statistics for managers using microsoft® excel 7th edition chapter 9 fundamentals of hypothesis...
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Statistics for ManagersUsing Microsoft® Excel
7th EditionChapter 9
Fundamentals of Hypothesis Testing:
One Sample Tests
Statistics for Managers Using Microsoft Excel, 7e © 2014 Pearson-Prentice-Hall, Inc. Philip A. Vaccaro , PhD
Learning Objectives
The Hypothesis
A hypothesis is a claim (assumption) about a population parameter: population mean ( μ )
pop. proportion ( π )
The Null Hypothesis, H0
3μ:H0
it is written in terms of the population
The Null Hypothesis, H0
The Alternative Hypothesis, H1
The Alternative Hypothesis, H1
The Hypothesis Testing Process
Claim: The population mean age is 50. H0: μ = 50, H1: μ ≠ 50
Sample the population and find sample mean.Population
Sample
The Hypothesis Testing Process
_
The Hypothesis Testing Process
Sampling Distribution of X
μ = 50If H0 is true
20X
The Test Statistic and Critical Values
The Test Statistic and Critical Values
Critical Values
Distribution of the test statistic
Region of Rejection
Region of Rejection
Do Not Reject Ho
Thecriticalvalues
are statedin either‘z’ or ‘t’values
Errors in Decision Making
Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is ‘’ Called level of significance of the test Set by researcher in advance:
Type II Error
Failure to reject false null hypothesis The probability of a Type II Error is ‘β’
within your control or dictated by management
Errors in Decision Making
Level of Significance, α
H0: μ ≥ 50
H1: μ < 500
H0: μ ≤ 50
H1: μ > 50
a
a
Lower-tail test
0
Upper-tail test
Two-tail test
0
H0: μ = 50
H1: μ ≠ 50
Claim: The population mean age is 50.
/2a
/2a
The probability of rejecting Ho when it is true
typical values
areα =.01α =.02α =.05α =.10
Hypothesis Testing: σ Known
_
Hypothesis Testing: σ Known
Do not reject H0 Reject H0Reject H0
/2
-Z0
H0: μ = 3
H1: μ ≠
3
+Z
/2
Lower critical value
Upper critical value
3
Z
X
Hypothesis Testing: σ Known
Hypothesis Testing: σ Known
_
Hypothesis Testing: σ Known
Is the test statistic in the rejection region?
Reject H0 Do not reject H0
a = .05/2
-Z= -1.96 0
= .05/2
Reject H0
+Z= +1.96
Here, Z = -2.0 < -1.96, so the test statistic is in the rejection region
a = .05level of
significance
Z 0.06
- 1.9 .0250
Z 0.06
+ 1.9 .9750
.025 .025
Hypothesis Testing: σ Known
Hypothesis Testing: σ Known
Hypothesis Testing: σ Known
Hypothesis Testing: σ Known ,the ‘p’- Value Approach
* Because we can reject or not reject Ho “at a glance“ .
Hypothesis Testing: σ Knownp-Value Approach
Hypothesis Testing: σ Knownp-Value Approach
.0228
/2 = .025
-1.96 0-2.0
Z1.962.0
.0228
/2 = .025
Z 0.00
- 2.00 .0228
Z 0.00
+ 2.00 .9772
Z = 2.84 - 3.00 .8 √100
= - .16 = - 2.00 .08
Z = 3.16 - 3.00 .8 √100
= +.16 = + 2.00 .08
Ho: μ = 3.00
2.84 3.16
(.9772)
Hypothesis Testing: σ Knownp-Value Approach
Compare the p-value with If p-value < , reject H0
If p-value , do not reject H0
.0228
/2 = .025
-1.96 0
-2.0
Z1.96
2.0
.0228
/2 = .025
The probabilityof seeing a
sample meanof 2.84 or less,
or 3.16 or more , if
the populationmean is really 3.0
is only 4.56%
Hypothesis Testing: σ KnownConfidence Interval Connections
Simpler,faster
hypothesistest
Canonly be
used witha two-tailhypothesis
test !
Ifa = .05,
constructa 95%
confidenceinterval
Hypothesis Testing: σ KnownOne Tail Tests
H0: μ ≥ 3
H1: μ < 3
H0: μ ≤ 3
H1: μ > 3
Hypothesis Testing: σ KnownLower Tail Tests
There is only one critical value, since the rejection area is in
only one tail.
Reject H0 Do not reject H0
α
-Z
μ
Z
XCritical value
Hypothesis Testing: σ KnownUpper Tail Tests
There is only one critical value, since the rejection area is in
only one tail.
