Hypothesis Testing for Population
Means and Proportions
Topics
• Hypothesis testing for population means:
– z test for the simple case (in last lecture)
– z test for large samples
– t test for small samples for normal distributions
• Hypothesis testing for population proportions:
– z test for large samples
z-test for Large Sample Tests
• We have previously assumed that the population
standard deviationσis known in the simple case.
• In general, we do not know the population standard
deviation, so we estimate its value with the standard
deviation s from an SRS of the population.
• When the sample size is large, the z tests are easily
modified to yield valid test procedures without
requiring either a normal population or known σ.
• The rule of thumb n > 40 will again be used to
characterize a large sample size.
z-test for Large Sample Tests (Cont.)
• Test statistic:
X  0
Z
s/ n
• Rejection regions and P-values:
– The same as in the simple case
• Determination of β and the necessary sample size:
– Step I: Specifying a plausible value of σ
– Step II: Use the simple case formulas, plug in theσ
estimation for step I.
t-test for Small Sample Normal
Distribution
• z-tests are justified for large sample tests by the fact that: A
large n implies that the sample standard deviation s will be
close toσfor most samples.
• For small samples, s and σare not that close any more. So
z-tests are not valid any more.
• Let X1,…., Xn be a simple random sample from N(μ, σ). μ
and σ are both unknown, andμ is the parameter of interest.
• The standardized variable
x
T
~ t n 1
s n
The t Distribution
• Facts about the t distribution:
– Different distribution for different sample sizes
– Density curve for any t distribution is symmetric about
0 and bell-shaped
– Spread of the t distribution decreases as the degrees of
freedom of the distribution increase
– Similar to the standard normal density curve, but t
distribution has fatter tails
– Asymptotically, t distribution is indistinguishable from
standard normal distribution
Table A.5
Critical Values for t
Distributions
α = .05
Degrees of Freedom
1
2
.
.
20
.
.
200
z*
0.1
3.078
1.886
.
.
1.325
.
.
1.286
1.282
0.05
6.314
2.92
.
.
1.725
.
.
1.653
1.645
0.025 0.01
12.706 31.821
4.303
6.965
.
.
.
.
2.086
2.528
.
.
.
.
1.972
2.345
1.96
2.326
0.005
63.657
9.925
.
.
2.845
.
.
2.601
2.576
t-test for Small Sample Normal
Distribution (Cont.)
• To test the hypothesis H0:μ = μ0 based on an SRS
of size n, compute t test statistic
x  0
T
s
n
• When H0 is true, the test statistic T has a t
distribution with n -1 df.
• The rejection regions and P-values for the t tests
can be obtained similarly as for the previous
cases.
Case 1 : H a :    0 (two tailed - test) . Then H 0 should be
rejected if x is too far away from 0.
- - The rejection region is | T | t / 2, n 1.
- - The P - value is 2 P (T  t ).
Case 2 : H a :    0 (upper - tailed test). Then H 0 should be
rejected if z is much larger tha n 0.
- -The rejection region is T  t , n 1.
- - The P - value is P(T  t ).
Case 3 : H a :    0 (lower - tailed test). Then H 0 should be
rejected if z is much smaller th an 0.
- -The rejection region is T  t , n 1.
- - The P - value is P(T  t ).
Recap: Population Proportion
• Let p be the proportion of “successes” in a population. A
random sample of size n is selected, and X is the number of
“successes” in the sample.
• Suppose n is small relative to the population size, then X
can be regarded as a binomial random variable with
E ( X )   X  np
Var ( X )   X2  np (1  p )
 X  np (1  p )
Recap: Population Proportion (Cont.)
• We use the sample proportion pˆ  X / n as an estimator of
the population proportion.
• We have
E ( pˆ )   pˆ  p
p (1  p )
Var ( pˆ )   
n
p (1  p )
 pˆ 
n
2
pˆ
• Hence p̂ is an unbiased estimator of the population
proportion.
Recap: Population Proportion (Cont.)
• When n is large, p̂ is approximately normal. Thus
z
pˆ  p
p(1  p) / n
is approximately standard normal.
• We can use this z statistic to carry out hypotheses for
H0: p = p0 against one of the following alternative
hypotheses:
– Ha: p > p0
– Ha: p < p0
– Ha: p ≠ p0
Large Sample z-test for a Population
Proportion
• The null hypothesis H0: p = p0
• The test statistic is
pˆ  p0
z
p0 (1  p0 ) / n
Alternative
Hypothesis
Ha: p > p0
P-value
P(Z ≥ z)
Rejection Region
for Level α Test
z ≥ zα
Ha: p < p0
P(Z ≤ z)
z ≤ - zα
Ha: p ≠ p0
2P(Z ≥ | z |)
| z | ≥ zα/2
Determination of β
• To calculate the probability of a Type II error, suppose
that H0 is not true and that p = p instead. Then Z still
has approximately a normal distribution but
E (Z ) 
p  p'
p0 (1  p0 ) / n ,
p ' (1  p ' ) / n
V (Z ) 
p0 (1  p0 ) / n
• The probability of a Type II error can be computed by
using the given mean and variance to standardize and
then referring to the standard normal cdf.
Case 1 : H a : p  p0 .
- - The Type II error probabilit y  ( p ' ) is :
(
p0  p '  z / 2 p0 (1  p0 ) / n
p (1  p ) / n
'
'
)  (
p0  p '  z / 2 p0 (1  p0 ) / n
p (1  p ) / n
'
'
).
Case 2 : H a : p  p 0 .
- -The Type II error probabilit y  (  ' ) is :  (
p0  p '  z / 2 p0 (1  p0 ) / n
p (1  p ) / n
'
'
)
Case 3 : H a : p  p0 .
- -The Type II error probabilit y  (  ' ) is : 1 -  (
p0  p '  z / 2 p0 (1  p0 ) / n
p (1  p ) / n
'
'
).
Determination of the Sample Size
• If it is desired that the level αtest also have β(p ) = β
for a specified value of β, this equation can be solved
for the necessary n as in population mean tests.
2

'
' 
 z p0 (1  p0 )  z  p (1  p )  , one - tailed test

p '  p0


n
2
'
'
 z
p0 (1  p0 )  z  p (1  p ) 
 /2
 , two - tailed test

'
p  p0


