Introduction to Hypothesis Testing
Download
Report
Transcript Introduction to Hypothesis Testing
Chapter 8 Hypothesis
Testing for Means: Part 2
Type II Error Probabilities,
Required Sample Size for
Specified Power
1
Hypothesis Testing for , Type II Error
Probabilities (Right-tail example)
• Example
– A new billing system for a department store will be costeffective only if the mean monthly account is more than
$170.
– A sample of 400 accounts has a mean of $174 and s =
$65.
– Can we conclude that the new system will be cost
effective?
2
Example (cont.)
• Hypotheses
– The population of interest is the credit accounts at
the store.
– We want to know whether the mean account for all
customers is greater than $170.
H0 : m = 170
HA : m > 170
– Where m is the mean account value for all customers
3
Example (cont.)
– Test statistic:
H0 : m = 170
HA : m > 170
x 174, s 65
t
x m
s
n
174 170
65
1.23
400
4
Example (cont.)
P-value: The probability of observing a value
of the test statistic as extreme or more
extreme then t = 1.23, given that m = 170 is…
P -value P ( t 399 1.23) .1097
t399
Since the P-value > .05, we conclude
that there is not sufficient evidence to
reject H0 : =170.
0
t 1 .2 3
Type II error is possible
5
Calculating , the Probability of a
Type II Error
• Calculating for the t test is not at all
straightforward and is beyond the level of this
course
– The distribution of the test statistic t is quite
complicated when H0 is false and HA is true
– However, we can obtain very good approximate
values for using z (the standard normal) in place
of t.
6
Calculating , the Probability of a
Type II Error (cont.)
• We need to
1. specify an appropriate significance level ;
2. Determine the rejection region in terms of z
3. Then calculate the probability of not being in the
rejection when = 1, where 1 is a value of
that makes HA true.
7
Example (cont.) calculating
– Test statistic:
H0 : m = 170
HA : m > 170
Choose = .05
Rejection region in terms of z: z > z.05 = 1.645
rejectio n reg io n in term s o f x :
z
x 170
65
a = 0.05
1 .6 4 5
400
x 1 7 0 1 .6 4 5
65
400
1 7 5 .3 4 .
170
175.34
8
Example (cont.) calculating
– The rejection region with a = .05.
Express the rejection
region directly, not in
standardized terms
x 175 . 34
– Let the alternative value be m = 180 (rather than just
m>170)
H : m = 170
0
HA: m = 180
Do not reject H0
a=.05
m= 170
1 7 5 .3 4
Specify the
alternative value
under HA.
m180
9
Example (cont.) calculating
– A Type II error occurs when a false H0 is not
rejected. Suppose =180, that is H0 is false.
A false H0…
…is not rejected
H0: m = 170
H1: m = 180
x 175 . 34
m= 170
1 7 5 .3 4
a=.05
m180
10
Example (cont.) calculating
(180) P ( x 175.34 given that H 0 is false )
P ( x 175.34 given that m 180)
P(z
175 . 34 180
65
400
) . 0764
H0: m = 170
Power when =180
= 1-(180)=.9236
H1: m = 180
m= 170
1 7 5 .3 4
m180
11
Effects on of changing a
• Increasing the significance level a, decreases
the value of , and vice versa.
2 < 1
m= 170
a2 > a1
m180
12
Judging the Test
• A hypothesis test is effectively defined by the
significance level a and by the sample size n.
• If the probability of a Type II error is judged to
be too large, we can reduce it by
– increasing a, and/or
– increasing the sample size.
13
Judging the Test
• Increasing the sample size reduces
R e call R R : z
xm
s
n
z a , or
x m za
s
n
By increasing the sample size the
standard deviation of the sampling
distribution of the mean decreases.
Thus, the cutoff value of for the
rejection region decreases.
14
Judging the Test
• Increasing the sample size reduces
R e call R R : z
xm
s
n
z a , or
x m za
s
n
Note what happens when n increases:
a does not change,
but becomes smaller
m= 170
xxxLLxLxLxLL m180
15
Judging the Test
• Increasing the sample size reduces
• In the example, suppose n increases from 400 to
1000.
x m za
P (Z
s
170 1.645
n
173.38
1000
173.38 180
65
65
) P ( Z 3.22) 0
1000
• a remains 5%, but the probability of a Type II
drops dramatically.
16
A Left - Tail Test
• Self-Addressed Stamped Envelopes.
– The chief financial officer in FedEx believes that
including a stamped self-addressed (SSA) envelop
in the monthly invoice sent to customers will
decrease the amount of time it take for customers to
pay their monthly bills.
– Currently, customers return their payments in 24
days on the average, with a standard deviation of 6
days.
– Stamped self-addressed envelopes are included with
the bills for 75 randomly selected customers. The
number of days until they return their payment is
recorded.
17
A Left - Tail Test: Hypotheses
• The parameter tested is the population mean
payment period (m) for customers who receive
self-addressed stamped envelopes with their bill.
• The hypotheses are:
H0: m = 24
H1: m < 24
• Use = .05; n = 75.
