程序代写案例-LEC0101
时间:2021-11-15
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ



1. 㘳䈅ؑ᚟
ᰦ䰤˖к䈮ᰦ䰤
Singh ᮉᦸ
x LEC0101: 11/15 10:10-11:00am
x LEC5101: 11/15 6:10-7:00pm
Eadie ᮉᦸ
x 11/6 11:10AM-12:00
㘳䈅޵ᇩ
Module 4: Sampling distributions
Module 5: Data collection
Module 8: Statistical test
Module 9: The effective Use of Statistical Tests




























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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Module 4: Sampling distributions

1. Binomial model ༽Ґ
Bernoulli variable: ਚᴹє⿽㔃᷌Ⲵਈ䟿
Binomial˖Bernoulli 䟽༽ n ⅑





2. Normal Model








(1) Standard normal model
ਜ਼ѹ˖ߤ ൌ 0,ߪ ൌ 1
ܺ~ܤ݅݊ሺ݊,݌ሻ
n:s # of trialˈ䈅ཊቁ⅑
p:ᡀ࣏Ⲵᾲ⦷
ܲሺܺ ൌ ݇ሻ ൌ ቀ
݊
݇ቁ ݌
௞ሺ1 െ ݌ሻ௡ି௞
ܧሺܺሻ ൌ ݊݌
ܸܽݎሺܺሻ ൌ ݊݌ሺ1 െ ݌ሻ
Q䈅傼ཊቁ⅑
Nᴹࠐ⅑ᡀ࣏
Sᡀ࣏Ⲵᾲ⦷
N ~ (ߤ,ߪଶ)
ߤ: ݌݋݌ݑ݈ܽݐ݅݋݊ ݉݁ܽ݊
ߪ: ݌݋݌ݑ݈ܽݐ݅݋݊ ݏݐܽ݊݀ܽݎ݀ ݀݁ݒ݅ܽݐ݅݋݊
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(2) Non standard normal
䀓仈↕僔˖
1. ᢮ Ɋ઼ ɐ 2. ܼ ൌ ௑ିఓ

3. 䈫㺘᢮ሩᓄ Z


3. Poisson model
The number of hurricane per year follows a Poisson distribution with a mean of 2. What is the probability that there are less than 2 hurricanes occurring this year? Ans: 0.406

P(Zb)=1-A(b) P(cb)= P(Zb)=A(a)+1- A(b) A(Z)˖㺘ѝ䈫ࠪᶕⲴᮠˈᡆ⭘ excel/R 䇑㇇
ܲሺܺ ൌ ݔሻ ൌ
ߣ௫݁ିఒ
ݔ!
ܧሺܺሻ ൌ ܸܽݎሺܺሻ ൌ ߣ
P (⽕ 2) = P (X : 0) t p N = 1 )
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

(I) Sampling distributions
1. Population vs sample
ᴬ㒆 Population (Census) Sample
⵲♆㑯 ߤ ݔҧ
㒸Ⳮ ߪଶ ݏଶ
㜆⃅Ⳮ ߪ ݏ
㜶㘫㑯 N n

2. Central limit Theorem
ਜ਼ѹ˖ᖃањṧᵜ䏣ཏབྷᰦˈsample mean ઼ variance 䘁լҾ normal
䏣ཏབྷḷ߶˖ а㡜ḷ߶˖n>30
㘱ᐸḷ߶˖݊݌ ൒ 10 ݊ሺ1 െ ݌ሻ ൒ 10


↕僔
1. 傼䇱ᱟ੖ਟԕ⭘ central limit theorem
2. ≲ࠪ E(X)઼ SD
3. ྲ᷌㔉Ⲵᱟᾲ⦷˖⭘ normal
ྲ᷌㔉ࠪ standard deviation: ⭘ t distribution

As n approaches λ
ܧሺ തܺሻ ൌ ߤ
ߪ௫ҧ ൌ
ߪ
ξ݊

݌~ܰሺ݌,݌ሺ1 െ ݌ሻ
݊

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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

ֻ仈 1˖ A repair shop specializes in repairing swibbles. Swibbles are big business (everyone has one!) so under normal circumstances, 200 swibbles are repaired by the shop each day. One out of every six swibbles can be repaired by replacing a broken widget. If the shop has 47 widgets in stock, what is, approximately, the probability that they run out of widgets? Ans˖0.0048

o np = 200 ㄨ t > 1 0 n ( 1 - p) ⼆ zoox 号 > 1 0 - ✓
µ = 亡 ⼦ : (⼀是 = 0 02 63
1
5
= ⿏ 。 235
P ( P > 思) = P (
中寺 > 2,6
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ֻ仈 2 A certain machine produces computer chips. The probability that the machine produces a defective chip is equal to 0.05. If the machine produces 1000 computer chips, then what is the probability that more than 900 will be non-defective? Ans˖1