Reject H0
Do not reject H0
α
Zμ
Critical value
Z
X
Hypothesis Testing: σ KnownUpper Tail Test Example
Form the hypothesis test:
Hypothesis Testing: σ KnownUpper Tail Test Example Suppose that = .10 is chosen for this test Find the rejection region:
Reject H0Do not reject H0
= .10
Z0
Reject H0
1- = .90
Z 0.08
+ 1.2 .8997
Theentirea = .10goes to theupper
tail
Hypothesis Testing: σ KnownUpper Tail Test Example
What is Z given a = 0.10?
Z .07 .09
1.1 .8790 .8810 .8830
1.2 .8980 .9015
1.3 .9147 .9162 .9177z 0 1.28
.08a = .10
Critical Value = 1.28
.90
.8997
.10
.90
Hypothesis Testing: σ KnownUpper Tail Test Example
-
Hypothesis Testing: σ KnownUpper Tail Test Example
Reach a decision and interpret the result:
= .10
1.280
Reject H0
1- = .90
Z = .88
Do Not Reject Ho
Hypothesis Testing: σ KnownUpper Tail Test Example
Calculate the p-value and compare to ‘’
Reject H0
= .10
Do not reject H0
1.28
0
Reject H0
Z = .88 .1894
.810610.88)P(Z
6410/
52.053.1ZP
53.1)XP(
p-value = .1894Z 0.08
+ .8 .8106
Z = 53.1 - 52.0 = 1.1 10 / √ 64 1.25
= .88
1 - .8106 = .1894
“Talk”is notcheap,
especiallywhen
you arebeing
cheated !
Hypothesis Testing: σ Unknown
df =1df =2df =3df =4df =5df =6
Hypothesis Testing: σ Unknown
Recall that the ‘t’ test statistic with n-1 degrees of freedom is:
Hypothesis Testing: σ Unknown Example
_
H0: μ =
168 H1: μ ¹ 168
_
Hypothesis Testing: σ Unknown Example
H0: μ = 168
H1: μ ≠ 168
Reject H0Reject H0
α/2=.025
-t n-1,α/2
Do not reject H0
0
α/2=.025
-2.0639 2.0639
t n-1,α/2
Determine the regions of rejection
t .025
24 df 2.0639
Hypothesis Testing: σ Unknown Example
a/2=.025
-t n-1,α/2 0
a/2=.025
-2.0639 2.0639
t n-1,α/21.46Since t df = 24, a=.025 = 1.46 < 2.0639,
Do Not Reject HoRejectHo
RejectHo
Hypothesis Testing: Connection to Confidence Intervals
For X = 172.5, S = 15.40 and n = 25, the 95% confidence interval is:
166.14 ≤ μ ≤ 178.86
Since this interval contains the hypothesized mean (168), you do not reject the null hypothesis at = .05 _
Hypothesis Testing: σ Unknown
Hypothesis Testing:Proportions
Involves categorical variables ( yes/no , male/female )
Two possible outcomes “Success” (possesses a certain characteristic)
“Failure” (does not possesses that characteristic)
Fraction or proportion of the population in the “success” category is denoted by π
Hypothesis Testing:Proportions
sizesample
sampleinsuccessesofnumber
n
Xp
pμn
)(1σ
p
Sample proportion in the success category is denoted by p
When both nπ and n(1-π) are at least 5, p can be approximated
by a normal distribution with mean and standard deviation
Hypothesis Testing:Proportions
The sampling distribution of p is approximately normal, so the test statistic is a ‘Z’ value:
n
pZ
)1(
Hypothesis Testing:Proportions Example
A marketing company claims that it receives 8% responses from its mailings. To test this claim, a random sample of 500 were surveyed with 30 responses. Test at the = .05 significance level.
Hypothesis Testing:Proportions ExampleH0: π = .08 H1: π ≠ .08
α = .05
n = 500, p = .06
Critical Values: ± 1.96
z0
Reject Reject
.025.025
1.96-1.96
Determine region of rejection
Do Not RejectHo
30 / 500 = .06 = p (sample proportion)
Z 0.06
- 1.9 .0250
Z 0.06
+ 1.9 .9750
Hypothesis Testing:Proportions Example
Test Statistic: Decision:
Conclusion:
z0
.025.025
1.96-1.96-1.646
Do Not RejectHo
Potential Pitfalls and Ethical Considerations
Chapter Summary
In this chapter, we have