18
A Left - Tail Test: Rejection Region
• The rejection region:
• t < t.05,74 = 1.666
• Results from the 75 randomly selected
customers:
x 22.95 days, s 6 days
19
A Left -Tail Test: Test Statistic
• The test statistic is:
t
xm
s
n
22.95 24
6
1.52
75
Since the rejection region is t t t 1.666
a
.05
We do not reject the null hypothesis.
Note that the P-value = P(t74 < -1.52) = .066.
Since our decision is to not reject the null hypothesis,
A Type II error is possible.
20
Left-Tail Test: Calculating , the
Probability of a Type II Error
• The CFO thinks that a decrease of one day in the
average payment return time will cover the costs of
the envelopes since customer checks can be
deposited earlier.
• What is (23), the probability of a Type II error when
the true mean payment return time is 23 days?
21
Left-tail test: calculating
(cont.)
– Test statistic:
H0 : m = 24
HA : m < 24
Choose = .05
Rejection region in terms of z: z < -z.05 = -1.645
rejectio n reg io n in term s o f x :
z
x 24
6
1 .6 4 5
a = 0.05
75
x 2 4 1 .6 4 5
6
75
2 2 .8 6 .
2 2 .8 6
24
22
Left-tail test: calculating (cont.)
– The rejection region with a = .05.
Express the rejection
region directly, not in
standardized terms
x 22.86
– Let the alternative value be m = 23 (rather than just
m < 24)
H : m = 24
0
HA: m = 23
Specify the
alternative value
under HA.
Do not reject H0
a=.05
2 2 .8 6
m= 23
m24
23
Left-tail test: calculating (cont.)
(23) P ( x 22.86 given that H 0 is false )
P ( x 22.86 given that m 23)
22.86 23
Pz
.718
6 75
H0: m = 24
Power when =23 =
1-(23)=.282
H1: m = 23
a=.05
2 2 .8 6
m= 23
m24
24
A Two - Tail Test for
• The Federal Communications Commission
(FCC) wants competition between phone
companies. The FCC wants to investigate if
AT&T rates differ from their competitor’s rates.
• According to data from the (FCC) the mean
monthly long-distance bills for all AT&T
residential customers is $17.09.
25
A Two - Tail Test (cont.)
• A random sample of 100 AT&T customers is
selected and their bills are recalculated using a
leading competitor’s rates.
• The mean and standard deviation of the bills
using the competitor’s rates are
x $17.55, s $3.87
• Can we infer that there is a difference between
AT&T’s bills and the competitor’s bills (on the
average)?
26
A Two - Tail Test (cont.)
• Is the mean different from 17.09?
H0: m = 17.09
H A : m 17.09
• n = 100; use = .05
27
A Two – Tail Test (cont.)
Rejection region
t t .025 , 99 or t t .025 , 99
t 1.9842 or t 1.9842
t
x m
s
n
17.55 17.09
3.87
1 .1 9
t99
100
a/2 0.025
-ta/2 = -1.9842
a/2 0.025
0 ta/2 = 1.9842
Rejection region
28
A Two – Tail Test: Conclusion
There is insufficient evidence to conclude that there is a
difference between the bills of AT&T and the competitor.
Also, by the P-value approach:
The P-value = P(t < -1.19) + P(t > 1.19)
= 2(.1184) = .2368 > .05
t
x m
s
n
17.55 17.09
3.87
1.19
100
A Type II error is possible
a/2 0.025
a/2 0.025
-1.19 0 1.19
-ta/2 = -1.9842
ta/2 = 1.9842
29
Two-Tail Test: Calculating , the
Probability of a Type II Error
• The FCC would like to detect a decrease of $1.50 in
the average competitor’s bill. (17.09-1.50=15.59)
• What is (15.59), the probability of a Type II error
when the true mean competitor’s bill is $15.59?
30
Two – Tail Test: Calculating (cont.)
Rejection region
rejectio n reg io n in term s o f x :
z
x 1 7 .0 9
1 .9 6
z z .025 or
z z .025
z 1.96 or
z 1.96
3 .8 7
100
x 1 7 .0 9 1 .9 6
3 .8 7
100
a/2 0.025
x 1 6 .3 3
z
x 1 7 .0 9
a/2 0.025
Do not reject H0
1 .9 6
3 .8 7
100
x 1 7 .0 9 1 .9 6
3 .8 7
1 6 .3 3
17.09
17.85
Reject H0
100
x 1 7 .8 5
31
Two – Tail Test: Calculating (cont.)
(15.59) P (16.33 x 17.8 5 given that m 15.59)
16.33 15.59
x 15.59
17.85 15.59
P
3.87 100
3.87 100
3.87 100
P (1.912 z 5.84) .028
H0: m = 17.09
HA: m = 15.59
Power when
=15.59 = 1(15.59)=.972
a=.05
m17.09
m= 15.59
1 6 .6 3
1 7 .8 5
32
General formula: Type II Error
Probability (A) for a Level Test
H A :m m0
m0 m A
P z za
n
H A :m m0
m0 m A
1 P z za
n
H A :m m0
m0 m A
m0 m A
P z za / 2
P z za / 2
n
n
33
Sample Size n for which a level
test also has (A) =
(z z ) 2
a
for a 1-tailed (right or left) test
m0 m A
n
2
( za / 2 z )
for a 2-tailed test (approx. solution)
m0 m A
34