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Unformatted text preview: 175 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 176 Recall from last time: If we plan to take a random
from a population with mean £ ¡ standard deviation ¥§
¦¤ (p. 255) ¡ #!£ £ ¤
" ¨
¢
¢ ¥ § ©¢ ¤
¨ Today: P. 265 – 271 . So . (p. 266, 261) of For Tuesday: Exercises 6.15, 6.21, 6.24, 6.28, is called its sampling distribution. is an unbiased estimator (p. 266), so dist. of ¢ concentrated around for larger sample sizes. ¥§ 6.33, 6.37, 6.38, 6.41–46 is more is called the standard error of .(p. 266) ¤ For Wednesday: P. 280 – 285
For Thursday: Exercises 7.1, 7.3–5, 7.10, 7.11, New terms ¨
$£ ¤ Distribution of Assignments : and is a random variable. ¥§ hair! , ¥§ run the country are busy driving taxicabs and cutting ¥§ Thought: Too bad that all the people who know how to ¢ sample of size – Parameter – meaningful number assoc. with a
Population. (p. 254) 7.15–20 – Statistic – meaningful number assoc. with a
Sample. (p. 254) 177 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 178 Know the mean and standard error of ,
HOW ABOUT THE DISTRIBUTION ? % ¤ If the population has a normal distribution, the § sampling distribution of is normal, with mean ¡"
£ ¢ £¢ and standard deviation (standard error)
distribution of x when n = 50. . ¡ ¤ Central Limit Theorem : (p. 267) For large , regardless of the shape of the population dist. § the sampling dist. of is : & approximately normal & standard error 3)
4120¡
(© ¡
¡"
£ ' £
¢ ' ¢
& with mean ¤ Larger sample size, For most populations, . better approximation.
is “large enough”. original population distribution distribution of x when n = 35. ¤ 179 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 180 Ex. : Time spent at the cashier at a movie theater : 1£ ¡
¢ seconds, with is “large” has an approximately distribution.
¢ & Standard Error : ©§
¨ score for 30 : ¤& & ¥§ Sampling Dist. of £ Want Sample Size on ¤ larger n 3 1 ¦§
¥ ¤ Effect of Increasing & ¥§ probability. ¢ are close to tickets for all 36 buyers is less than 30 seconds? £ Values of . ¡ ¤ Smaller standard error : What is the probability that the average time to buy smaller n . £ : ¡"
£ £ ¡ Bigger §( ¤
¢ buyers. seconds. Thirty six ticket µ 181 ¥§ STA 2023 c B.Presnell & D.Wackerly  Lecture 15 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 182 Ex. : (#6.41, p. 275) Study relating IQ and juvenile
delinquency. For ALL juveniles, IQ : 3 ¢
a. What shape for the sampling distribution of ? £ has an approximately distribution. ¡ £ " ' £
&
¢ & Standard Error : Does your answer depend on the shape of the
distribution of IQ scores for juveniles? £ ¤ sec min ¡ min is “large” §( time until all 36 have tickets
¥
¦§ ¤
¥¦§ ¤
§ ¤
¥
¦ min juveniles given IQ test. ¥§ ¡ #
¢ before the movie starts? £ the probability that all 36 ticket buyers get tickets !
" There are 18 minutes until the movie starts. What is
¥ &
¤ ¤ 183 STA 2023 c B.Presnell & D.Wackerly  Lecture 15 STA 2023 c B.Presnell & D.Wackerly  Lecture 16 184 Thought: An honest politician is one who, when bought,
stays bought.—(Simon Cameron) b. ASSUME : Nondelinquent juveniles have same 3 ) § score for Assignments : . 3 ¤ Want ¡ nondelinquents: #
¢ mean IQ with same std. dev. Look at Today: P. 280 – 285
: For Thursday: Exercises 7.1, 7.3–5, 7.10, 7.11,
7.15–20 & ©¨
§ For Monday: P. 299–304, COMPUTER
DEMONSTRATION
For Tuesday: Exercises 7.37, 7.42, 7.44–46 nondelinquents, got Last Time: SEE FORMULA SHEET
,
Estimator
¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 16 the time that an interval estimator actually encloses
a parameter when we compute a large number of
intervals (based on diff. samples). Ex., Sample etc. ¤ , Conﬁdence Level (p. 256) : conﬁdence coefﬁcient 3 ¢ ¤
¤ Point estimator (p. 261) : formula for a SINGLE £¡ expressed as percentage. Ex., ¡ Have : Large Sample from Population with ﬁxed £¡ ¤ Conﬁdence Coefﬁcient (p. 282) : the proportion of Chapter 7 : Estimation based on a Single but Unknown mean, 186 3 &
185 approximately NORMALLY ¡
¡! ¢ ¡! STA 2023 c B.Presnell & D.Wackerly  Lecture 16 ¥§ § 3 ¥§ For “large” sample size,
distributed. Standard Error of Est. ¡" Parameter £ (like for ALL juveniles)? 3 . Do you think true popn. mean IQ is
std. dev. #
¢ 1 ¢ ) § ¤ &
! c. Actually took sample of , etc. Useful new idea! NUMBER that is intended to be “close” to the ¤¥ 3 ¥¤
¡ !¤ ¦ § ©§
¨
¤ ¥§ ¢ ¢
:
¥ is a point estimator for , let ¦
§§ Notation: for any parameter value. be such that A §& Interval estimator (p. 282) : formula that tells us
how to compute TWO NUMBERS that are intended
0 to ENCLOSE the value of the parameter zA BETWEEN THEM
.5.025=.475
.025 0 z.025
§ ¤ 187 STA 2023 c B.Presnell & D.Wackerly  Lecture 16 STA 2023 c B.Presnell & D.Wackerly  Lecture 16 Conﬁdence
.5.05=.45 (alpha) Coefﬁcient ¢ © z ¢ .05 0 188 .99 .05
§ .98
.95
Note: .90 ! ¡!