P ( P < 0 - 1)
⼏ P > lo
h ( 1- 1》 > 1 0
µ = o . 05 5
=酒 ⼆ 命器
P ( P < 0 1) = p l
中寻 < ""等 )
= P ( Z <7 25) ~~ 1
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Module 5: Data collection
⸕䇶⛩ 1˖Observationalstudy
1. Observational study vs experiment
ޡ਼⛩˖⹄ウ x(explanatory variable)઼ y(response variable)Ⲵޣ㌫ Recall: explanatory variable=predictor=covariate=independent variable response variable=dependent variable=outcome Observational study Experiment
ᮠᦞᱟ⧠ᴹᆈ൘Ⲵ Data are measurements of existing characteristics ᮠᦞᱟڊᇎ傼ᗇࠪⲴˈᱟڊ㔃䇪Ⲵᴰྭᯩ⌅The best way we have to make causal conclusions
⹄ウӪઈਚ㿲⍻ᮠᦞˈн᧗ࡦ Investigator/Scientist has no control over what might have caused the data to have certain relationships
⹄ウӪઈ᧗ࡦᮠᦞ Investigator/Scientist has applied a “treatment” to see how it affects the response variable



2. Mechanism Causation: x ઼ y ᱟഐ᷌ޣ㌫ ഐѪ x ᡰԕ y Reverse causation˖x ઼ y ᱟ৽ഐ᷌ޣ㌫ ഐѪ y ᡰԕ x Association˖ᐗਸˈx ઼ y ⋑ޣ㌫ Common cause˖ሬ㠤 x ઼ y Ⲵޡ਼৏ഐ Confounding variable˖ᒢᢠഐ㍐ˈн㜭⺞ᇊ y ⲴਈॆᱟഐѪ x 䘈ᱟഐѪ confounding variable Lurking variable˖՚⴨ޣˈྲ࡫ᶯঠ䊑˖ྣਨᵪнՊᔰ䖖


⸕䇶⛩ 2: Experiment
1.TermExperiment ᇊѹ˖⹄ウӪઈᯭ࣐ᖡ૽Ⲵ䈅傼 Researcher manipulates or applies a treatment to the explanatory variables, then observes the response variable. Experimental unit: 䈅傼অս
2.FactorandlevelFactor: ഐ㍐Ⲵ⿽㊫ˈྲᙗ࡛ˈ㦟Ⲵ㊫࡛ Level: Factor Ⲵ٬ˈྲ⭧ᙗྣᙗˈ㦟⢙ Aˈ㦟⢙ Bˈᆹហࡲ˄Placebo˅
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

3.Extraneousfactor
ѝ᮷˖ཆ⭼ഐ㍐
x ਟ㜭Պᖡ૽ yˈնᱟнᱟ⹄ウሩ䊑 not of interest in the current study, but may affect the response
x Extraneous factor 䴰㾱㻛᧗ࡦˈ⇿㓴Ⲵਈ䟿ᓄ䈕аṧ(Blocking)
4.Blocking
ਜ਼ѹ˖֯ཆ⭼ഐ㍐؍ᤱа㠤
ֻྲ˖⍻䈅 bell ઼ rogers Ⲵ㖁䙏ˈࡉ䴰㾱᧗ࡦ⇿њ㖁㔌лᴹ⴨਼ⲴӪᮠ֯⭘㖁㔌ˈфڊⲴһᛵ⴨

5.Experiment ৏ࡉ
᧗ࡦᡰᴹⲴ extraneous factor
䲿ᵪ࠶䝽ᇎ傼ሩ䊑(No selection bias)
䟽༽䈅傼
㤡᮷䀓䟺˖ Control any extraneous factors Randomly assign experimental units to treatment group Replicate, by applying each treatment to many experimental units
6.CausalStatementsandExperimentalDesign


7. ᾲᘥ(1) Blinding Single blind:ᇎ傼ሩ䊑н⸕ᛵDŽ∄ྲ⌘ሴ⯛㤇ˈн੺䇹༷⌘ሴⲴӪ⌘ሴⲴᱟ⯛㤇䘈ᱟ⭏⨶ⴀ≤
-
>
conusiusf.tl
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Double blind: ᇎ傼ሩ䊑઼ᇎ傼㘵䜭н⸕ᛵˈ⌘ሴ⯛㤇⌘ሴⲴӪ઼५⭏䜭н⸕䚃ᱟ⯛㤇䘈ᱟ⭏
⨶ⴀ≤ (2) Placebo Effect
ѝ᮷˖ᆹហࡲ᭸ᓄ
Ѯֻ˖⌘ሴ⭏⨶ⴀ≤Ⲵ䛓а㓴Ӫҏ㿹ᗇ㠚ᐡᣥᣇ࣋ਈᕪҶ
(3)Controlgroup
ѝ᮷˖᧗ࡦ㓴ˈሩ➗㓴
ਜ਼ѹ˖᧕ਇᆹហࡲⲴ䛓а㓴ˈ⭘ᶕоⵏ↓ᇎ傼ⲴӪ֌ሩ∄

