Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly  Lecture 17 194 Thought: A committee is a group of people who,
individually, can do nothing, but collectively can meet,
and decide that nothing can be done. Assignments :
Today: P. 299 – 302, COMPUTER DEMONSTRATION
For Tuesday: Exercises 7.37, 7.42, 7.44–46, 7.72,
7.77, 7.79, 7.82, 7.91
Wednesday: P. 306 – 308, 322 – 326
For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67,
8.7, 8.9, 8.10, 8.13
Monday: OPTIONAL review day
Tuesday: EXAM 2 STA 2023 c B.Presnell & D.Wackerly  Lecture 17 195 Last Time: Conﬁdence Coefﬁcient Conﬁdence Level (p. 282) ¨
©§ ¤ £¡
¢ ¨
¦ §
§¡
such that Interval Estimator (p. 282)
¦¥ Point Estimator (p. 261) (p. 282) . STA 2023 c B.Presnell & D.Wackerly  Lecture 17 196 §§
©¨¡ Conﬁdence Interval for ¦¥¤ £ ¢ ¡ (large sample) p. 283
" # !§
&
'% ¥ is the z value that cuts off an area of in
$ § the upper tail of a standard normal distribution α/2 α/2 1−α §
§
§ ¤
( NOTE : the basic form of this interval is
¡ $§
table formula sheet 53 1
40 1
) 3 40 ) 3 1
20 ) formula sheet standard errors ¦ estimator STA 2023 c B.Presnell & D.Wackerly  Lecture 17 197 Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger
than steel, but brittle at high temps. Want to estimate
the mean temp. at which metalic glass becomes brittle.
36 specimens randomly selected, measured temp. ( ¥£
¤
¦ at which each became brittle : and ¡
¢ ¡ ¢¢ ¦ true mean temp. metalic glass gets brittle is § unknown,
¢¢ ( £
¤ ¦ (¢
5 5
# ¢
£
¢¢ ¦ & £¡
£¢ ( £
¤ 5
© 5
5
¡ ¡ ¦ & ( 43¤ 10(¡
©¤£2
)£
¦&( 5 ©¤ £2
¥98¤
75 )£
65(¡ : ¦ !
" § § 5
& ¢
£ #
" 5
& ¢
£ ¦ (
5
¥(
¥ 5 ¨ ¤¢ ¥
&
$
%(
$
%( conﬁdence interval for ¨ ¥
© CI 5
# ¢
£ and ¨
5
&( ©
( ¥£
¤ '
STA 2023 c B.Presnell & D.Wackerly  Lecture 17 $¤ conﬁdence interval for § . ¥ ¥
© ¦
¨
( ¥£
¤ &
$¤
Note : Higher conﬁdence coefﬁcient (.98) 5
)
& £ ' ¥ CI ( ¥£
¤ ¤¢
and
¢¢ Find a 198 interval!! 5
5
£¤ £2 ©
#¥981) 5(¡
)£ § $¤ – the region of “believable” values for
conﬁdence level. 475.73 484.27 § : ¦ ¥&
) conﬁdence interval for at the . $¤
STA 2023 c B.Presnell & D.Wackerly  Lecture 17 &)
'0£
5 § this claim at the
is believable values for . 5 &)
'0£ § Claim : 475.73
472 because
interval. this claim at the 484.27 $¤ – conﬁdence level in the region of § because &)
'0£ – 484.27 475.73
472 $¤ Claim : 199 conﬁdence level
is in the STA 2023 c B.Presnell & D.Wackerly  Lecture 17 conﬁdence level because ALL believable values for this claim at the
is § – 484.27
487 $¤ 475.73 5 ) ¥£
¤ Claim : § this claim at the – 484.27
487 $¤ § 475.73 5 ) ¥£
¤ Claim : 200 in the interval. conﬁdence level: ) ¥£
¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 17 5 ¤)
60£
§ Claim : 201 475.73 484.27
478 $¤ conﬁdence level: ¤)
65£
5
5
75
2&¢ &43© 6542 5£
£¤£2 ) £
)
§ – At the of ( so are etc.). is STA 2023 c B.Presnell & D.Wackerly  Lecture 17 202 ©
&& ©
¨ Ex.: #7.79 p. 316 Survey of Estimation of a Proportion, postretirees. &)¢ said that they stayed away from home between 4 $
%(
and 7 nights on trips. Find a conﬁdence interval for the true proportion of postretirement travelers who
stay between 4 and 7 nights on a typical trip. Have: a popn. where the proportion with a particular
attribute (“ ”) is .
