Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly  Lecture 19 Thought: Eagles may soar, but weasels aren’t sucked
into jet engines. Assignments Today : P. 328 – 332
For Tomorrow : Exercises 8.18, 8.21–23, 8.25, 8.27
Monday : P. 334 – 338, 347–351,
Tuesday : Exercises 8.29, 8.33, 8.34, 8.38–41, 8.59,
8.61, 8.67–69 217 STA 2023 c B.Presnell & D.Wackerly  Lecture 19 218 Last Time:
¤ £¡
¢ to within “B” units with ¨¨¢
©§¦ ¥ Estimate conﬁdence.
and solve for . (p. 307)
% $ " !
# Parts of a statistical test. (p. 322)
& The hypothesis of MAIN INTEREST is the
ALTERNATIVE or RESEARCH hypothesis, –
1 0 ( , ¦642
5753¢ '
)¡ ( ' light bulb ex. . (p. 322) What we are “trying to prove” in an objective, fair
manner
The “other” hypothesis is called the NULL
8 ' & HYPOTHESIS, – , light bulb ex. (p. 322) ¦642
575©¢ # 0 8 '
)¡ Errors: p. 325
Reality
Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct STA 2023 c B.Presnell & D.Wackerly  Lecture 19 (p. 323), SIGNIFICANCE
¦ Type I error 219 ¡ & #
¥ LEVEL of the test.
(p. 325)
¦ Type II error ¡£¤ ¡ ¡
# ( ' ¥ &
¥ & 642
75©¢ ¥ # & ¥ The test that we will discuss have the SMALLEST
¥ for the ’s that we pick. ¡ when ¦642
§5©¢ 1 saying & ¥ In our lightbulb example,
# & ¥ ¥ #
¡£¢ saying what we “want” to say when we should not ’s ¦ ' ¤ # true
8 when ¦ ¥ accepting ¢ and/or STA 2023 c B.Presnell & D.Wackerly  Lecture 19 220 Parts of a Statistical Test (p. 326)
.
' ' 2. Alternative Hypothesis : ( 8 1. Null Hypothesis : . 3. Test Statistic : (TS)
& computed from the sample data using a formula
& forms the basis for our decision. 4. Rejection Region : (RR)
&
&
&
& Get data
& Compute value of TS
& Make decision ¥ Do experiment Then ' depends on the choice of 8 gives values of TS for which is REJECTED STA 2023 c B.Presnell & D.Wackerly  Lecture 19 221 Decision :
& If the value of the TS is in the REJECTION
' and ( REGION, we . '
8 If the value of the TS is NOT in the REJECTION
REGION, we DO NOT REJECT.
because we usually do
8 ' – We do not ACCEPT not know the probability of making an error if we
do so.
8 ' – If we accept , what kind of error could we make?
– What is the probability of a TYPE II error? usually
' judgement 8 Don’t want to accept , so we reserve ' that is really true ( depends on the value of the parameter in & STA 2023 c B.Presnell & D.Wackerly  Lecture 19 222 Courtroom Analogy ( & 8 : innocent
: guilty '
&
'
& Experimenter : Prosecutor
& Put burden of proof on Prosecutor : Experimenter
& Proof “beyond a reasonable doubt” : small.
¥ STA 2023 c B.Presnell & D.Wackerly  Lecture 19 223 Large Sample Hypothesis Testing 8 8 # 1 0
0 8
( '
' 1 0
0 8
( '
'
#8 &
& Rejection Region, RR 62
4 3¢ # Test Statistic, TS 62
4 3¢ Need . a ﬁxed particular value of Lightbulb Example : In this case, about a Population Mean, STA 2023 c B.Presnell & D.Wackerly  Lecture 19 224 ¦ 2
¨ Recall : Large Sample %¡ & has an APPROXIMATE distribution ¡ £
¢ %
¤ ¤ ¡ ¤
¦ £ # #
£ % ¢ # & & That is
¡ ¤
¥ ¢ £ is a
is true
¤ ¤ 0 8 ' & ¡ 8
# ¦ # ¤ ¡ 8 8 if ¤
¥ £ % ¢
# £ is a TEST STATISTIC
is true,
¤ 8 ¦ % ¤
¡ ¤
£ ' # 8 ¡
¥ ¤
¢ £ # has a 8 If distribution & STA 2023 c B.Presnell & D.Wackerly  Lecture 19 is close to the true value of ¡ FACT: 225 , whatever that true value is.
