an F-test you need to know the degrees of freedom of the populations, and the selected significance level alpha (pick alpha = 5%).
The critical value comes from looking these parameters up in the appropriate table, or use a statistical calculator.
Do you have such a table available?

For (b) you did not use quite the right formula.
And yes, its okay to do a 95% confidence interval.
That means that you have chosen a significance level alpha of 5%, which is usually chosen.

Btw, a program like SPSS does all these things for you.
Are you supposed to use for instance SPSS?

Last edited: Jul 24, 2011
3. Jul 24, 2011

### Maybe_Memorie

Hello again

For unequal variances
[(x1 - x2) - (u1 - u2)]/root[(s1^2 - s2^2)/n)

I'm using n as a common denominator since the number in A and B is the same.

Equal variances, I don't have my book with me at the moment so I can't state the formula off-hand, but I know there is a formula for a pooled t-test.
Btw, I'm using An Introduction to the Practice of Statistics by Moore and McCabe.

So I'm assuming the appropriate formula to use would be determined by the F-test?

Each population as 11 members, so 10 degrees of freedom.
Yes, I have two available tables, one for 10% critical values and one for 5% critical values.
Using the 5% one, and 10 degrees of freedom, should my critical value be 3?

Unfortunately no. In my exam I have to get all information from the supplied tables.

4. Jul 24, 2011

### I like Serena

I'm sorry to say that I don't have a good book available, so I'm dependent on Google.
It's best if you pick the right formulas from your own book.

My problem with your formula is that you subtract the sample variances.
Variances are never subtracted, they should be added.

Yes.

An F-test has 2 parameters that are both degrees of freedom (df).
They are called the df of the nominator and the df of the denominator.
I believe in this case you need 10 and 10, which comes out at alpha=5% as F*=2.97.

So what is your conclusion for the F-test?

5. Jul 24, 2011

### Ray Vickson

For part (a): if the null hypothesis $\sigma_1 = \sigma_2$ holds then both s1^2/s2^2 and s2^2/s1^2 should have the so-called F(10,10) distribution. (Here, the '10s' are the "degrees of freedom" (n-1) used in computing the s^2 values.) In other words, we want to know how likely it would be to see a value of s1^2/s2^2 as large as 2.29, or a value of s2^2/s1^2 as small as 1/2.29 = 0.437, from a random variable with distribution F(10,10). You need to look at F-tables (or use an on-line calculator, or spreadsheet, or whatever) to look at the corresponding probabilities.

Note: you may be confusing yourself by using sloppy notation: s1 is a sample standard deviation, not a true "population" standard deviation. We are not testing whether s1 = s2 because this is obviously not true: one of them is about 1.5 times the other. However, the underlying standard deviations $\sigma_1$ and $\sigma_2$ may, or may not be equal, or nearly so. That is what we are trying to find out.

RGV

6. Jul 25, 2011

### Maybe_Memorie

My book says to double F* obtaining 6. (in my tables the value for F(10,10) is 3 so I'm just going to use that.)

This is greater than the 5% level of significance, so is significant in our test.
So the null hypothesis fails?

7. Jul 25, 2011

### I like Serena

Hmm, doubling F*?
I've never heard of that.
Aren't you mixing it up with the t-test?
There's doubling involved in the t-test..... but we'll get to that later.

Either way, your F was less then 3 (I assume this is a rounded value), so whatever the doubling, the F is less than the critical F*.

Now concerning the null hypothesis.
Suppose the null hypothesis was perfectly true, what F-value would you get?
And if the null hypothesis would be very untrue, what kind of F-value would you have then?

8. Jul 25, 2011

### Maybe_Memorie

If it was perfectly true, F would equal 1, if it was very untrue, F would be very large.
Since my F is smaller than 5, can I take it that the variances are equal and then in (b) use
t = (x1 - x2)/root[(s1^2 + s2^2)/n) ?

9. Jul 25, 2011

### I like Serena

Yes.
To be a little more accurate, according to the F-test you do not have enough proof that you can reject the null hypothesis, so we'll assume equal variances.

And so yes, you can use the formula for the independent two-sample t-test for equal variances and equal sample sizes.
But usually, before you do that, you should first state the null hypothesis and the alternative hypothesis.
In particular you need to decide whether you're testing 1-sided or 2-sided...

10. Jul 25, 2011

### Maybe_Memorie

The null hypothesis is that u1 = u2
The alternative is that u1 =/= u2.

My book only mentions and uses the 1-sided test, I can't find a mention of a 2-sided test...

11. Jul 25, 2011

### I like Serena

Btw, I noticed that now you are referring to 5.
Any particular reason?

Actually, I was not entirely sure about the degrees of freedom being 10 and 10.
They might well be 2-1 resp 11-1, in which case you would get 4.9647....
Pity I do not have a good reference book available. :(

12. Jul 25, 2011

### I like Serena

Well, you just defined a 2-sided test!

Basically you have three choices for the alternative hypothesis Ha.
1. u1 < u2
2. u1 > u2
3. u1 ≠ u2

The first and second are 1-sided, the third is 2-sided.

You should always carefully read the problem statement to find out what the alternative hypothesis should be.
Did you?
Anyway, let's assume a 2-sided test, which is what you correctly selected!

13. Jul 25, 2011

### Maybe_Memorie

When I said 5 I was referring to my alpha.
I'm assuming this is wrong?

My book said when the populations have n1 and n2 members, then my degrees of freedom are n1 - 1 and n2 -1

14. Jul 25, 2011

### I like Serena

You are wrong if you were comparing your F-value with the significance level alpha of 5%.
In all these statistical theories you're dealing each and every time with two types of variables: test-statistics (like t-values and F-values) and probabilities (like significance level alpha and p-values).
Never mix these two types! :surprised

Okay, then we were and are on track!

15. Jul 25, 2011

### Maybe_Memorie

Oh right I see!
So I should be using 2-sided because the question said "compare the sample means", not investigate which is bigger?

16. Jul 25, 2011

### Maybe_Memorie

Right... so what should I have been comparing my F-value to in order to not be able to reject Ho?

17. Jul 25, 2011

### I like Serena

Almost!
As you formulate it, they would both be 2-sided tests.
"Investigate which is bigger" means you have no clue which one it is.

What you're looking for, is for instance a test to find out if people become more intelligent after taking a course.
That would be typically 1-sided.
Or for instance if the weight of rats becomes less after taking a diet.
That is, if the problem statement uses the word "more" or "less".

18. Jul 25, 2011

### I like Serena

You should compare your F-value with the critical F-value F*, which you found was 3 (or 6?!?).

Alternatively you can look up your F-value in a table to find a so called p-value.
In that case you would compare your p-value with the significance level alpha of 5%.

19. Jul 25, 2011

### Maybe_Memorie

Oh! Right!

Sorry I got a little lost there.

Okay, I understand the F test.

So in my t-test, I'm getting t = 3.49.

My critical value is 2.26, using 10 degrees of freedom and alpha = 5%

Because my t is greater than t* I can reject the null hypothesis.

Is this right?

20. Jul 25, 2011

### I like Serena

Good!

Wait! Stop!

Where did you find 2.26?
I can't find it.

Btw, this is the point where there is possible doubling or halving in the t-test.
For these calculations a 1-sided alpha of 2.5% corresponds to a 2-sided alpha of 5%.
Do you have references to that in your table?

Then the degrees of freedom.
I think you should not use 10 degrees of freedom.
Does your book say anything about the degrees of freedom in a t-test with equal variances assumed?
(I think it should be 20 degrees of freedom.)