¢ and get ¨ ¡ Take: a random sample of size with the attribute.
¨
£ ¢ Estimate for : , the sample proportion with ¢ the attribute. is a random variable.
¢
¤ ¥ § is an unbiased estimator for , . (p. 300) ¢ ¢ £ £
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 17 203 ¨ ¤
¥ : ¢ £ ¢ Standard error of
sheet)
¢ ¨ is “large”, is approximately normally
£ If (p. 300,formula distributed. (p. 300)
LOOKS FAMILIAR!!! Conﬁdence
¡ § standard errors table formula sheet ¡
¢ 1
20 ¡ and 3 1
) 3 20 ) 3 1
40 ) formula sheet ¦ estimator £ ¢ ¡ Interval for a Proportion, ¤¢ §§¡
©!¥ ¤ Large Sample . ¡ ¡ ¨
¡ ¢ 2
¨ ¢
$ §
¡
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 17 ©
&& ©
¨ Ex.: #7.79 p. 316 Survey of 204 postretirees. &)¢ said that they stayed away from home between 4 $
%(
and 7 nights on trips. Find a conﬁdence interval for the true proportion of postretirement travelers who
stay between 4 and 7 nights on a typical trip.
¥
& ¥
¨
£ Conﬁdence interval : ¢
or equivalently )£(
5 & © 5 .
Thus, the believable values, at the
¢ conﬁdence level for the true proportion of postretirement travelers who stay between 4 and 7
nights on a typical trip are those between
and . STA 2023 c B.Presnell & D.Wackerly  Lecture 18 205 Thought: When a man is wrapped up in himself, he
makes a pretty small package – (John Ruskin) Assignments :
Today: P. 306 – 308, 322 – 326
For Thursday: Exercises 7.51, 7.53, 7.59, 7.64, 7.67,
8.7, 8.9, 8.10, 8.13
Monday: OPTIONAL review day
Tuesday: EXAM 2
Wednesday: P. 328 – 332 Tuesday, 4/3/01: EXAMS RETURNED
We cannot give out grades over the phone. STA 2023 c B.Presnell & D.Wackerly  Lecture 18 206 Last Time:
Conﬁdence Interval for a $
%( ( ¢ ¦¥ ¤ £¡
¢ PARAMETER
¡ $§ formula sheet 1
40 1
) 3 40 ) 3 1
20 α/2 1−α §
§
§ ¤
(
§ (P. 283), LARGE sample: ¨
' ¢ ¨
$ § – Population Proportion,
§ – Population mean, ¦ ) α/2 53 formula sheet ¦ standard errors table estimator (P. 300), LARGE sample:
¡ ¡ 5 ¢ ¨
$ §
¡
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 207 Choosing the Sample Size
Want: § Estimate the population mean, , by using the sample mean, . © $
with units of the true value of § Estimate to be within conﬁdence. Question : If the population standard deviation is ¨ , how large should “ ”, the sample size be? &¢
.025 .025
.95
5 ¢ §
§
§ of "# # "#
5 ¢
75 ¢ is within ¤ §
That is : of the time STA 2023 c B.Presnell & D.Wackerly  Lecture 18 ¢ 95% of time. © © # © ¨
¨
¨
&
5 ¢
75 ¨
5
)
0£ ¢ © units of 95% of the time. of §
75 Want : is within # Want : to be within § Know: 208 5
£
¦ &¤ ) ¡
5
£&¤ )
©
5
& &&
©
¨
5
& ©&&
¨
¨ ¢
5 ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 in prev. example) § to within “ ” units with
¥ (p. 307) ¤¢ ¨
$ §
¨ Need : at least a “ballpark value for
from past study or pilot study if available.
