1. Basic t-test
10/6

2. Hypothesis Test for Population Mean
Goal: Infer m from M
Null hypothesis: m = m0, usually 0
Change scores
Memory improvement, weight loss, etc.
Sub-population within known, larger population
IQ of CU undergrads
Approach:
Determine likelihood function for M, using CLT
Use actual sample mean to get p-value

3. Likelihood Function for M
Probability distribution for M, according to H0
Central Limit Theorem:
Mean equals population mean: mM = m0
Standard deviation:
Shape: Normal
z-score .
p-value: p(Z > z) m0
Probability sM p M

4. Example: IQ Are CU undergrads smarter than population?
Sample size 100, sample mean 103
Likelihood function
If no difference, what are probabilities of sample means?
Null hypothesis: m0 = 100
CLT:
z-score: ( )/1.5 = 2
mM = 0
sM = s/√n = 15/10 = 1.5
> 1-pnorm(2)
[1] .0228 p M zM

5. Problem: Unknown Variance?
Test statistic depends on population parameter
Can only depend on data or values assumed by H0
Could include s in null hypothesis
H0: m = m0 & s = s0
Usually no theoretical basis
Cannot tell which assumption fails
Change test statistic
Replace population SD with sample SD
Depends only on data and m0 .

6. t Statistic Invented by “Student” at Guinness brewery .
Deviation of sample mean divided by estimated standard error
Depends only on data and m0
Sampling distribution depends only on n

7. t Distribution Sampling distribution of t statistic
Derived from ratio of Normal and (modified) c2
Depends only on sample size
Degrees of freedom: df = n – 1
Invariant with respect to m, s
Shaped like Normal, but with fatter tails
Reflects uncertainty in sample variance
Closer to Normal as n increases
Critical value decreases as n increases
a = .05 df
tcrit 1
6.31 5
2.02 2
2.92 10
1.81 3
2.35 30
1.70 4
2.13 ∞
1.64

8. Steps of t-test State clearly the two hypotheses
Determine null and alternative hypotheses
H0: m = m0
H1: m ≠ m0
Compute the test statistic t from the data .
Determine likelihood function for test statistic according to H0
t distribution with n-1 degrees of freedom
Get p-value
R: 1-pt(t,n-1)
Choose alpha level
7a. p > α: Retain null hypothesis, μ = μ0
7b. p < α: Reject null hypothesis, μ ≠ μ0

9. Example: Rat Mazes Measure maze time on and off drug
Difference score: Timedrug – Timeno drug
Data (seconds): 5, 6, -8, -3, 7, -1, 1, 2
Sample mean: M = 1.0
Sample standard deviation: s = 5.06
Standard error: SE = s/√n = 5.06/2.83 = 1.79
t = (M – m0)/SE = (1.0 – 0)/1.79 = .56
p = p(tn – 1 ≥ t) = .30

10. Two-tailed Tests Interested in m < m0 or m > m0
Leads to t < 0 or t > 0
Need critical region to cover both tails
p-value counts both tails
p(|tn-1| > |t|)
2p(tn-1 > t) if t > 0
2p(tn-1 < t) if t < 0