is POSITIVE and LARGE 0 8 & ' in favor of 0
1 “something”
¥ # ( ¦ ' zα α 1 1 0 ( ' ¥ Rejection region : type I error & Rejection region : 0 # If we are interested in & Want than
& Should is probably &
¡ The true value of by a “lot” of standard errors. than 8 is 8 If . & STA 2023 c B.Presnell & D.Wackerly  Lecture 19 226 ¤ ¡ ¤ £
¦ % ¢ 8 # 8 1 0 8
0 ¤ '
( ' ¡ ¤
¦ £ %
#
1 ¡ 0
0 ¤¤ “Upper Tail test”, “One Tail Test” (p. 329) Ex. : Lightbulb Example
true mean lifelength of ALL BULBS # 642
75©¢
1 642
75©¢ 0 8
0 # # 2
2 4 ¦
4 £
2
2 # ¨ ¦ '
( '
¡ 2
3¢ # & # ¦
§¥
# 6
5¨
% # &
¥ Data : level test, RR : # & STA 2023 c B.Presnell & D.Wackerly  Lecture 19 227 & Conclusion :
AT THE
( ' in favor of
8 ' LEVEL!! 6
5¨ # ¥
& In terms of this problem: 642
75©¢ ¦ conﬁdence ).
56 ¢¨ # ¥ If we wanted , is level of 6¨ signiﬁcance” ( or with at the “ Claim that the mean lifelength of all bulbs, – RR :
4 1
6 ¦ ¢ # ¢¨ at the – 2
72 – Conclusion : Is ? level '
8 – In terms of this problem
claim that the mean lifelength of all
642
5©¢ conﬁdence ).
¦¦ with at the bulbs is larger than ¢¨ “ – NOTE: This DOES NOT mean that
is true!! level” ( or 642
75©¢ # 0 ' & STA 2023 c B.Presnell & D.Wackerly  Lecture 19 228 Ex. : #8.24, p. 333 Manufacturer inspection equip. for
printed circuit boards claims that “product can inspect,
on average, at least 10 boards per second”. Evidence to % 6¨ # ¥ . ?
0 8 ( ' and # onesecond runs? Use ¥ “refute the claim” based on data for ' 0 8
( &
' ' If we are interested in : ¤ ¦ % ¡ ¤ £ ¢ 0 ¡ 8
0 ¤ '
( ' ¡ ¤
¦ % 8 # 8
£ #
0 ¡ ¤¤
0 α − Zα 0 “Lower Tail test”, (p. 329) STA 2023 c B.Presnell & D.Wackerly  Lecture 19 229 Back to #8.24:
& Data:
10 9 10 10 11 9 12 8 7 10 11 9 9 13 9 10 9 9 9 7 12 6 9 10 11 12 10 0 10 11 12 9 8 9 6 10 11 10 12 8 10 8 7 9 7 9 9 10
2¨
73¢ £
# 4 4 ¦
4 ¦ ¡ # # £
level test 6
5¨ % # &
¥
& RR : ¨¤ 2
72
# # level of signiﬁcance. In terms of this 4¤ ¥ ¨
2 6¨ application: “There # ¨ at the enough evidence at the level to indicate that the mean number of circuit 6
5¨ boards inspected per second is less than 10 .” & ...
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 Spring '08
 Ripol
 Statistics, Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Type I and type II errors, Statistical Test

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