Transcript z - McGraw Hill Higher Education

Chapter 9
Hypothesis Testing
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Hypothesis Testing
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Null and Alternative Hypotheses and Errors in
Testing
z Tests about a Population Mean
s Known
t Tests about a Population Mean
s Unknown
z Tests about a Population Proportion
Type II Error Probabilities and Sample Size
Determination (Optional)
The Chi-Square Distribution (Optional)
Statistical Inference for a Population Variance
(Optional)
9-2
LO 1: Specify
appropriate null and
alternative hypotheses.
9.1 Null and Alternative Hypotheses
and Errors in Hypothesis Testing





Null hypothesis, H0, is a statement of the basic
proposition being tested
 Represents the status quo and is not rejected unless
there is convincing sample evidence that it is false
Alternative hypothesis, Ha, is an alternative accepted
only if there is convincing sample evidence it is true
One-Sided, “Greater Than” H0:   0 vs. Ha:  > 0
One-Sided, “Less Than” H0 :   0 vs. Ha :  < 0
Two-Sided, “Not Equal To” H0 :  = 0 vs. Ha :   0
where 0 is a given constant value (with the
appropriate units) that is a comparative value
9-3
LO 2: Describe Type I
and Type II errors and
their probabilities.
Error Probabilities

Type I Error: Rejecting H0 when it is true



 is the probability of making a Type I error
1 –  is the probability of not making a Type I error
Type II Error: Failing to reject H0 when it is false


β is the probability of making a Type II error
1 – β is the probability of not making a Type II error
State of Nature
Conclusion
Reject H0
Do not Reject H0
H0 True
H0 False
Type I
Error (α
Correct
Decision
Correct
Decision
Type II
Error (β
)
)
9-4
LO 3: Use critical values
and p-values to perform
a z test about a
population mean when s
is known.

9.2 z Tests about a
Population Mean: σ Known
Test hypotheses about a population mean
using the normal distribution


Called z tests
Require that the true value of the population
standard deviation σ is known

In most real-world situations, σ is not known



But often is estimated from s of a single sample
When σ is unknown, test hypotheses about a population
mean using the t distribution
Here, assume that we know σ
9-5
LO3
1.
2.
3.
4.
5.
6.
7.
Steps in Testing a “Greater
Than” Alternative
State the null and alternative hypotheses
Specify the significance level 
Select the test statistic
Determine the critical value rule for deciding
whether or not to reject H0
Collect the sample data and calculate the value of
the test statistic
Decide whether to reject H0 by using the test
statistic and the rejection rule
Interpret the statistical results in managerial terms
and assess their practical importance
9-6
LO 4: Use critical values
and p-values to perform
a t test about a
population mean.


Assume the population being sampled is
normally distributed
The population standard deviation σ is
unknown, as is the usual situation


9.3 t Tests about a Population
Mean: σ Unknown
If the population standard deviation σ is unknown,
then it will have to estimated from a sample
standard deviation s
Under these two conditions, have to use the t
distribution to test hypotheses
9-7
LO4





Defining the t Statistic: σ
Unknown
Let x be the mean of a sample of size n with
standard deviation s
Also, µ0 is the claimed value of the population mean
Define a new test statistic
x  0
t 
s n
If the population being sampled is normal, and s is
used to estimate σ, then …
The sampling distribution of the t statistic is a t
distribution with n – 1 degrees of freedom
9-8
LO4
t Tests about a Population
Mean: σ Unknown
Alternative
Reject H0 if:
p-value
Ha: µ > µ0
t > t
Area under t distribution to
right of t
Ha: µ < µ0
t < –t
Area under t distribution to
left of –t
Ha: µ  µ0
|t| > t /2 *
Twice area under t
distribution to right of |t|
tα, tα/2, and p-values are based on n – 1 degrees of freedom
(for a sample of size n)
* either t > tα/2 or t < –tα/2
9-9
LO 5: Use critical values
and p-values to perform
a large sample z test
about a population
proportion.
9.4 z Tests about a
Population Proportion
Alternative
Reject H0 if:
p-value
Ha: ρ > ρ0
z > z
Area under t distribution to
right of z
Ha: ρ < ρ0
z < –z
Area under t distribution to
left of –z
Ha: ρ  ρ0
|z| > z /2 *
Twice area under t
distribution to right of |z|
Where the test statistics is
* either z > zα/2 or z < –zα/2
z
p̂  p0
p0 1 p0 
n
9-10
LO 6: Calculate Type II
error probabilities and
the power of a test, and
determine sample size
(optional).

Want the probability β of not rejecting a false null
hypothesis




That is, want the probability β of committing a Type II error
1 - β is called the power of the test
Assume that the sampled population is normally
distributed, or that a large sample is taken
Test…



9.5 Type II Error Probabilities and
Sample Size Determination
(Optional)
H0: µ = µ0 vs
Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0
Want to make the probability of a Type I error equal
to α and randomly select a sample of size n
9-11
LO6
Calculating β

The probability β of a Type II error corresponding to
the alternative value µa for µ is equal to the area
under the standard normal curve to the left of
z* 


Continued
0  a
s
n
Here z* equals zα if the alternative hypothesis is
one-sided (µ < µ0 or µ > µ0)
Also z* ≠ zα/2 if the alternative hypothesis is twosided (µ ≠ µ0)
9-12
LO6
Sample Size




Population is normal, or a large sample is taken
H0:  = 0 vs. Ha:  < 0 or Ha:  > 0 or Ha:  ≠ 0
Want the probability of a Type I error equal to  and
probability of a Type II error corresponding to
alternative value  for  equal to b
Then take a sample of size:

z *  zb 2 s 2
n
 0   a 2


z* equals z, if alternative hypothesis is one-sided and
z* equals z/2 if the alternative hypothesis is two-sided
zb is point on the standard normal curve that gives a
right-hand tail area equal to b
9-13
LO 7: Describe the
properties of the chisquare distribution and
use a chi-square table
(optional).

The chi-square ² distribution depends on the
number of degrees of freedom


9.6 The Chi-Square
Distribution (Optional)
See Table A.17 of Appendix A
A chi-square point ²α is the point under a chisquare distribution that gives right-hand tail area 
9-14
LO 8: Use the chisquare distribution to
make statistical
inferences about
population variances
(optional).




9.7 Statistical Inference for
Population Variance (Optional)
If s2 is the variance of a random sample of n
measurements from a normal population with
variance σ2
The sampling distribution of the statistic (n - 1) s2 /
σ2 is a chi-square distribution with (n – 1) degrees of
freedom
Can calculate confidence interval and perform
hypothesis testing
100(1-α)% confidence interval for σ2
 (n  1) s 2 (n  1) s 2 


, 2
2
1 / 2 
  / 2
9-15