Lecture 9 t-test
Z-test assumptions
- Random sampling
- Anormal sampling distribution
- Independent observations
- Constant
- Two of these are actually pretty hard to justify in the real world. Which?
• Normal sampling distribution and
Z-test assumptions
- Constant
• You often don’t know ²
• In fact, you often run experiments specifically so that you can estimate the population
parameters
- Anormal sampling distribution
• If you can get a large enough sample, that’s fine
• But what if you can’t?
• This problem prompted a pilgrimage which I undertook a few years ago
Gosset’s revolutionary discovery
- The mean of a small n sample, that produces an estimate of , can be compared to a
theoretical distribution of sample means that is NOT normally distributed
- The distribution would be relatively platykurtic for small n’s, but would still be
symmetrical and unimodal
- The distribution would approach the shape of a normal distribution as n increased
The student’s t-test
- Used instead of a z-test whenever you don’t know
- Used instead of a z-test whenever you have small samples and, as such, can’t be certain
that the distribution of sample means is normally distributed
“New” formulae
S = S
M √n SS
Where: S = √ n−1
M−µ
t= S
M
Let’s have a flashback…
- Remember in Quiz #1 when I asked you to find the missing X value given a
particular mean?
- Many of you found unique formulae that let you do this
• That’s great! You got full marks for that
- The problem is: those unique formulae missed the point of the question…
Find the missing X
- If X = {1, 2, 3, ???} and M = 5 what is the missing value?
- One of the values is NOT FREE TO VARY
- This means that, although most of the scores are free to vary, the sample has
lost one degree of freedom
- For any t-test: df = n – 1
df = n – 1
- A t distribution will approach a normal distribution as df increases
- So how do you increase df for a t-test?
The t-table
- The table is a list of critical values to be used in hypothesis testing
- This one is a bit more complicated than the z-table. Fortunately, you’ll only have
to deal with an abbreviated version:
• Only a few values are listed
- What critical t score would be associated with the following:
• I tested 10 people who I believe were made better by my treatment. = .05
• Do men perform differently than women? I tested 15 men. = .05
Let’s do a t-test!! - I suspect that students will do better on Test 2 than on Test 1 because they will
have had more exposure to the subject matter
- I get a sample of 5 test takers
- The mean grade for test 1 was µ = 5
- My sample had the following scores:
• 6, 8, 4, 9, 8
- Test my hypothesis, maintaining a Type I error rate of 2.5%
Getting the necessary values
X (X-M)2
6 1 s2 = 16/(5-1) = 4
8 1 s = 2
4 9
9 4
8 1
M = 7 SS = 16
Step 1
- State the hypothesis
H µ
- 0 = treatmen≤ 5
- H 1 = µtreatmen> 5
Step 2
- State your critical value
- At = .025 one-tailed, and with df = 4, tcritic= 2.776
Step 3
- Run the test
- T = (7 – 5) / (2 / 5)
- = 2 / .89
- 2.2361
- 2.24
Step 4 - Compare obtained and critical values
- T- observed (2.24) is less extreme than

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