Download Quantitative Methods HYPOTHESIS TESTING

Quantitative Methods
HYPOTHESIS TESTING
Notation for one group
X̄ = sample mean
s = sample standard deviation
p̂ = sample proportion
µ = population mean
σ = population standard deviation
p = population proportion
n = sample size
α = significance level = probability of making a type I error
GENERAL PROCEDURE
1. State the null (H0 ) and the alternative (Ha ) hypotheses.
2. Calculate the test statistic using the appropriate formula.
3. Find the p-value using the appropriate distribution:
(a) find the area in the tail of the appropriate distribution
(b) double this area in the case of a two-tail test (6=)
4. Reach a conclusion:
(a) if p < α, reject the null hypothesis
(b) if p > α, fail to reject the null hypothesis
Hypothesis testing for a mean
Hypotheses:
H0 : µ = µ0
Ha : µ 6= µ0 (two-tail)
or
µ < µ0 (one-tail)
Test statistic:
t∗ =
or µ > µ0 (one-tail)
X̄ − µ0
√
s/ n
Distribution: use the t-distribution with df = n − 1.
Hypothesis testing for a proportion
Hypotheses:
Test statistic:
H0 : p = p0
Ha : p 6= p0 (two-tail)
or
p < p0 (one-tail)
p̂ − p0
z∗ = p
p0 (1 − p0 )/n
Distribution: use the normal distribution.
or p > p0 (one-tail)
Notation for comparing two groups
n1
X̄1
s1
p̂1
=
=
=
=
size of first group
sample mean in first group
standard deviation in first group
proportion in first group
n2
X̄2
s2
p̂2
=
=
=
=
size of second group
sample mean in second group
standard deviation in second group
proportion in second group
Hypothesis testing for comparing two means
Hypotheses:
H0 : µ1 = µ2
Ha : µ1 6= µ2 (two-tail)
Test statistic:
or
µ1 < µ2 (one-tail)
or µ1 > µ2 (one-tail)
X̄1 − X̄2
t∗ = s
s21
s2
+ 2
n1 n2
Distribution: use the t-distribution with df equal to the minimum of n1 − 1 and n2 − 1.
Hypothesis testing for comparing two proportions
Hypotheses:
H0 : p1 = p2
Ha : p1 6= p2 (two-tail)
or
Test statistic:
z∗ = s
p1 < p2 (one-tail)
or p1 > p2 (one-tail)
p̂1 − p̂2
1
1
p̂(1 − p̂)
+
n1 n2
where p̂ is the pooled sample proportion, the proportion if we combine the two groups.
Distribution: use the normal distribution.
Notation for test of independence
r = number of rows
c = number of columns
fo = observed frequencies
fe = expected frequencies =
row total × column total
table total
Hypothesis testing for the independence between variables
Hypotheses:
H0 : the two variables are independent (no relationship)
Ha : the two variables are dependent (always one-tail)
Test statistic:
χ2 =
X (fo − fe )2
fe
Distribution: use the chi-square distribution with df = (r − 1) × (c − 1).