Chapter 9 - Hypothesis testing. After mastering the material in this chapter, you will be able to: Set Up appropriate null and alternative hypotheses, describe Type I and Type II errors and their probabilities, use critical values and p-values to perform a z test about a population mean when s is known,...
Hypothesis Testing
9.1 Null and Alternative Hypotheses and
Errors in Testing
9.2 z Tests about a Population Mean
σ Known
9.3 t Tests about a Population Mean
σ Unknown
9.4 z Tests about a Population Proportion
9.5 Type II Error Probabilities and Sample
Size Determination (Optional)
92
LO9-1: 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
OneSided, “Greater Than” H0: μ μ0 vs. Ha: μ > μ0
OneSided, “Less Than” H0 : μ μ0 vs. Ha : μ
LO9-1
Types of Decisions
As a result of testing H0 vs. Ha, will decide either of
the following decisions for the null hypothesis H0:
◦Do not reject H0 or reject H0
To “test” H0 vs. Ha, use the “test statistic”
x 0 x 0
z
x n
z measures the distance between μ0 and on the
sampling distribution of the sample mean
If the population is normal or n is large*, the test
statistic z follows a normal distribution
94
LO9-2: 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
Table 9.1 95
LO9-2
Typical Values
Usually set to a low value
◦So there is a small chance of rejecting a true H0
Typically, = 0.05
◦Strong evidence is required to reject H0
◦Usually choose α between 0.01 and 0.05
= 0.01 requires very strong evidence to reject H0
Tradeoff between and β
◦For fixed sample size, the lower , the higher β
And the higher , the lower β
96
LO9-3: Use critical
values and p-values to
perform a z test about a
population mean when
σ 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 realworld 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
97
LO9-3
Steps in Testing a “Greater Than”
Alternative
1. State the null and alternative hypotheses
2. Specify the significance level α
3. Select the test statistic
4. Determine the critical value rule for deciding
whether or not to reject H0
5. Collect the sample data and calculate the value of
the test statistic
6. Decide whether to reject H0 by using the test
statistic and the rejection rule
7. Interpret the statistical results in managerial terms
and assess their practical importance
98
LO9-4: Use critical
values and p-values to
perform a t test about a
population mean when
σ is unknown.
9.3 t Tests about a Population Mean:
σ Unknown
Assume the population being sampled is
normally distributed
The population standard deviation σ is
unknown, as is the usual situation
◦If the population standard deviation σ is
unknown, then it will have to estimated from a
sample standard deviations
Under these two conditions, have to use the t
distribution to test hypotheses
99
LO9-5: Use critical
9.4 z Tests about a Population
values and p-values to
perform a large sample
z test about a
Proportion
population proportion.
Alternative Reject H0 if: pvalue
Ha: ρ > ρ0 z > z Area under t distribution to
right of z
Ha: ρ zα/2 or z
LO9-6: Calculate Type II
error probabilities and
the power of a test, and
determine sample size
(Optional).
9.5 Type II Error Probabilities and
Sample Size Determination (Optional)
Want the probability β of not rejecting a false null
hypothesis
◦ 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…
◦H0: µ = µ0 vs
◦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
911
LO9-6
Calculating β Continued
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
0 a
z*
n
Here z* equals zα if the alternative hypothesis
is onesided (µ µ0)
Also z* ≠ zα/2 if the alternative hypothesis is
twosided (µ ≠ µ0)
912
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