, and use
or
¡ – guess the range,
¡ ¦ – use ' and solve for conﬁdence level If I want to estimate bound I want ( © “” 209 £
' ¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 § Ex. Want to estimate 210 average cost of freshman year to within $500 with prob. .95. Know – cost between
have to be?
¨ $4800 and $13,000. How large does ¥
& ¥
! §
' (
'¥( &
£ ' ¡ ¨
(
'( &
¨ 5
¦ © ( ¤ ¡
5
© ( ¤ ¨
( '
(
¨
¨
75 ¢
75 ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 211 Tests of Hypotheses: Section 8.1
Ex. : A manufacturer of light bulbs would like to claim
that the average lifetime of our bulbs exceeds 1325
hours. £
¨ Put a random sample of bulbs on test, record time to burnout for each. ©©
&& ¦ ( ©¢
Data : Can the claim be made? (VALIDLY) Choose between two HYPOTHESES. 5 &©¢ (NULL hypothesis) &©¢ (ALTERNATIVE hypothesis)
§
§
§ ¢
¢ ¡
£ We do not know the true value of true mean lifelength of the manufacturer’s bulbs. ( FIXED, but
UNKNOWN )
Decision is to be made using sample data. STA 2023 c B.Presnell & D.Wackerly  Lecture 18 212 Sample DOES NOT contain ALL of the information
about the POPULATION. Using sample data to make decision – always the
possibility of making an ERROR. (See Table 8.1,
p. 325)
Reality
Decision Ho true Ha true Accept Ho Correct Type II error Reject Ho Type I error Correct Consider the lightbulb example Type I Error
– Claim when really Type II Error
– Claim when really STA 2023 c B.Presnell & D.Wackerly  Lecture 18 213 (p. 323) 5
¡ Type I Error
¥ 3 40 )
1 alpha
(p. 325) 5
¡ Type II Error
¢ 3 40 )
1 beta
¥ and are PROBABILITIES
¢ Note: ( numbers between 0 and 1 ) ¨ is a ﬁxed sample size.
¥ Suppose Ideally, want BOTH How about both very small?
Type I Error we
¡ ¡ ¥ ” OFTEN. ¡ – Thus, “accept must decide to “reject . – If we want to REDUCE to be
¢ and ” – Leading to an OFTEN.
in
¡ Type II Error
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 ¨ For ﬁxed 214 . 2
2
¢ move in
¢ ¥ ¥ ¥ ¢ ¡ and directions. Our strategy
¥ Choose small enough to be “CONVINCING” – If we decide that the ALTERNATIVE IS
CORRECT
The ONLY type of error possible is a
¢ ERROR.
The PROBABILITY of making a TYPE I error
¥ – By making .
SMALL we our conﬁdence in our conclusion to ACCECT £ ¢ is . STA 2023 c B.Presnell & D.Wackerly  Lecture 18 215 The hypothesis of MAIN INTEREST is the
ALTERNATIVE HYPOTHESIS, or the Research
Hypothesis.
– What we are “trying to prove” in an objective, fair
manner
£ – Denote by . (p. 322) The “other” hypothesis is called the NULL
HYPOTHESIS
¡ – Denote by . (p. 322) In out setups
The NULL hypothesis is always that a
PARAMETER EQUALS the value used in
specifying the ALTERNATIVE or “RESEARCH”
hypothesis. &©¢ (ALTERNATIVE hypothesis) (1) &©¢ (NULL hypothesis) (2) §
§ ¢
¢ £
¡ STA 2023 c B.Presnell & D.Wackerly  Lecture 18 216 ¡ Type I error
¥ £ when true
¡ ¡ accepting
¥ ¥ ¥ when
¡ saying
SIGNIFICANCE LEVEL of the test, LEVEL of ¥ the test. ¥ for the ’s that we pick. ¢ The test that we will discuss have the SMALLEST ’s ¡ saying what we would like to say when we should not ...
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This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.
 Spring '08
 Ripol
 Statistics

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