– 41 – STATS 101
STATS 101G
STATS 108
FORMULAE
Confidence intervals and t-tests
Confidence interval: estimate± t×se(estimate)
t-test statistic: t0 =
estimate− hypothesised value
standard error
Applications:
1. Single mean μ: estimate = x; df = n− 1
2. Single proportion p: estimate = p̂; df = ∞
3. Difference between two means μ1 − μ2: (independent samples)
estimate = x1 − x2; df = min(n1 − 1, n2 − 1)
4. Difference between two proportions p1 − p2:
estimate = p̂1 − p̂2; df = ∞
Situation (a): Proportions from two independent samples
Situation (b): One sample of size n, several response categories
Situation (c): One sample of size n, many yes/no items
The F -test (ANOVA)
F -test statistic: f0 =
s2B
s2W
; df1 = k − 1, df2 = ntot − k
The Chi-square test
Chi-square test statistic: χ2
0
=
∑
all cells in the table
(observed − expected)2
expected
Expected count in cell (i, j) =
RiCj
n
df = (I − 1)(J − 1)
Regression
Fitted least-squares regression line: ŷ = β̂0 + β̂1x
Inference about the intercept, β0, and the slope, β1: df = n− 2