α/2 § ¥ § ¥ § ©
¤ .025 α/2 1−α .025
.95 and the resulting interval the . “middle” (shaded) region? are in the & ¤ Thus, in repeated sampling, approximately ¤ If we obtain a value of in the “tails” (unshaded) & ¢ the value of § and the resulting interval
Plug that value into the formula ¡¡ ! § region. . . & What fraction of the time will this occur? £ £ £ .025 of the intervals will enclose the true value of % ¢ £ ¡¡ ¢ §
! ¤ What proportion of the values of § ¤ & Plug that value into the formula
value of with . ¥§ £
£ ¤ .95 in the “middle” (shaded) region. ¢ .025 If we obtain a value of 190 £ ¡¡ ! §
!
¡¡ ¢ © ¥§
©£ ¨§ STA 2023 c B.Presnell & D.Wackerly  Lecture 16 §
§ © 3 , use it!! If not, estimate ¡¡ ! If you know £ § § Recall not ! § ¦¤§ ©
¥£ 3 189 A 95% Conﬁdence Interval for ¦£ §
¥ STA 2023 c B.Presnell & D.Wackerly  Lecture 16 § NOTE : ¨ !
£ ¡¡ ¢ ¢
¢ ¨ !
£ ¡¡ ¢ © ¢ ¤ STA 2023 c B.Presnell & D.Wackerly  Lecture 16 ¥ ¨ ¦ ¥
©§¢¤ than steel, but brittle at high temps. Want to estimate (large sample) p. 283 36 specimens randomly selected, measured temp. ¨ ¥
£ ¦£ § ¥ § 3 # ¥ §
and is the z value that cuts off an area of $¢ the mean temp. at which metalic glass becomes brittle. ¤ ¦£ §
¥ 1¡¤ ¢¤
¢
¡ at which each became brittle : Conﬁdence Interval for Ex. : (#7.71, p. 314) Metalic Glass, 4 times stronger 192 191 STA 2023 c B.Presnell & D.Wackerly  Lecture 16 in the upper tail of a standard normal distribution true mean temp. metalic glass gets brittle is unknown, ¦£ § ©
¥
"
#! £¡ 3 ¤ £¡ 3
(
! 3 ¡ ! §
3! 1 3 #
¡ " # ¢ ¥ § £
!
¥§
¢
3 ! 1 # ¡ ¡ ! # ¢
476.98 ¡" ¨
¥ § £
: ¤ ¢
¢ © ¤ £¡ 3 3¤ £¡ – the region of “believable” values for
conﬁdence level. !
# ¢ ( conﬁdence interval for 483.02 £ ¢ ¢ interval!! at the $ Note : Lower conﬁdence coefﬁcient . $ #!
" £¡ CI conﬁdence ¦£ §
¥ ¡ ! 1 3 #
¡ " ¡¡ ¢ ¥ § £ !
¨ !
¥ § '£ ¡¡ ¢ ¥ § . and 193 table formula sheet standard errors "
#! ¤
conﬁdence $ ¡" interval for 3 !
¡¡ ¢ estimator STA 2023 c B.Presnell & D.Wackerly  Lecture 16 Ex. Back in Exercise 7.71, ﬁnd a NOTE : the basic form of this interval is formula sheet ! £ interval. HOW? 1−α ¥©£ § 3 # ¥§ Problem actually asks for a α/2 § CI α/2 £¡ 3 ¤ ...
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 Spring '08
 Ripol
 Statistics, Standard Deviation

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