Download Slides 2-7 Hypothesis Testing

BA 275
Quantitative Business Methods
Agenda
 Statistical Inference: Hypothesis Testing


Type I and II Errors
Power of a Test
 Hypothesis Testing Using Statgraphics
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Type I, II Errors, and Power
a = P( Type I Error ) = P( reject H0 given that H0 is true)
b = P( Type II Error ) = P( fail to reject H0 given that H0 is false)
Power = 1 – b = P( reject H0 given that H0 is false )
Refer to textbook p. 416
2
Example 1
 You want to see if a redesign of the cover of a mail-
order catalog will increase sales. Assume that the
mean sales from the new catalog will be
approximately normal with s = $50 and that the mean
for the original catalog will be m = $25. Given a
sample of size n = 900, you wish to test H0: m = 25
vs. Ha: m > 25. Rejection region is: reject H0 if the
sample mean > 26.
 Q1: Find a = P( Type I error ).
 Q2: Find b = P( Type II error ) given m = $28.
 Q3: Find b = P( Type II error ) given m = $30.
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Example 2
 A survey of 100 retailers was conducted to see if the
mean after-tax profit exceeded $70,000. Assume
that the population standard deviation is $20,000.




Q1. Let rejection region be: Rejection H0 if the sample
mean exceeds $73,290. Find a.
Q2. Calculate the power of the test given that the true
m = 75,500?
Q3. Suppose the sample mean is $75,000. Find the pvalue.
Q4. Estimate the true mean after-tax profit using a 95%
confidence interval.
4
Hypothesis Testing Using SG
 Steps are similar to what you did for Confidence
Interval Estimation.
 Use the right mouse button for more options. For
example, to change the significance level (the value
of a) and/or to select the type of tests (lower, upper,
or two-tailed test.)
 Refer to page 3 of the SG instruction for interval
estimation (filename: SG Instruction
Estimation.doc downloadable from the class
website) on how to select a subset of sample for
analysis.
5
Central Limit Theorem (CLT)
 The CLT applied to Means
If X ~ N ( m , s 2 ) , then X ~ N ( m ,
s2
).
n
If X ~ any distribution with a mean m, and variance s2,
then X ~ N ( m ,
What if s is unknown?
What if s is unknown
and n is small?
s2
n
) given that n is large.
With a large n, approximate s with s.
The CLT still holds.
Need to modify the CLT.
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Central Limit Theorem (CLT)
 s is unknown but n is large
s
2
s
If X ~ N ( m , s ) , then X ~ N ( m , ) .  N ( m , )
n
n
2
2
If X ~ any distribution with a mean m, and variance s2,
then X ~ N ( m ,
s2
n
) given that n is large.
s2
 N (m , )
n
7
Central Limit Theorem (CLT)
 s is unknown and n is small
s2
X ~ t ( m , ) with degrees of freedom  n  1.
n
If X ~ N ( m , s 2 ) , then X ~ N ( m ,
s2
).
n
If X ~ any distribution with a mean m, and variance s2,
then X ~ N ( m ,
s2
n
) given that n is large.
8
T distribution with degrees of freedom
5 vs. Normal(0, 1)
9
Example 3
 A random sample of 10 one-bedroom apartments




(Ouch, a small sample) from your local newspaper
gives a sample mean of $541.5 and sample standard
deviation of $69.16. Assume a = 5%.
Q1. Does the sample give good reason to believe
that the mean rent of all advertised apartments is
greater than $500 per month? (Need H0 and Ha,
rejection region and conclusion.)
Q2. Find the p-value.
Q3. Construct a 95% confidence interval for the
mean rent of all advertised apartments.
Q4. What assumption is necessary to answer Q1-Q3.
10
T Table (Table D)
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Answer Key to Examples Used
Example 1
Q1. a = P( reject H0 given that H0 is true ) = P(
26  25
= P( Z 
) = 0.2743.
50 / 900
X > 26 given that m = 25 )
Q2. b = P( fail to reject H0 given that H0 is false ) = P(
26  28
= P( Z 
) = 0.1151.
50 / 900
X < 26 given that m = 28 )
Q3. b = P( fail to reject H0 given that H0 is false ) = P( X < 26 given that m = 30 )
26  30
= P( Z 
) = 0.0082.
50 / 900
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Answer Key to Examples Used
Example 2
Q1. a = P( reject H0 given that H0 is true ) = P(
73290  70000
= P( Z 
) = 0.05.
20000 / 100
X > 73290 given that m = 70000 )
Q2. b = P(fail to reject H0 given that H0 is false) = P( X < 73290 given that m = 75500 )
73290  75500
= P( Z 
) = 0.1335. Power = 1 – b = 0.8665.
20000 / 100
Q3. p-value = P( getting a sample mean $75,000 or more extreme given that m = 70000 )
75000  70000
= P( Z 
) = 0.0062. Reject H0.
20000 / 100
20000
Q4. 75000  (1.96)(
)
100
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Answer Key to Examples Used
Example 3
Q1. H0: m = 500 vs. Ha: m > 500. Rejection region: reject H0 if t 
x  500
 1.833 .
s/ n
1.833 comes from Table D with a = 0.05 and df = n – 1 = 9. Since the sample
mean 541.5 has a t score = 1.8975 (in the rejection region), we concluded that m >
500, i.e., reject H0.
Q2. We rejected H0 with a = 0.05 but will not reject H0 if a = 0.025. This implies that
the p-value must fall between 0.025 and 0.05.
69.16
Q3. 541.5  2.262
10
Q4. all rents advertised follow a normal distribution, i.e., the population distribution is
normal X ~ N ( m , s 2 ).
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