Post on 04-Jan-2016
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CHAPTER 9
STATISTICAL HYPOTHESIS TESTING:HYPOTHESIS TESTS FOR A
POPULATION MEAN
The Nature of Hypothesis Testing
• In hypothesis testing, a statement—call it a hypothesis— is made about some characteristic of a particular population. A sample is then taken in an effort to establish whether or not the statement is true.
• If the sample produces a result that would be highly
unlikely under an assumption that the statement is true, then we’ll conclude that the statement is false.
Choosing the Null Hypothesis
• The status quo or “if-it’s-not-broken-don’t-fix-it” approach
• The good sport approach
• The skeptic’s “show me” approach
Standard Forms for the Null and Alternative Hypotheses
H0: > A (The population mean is greater than or equal to A)
Ha: < A (The population mean is less than A)
H0: < A (The population mean is less than or equal to A)
Ha: > A (The population mean is greater than A)
H0: = A (The population mean is equal to A)
Ha: ≠ A (The population mean is not equal to A)
Figure 9.1 The Sampling Distribution of the
Sample Mean
n
x
x
=
Figure 9.2 The “Null” Sampling Distribution = 5000
5000
x
Figure 9.3 Likely and Unlikely Sample Means in the “Null” Sampling Distribution
5000
x
xxx = 5 = 4997
= 4998
Significance Level ()
A significance level defines what we mean by unlikely sample results under an assumption that the null hypothesis is true (as an equality).
Figure 9.4 Setting the Boundary on the “Null” Sampling Distribution
z
5000
0zc = -1.65
= .05
REJECT H0
pounds scale
z scalex
Figure 9.5 Showing zstat on the Null Sampling Distribution
x5000
0zc = -1.65
= 4912
REJECT H0
pounds scale
z scale
x
zstat = -2.49
z
The Four Steps of Hypothesis Testing
Step 1: State the null and alternative hypotheses.
Step 2: Choose a test statistic and use the
significance level to establish a decision rule.
Step 3: Compute the value of the test statistic.
Step 4: Apply the decision rule and make your
decision.
Figure 9.6 Showing the Boundary, c, on the Null Sampling Distribution
5000
REJECT H0
zzc = -1.65
c = 4941.6
pounds scale
z scale
=.05
x
p-value
The p-value measures the probability that, if the null hypothesis were true, we would randomly produce a sample result at least as unlikely as the sample result we actually produced.
Figure 9.7 p-value for a Sample Mean of 4962
(pounds)
z -1.07
4962
p-value = .1423
0
x
p-value Decision Rule
If the p-value is less than , reject the null hypothesis.
Figure 9.8 Using the p-value to Make a Decision
(pounds)
z
zstat =-1.07
4962
p-value = .1423
4941.6
zc = -1.65
REJECT H0
x
Error Possibilities in Hypothesis Testing
TYPE I Error: Rejecting a true null hypothesis.
TYPE II Error: Accepting a false null hypothesis.
and the Risk of Type I Error
measures the maximum probability of making a Type I Error.
Figure 9.9 A Two-tailed Hypothesis Test
REJECT H0
zzcl
/2
zcu
REJECT H0
2
x
Figure 9.10 A Two-tailed Hypothesis Test for Montclair Motors
5000
REJECT H0
z-1.96
= .025
+1.96
REJECT H0
= .025
cL = 4930.7 cu = 5069.3 x
Test Statistic When s Replaces (9.2)
ns
x
/
tstat =
Figure 9.11 Testing with the t
Distribution
t
REJECT H0
tc
tstat