8.MeanvsproportionMean: ᒣ൷ᮠˈᱟ quantitative variable Proportion˖∄ֻᡆ categorical variable
㓳Ґ 1˖





施加影响
-
experimentobservationdsudyi-treatmenti.ua
riay
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㓳Ґ




Blocked-competey.amdoniud
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㓳Ґ 3





















nr-iexperiment.explanatorgaplacebo.iq
vc ,3gVC.3gVCtonsetofawldrespohsedumtionof.twwmencd
Yes
No
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-
experimentiblockchiaseedundplaeeboYes.su/No.
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ipositivehnearrdationsh.ir
随⼼ y T
N , 没控制变量 ,otherpossibleconfundmgvarneshhe.IQadfamìy income
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O-e-rnnnsimplemndomsamphngfstrat.fied䶈村庄
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Module 8 &9
⸕䇶⛩ 1˖Hypothesis
1. ⍱〻
ᨀࠪٷ䇮-傼䇱ٷ䇮-ᗇࠪ㔃䇪(ᤂ㔍䘈ᱟ᧕ਇ)

2. ᨀࠪٷ䇮
Null hypothesis
ㅖਧ˖ܪ଴
⢩⛩˖ᑖㅹਧ
Alternative hypothesis
ㅖਧ˖ HୟᡆHଵ
⢩⛩˖нᑖㅹਧ
Ѯֻ˖
Two sided example:
We would like to test the average year income of a Canadian is whether $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ് 50000

One sided example:
We would like to test the average year income of a Canadian is above $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ൐ 50000

We would like to test the average year income of a Canadian is below $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ൏ 50000

3. P value
ᇊѹ˖ٷ䇮 H0 ↓⺞ˈࠪ⧠䈕һԦⲴᾲ⦷DŽ
㔃䇪 1˖ᾲ⦷䎺վˈ䈤᰾ H0 ↓⺞Ⲵᾲ⦷䎺վˈࡉᤂ㔍 H0 (⌘᜿˖н㜭䈤 H1 ᱟሩⲴʽ)
Ѯֻ˖ྲ᷌䇱᰾⧠൘⭧৻ᱟ⑓⭧ˈнԓ㺘лањ⭧ᴻ৻нᱟ⑓⭧

㔃䇪 2˖ᾲ⦷䎺儈ˈ䈤᰾ H0 ↓⺞Ⲵᾲ⦷䎺儈ˈࡉ᧕ਇ(Fail to reject) H0
⌘᜿˖㘳䈅བྷ仈н㜭߉ accept 㾱߉ fail to reject


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4. Level of significance
ㅖਧ˖ߙ
ਜ਼ѹ:ањѤ⭼٬ˈབྷҾԆࡉ᧕ਇ˄do not reject˅ܪ଴ˈሿҾ 0 ࡉᤂ㔍ሺܴ݆݁݁ܿݐሻܪ଴




5. Estimating a proportion ݌Ƹ doing a hypothesis test
݌Ƹ˖㿲⍻ࠪⲴᾲ⦷
p: assume Ⲵᾲ⦷
step1: ߉ࠪ hypothesis
step2: Ự⍻ྲ᷌┑䏣 central limit theorem:
Recall:֯⭘ central limit theorem ᶑԦ˖np>10 and np(1-p)൒10
step3:
݌Ƹ ൌ
ݕ
݊

ܼ ൌ
݌Ƹ െ ݌
ට݌ሺ1 െ ݌ሻ
݊

step4a: Z ઼ܼ௖௥௜௧௜௖௔௟∄䖳
step4b˖㇇ P value ઼ significant level ∄䖳

⼘。 错误可能性
0
test 5
⼼状
"

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6. Critical value ܼ௖௥௜௧௜௖௔௟ᡆ Z*
⭡ᶕ˖Significance level ᡰሩᓄⲴ٬

઼㇇ࠪᶕⲴ Z ᰐޣʽ
઼㇇ࠪᶕⲴ Z ᰐޣʽ
઼㇇ࠪᶕⲴ Z ᰐޣʽ

ᑨ⭘ Critical value ٬
݂ܵ݅݃݊݅݅ܿܽ݊ܿ݁ ݈݁ݒ݈݁ ሺߙሻ One sided Two sided
0.1 1.28 1.645
0.05 1.645 1.96
0.01 2.33 2.576

7. Student t distribution
䘲⭘ᶑԦ˖仈ⴞѝ᰾⺞ᨀࡠ standard deviation

䀓仈↕僔
㊫լ Z test

step1: ࡇٷ䇮
step2: ܶ ൌ ௑
തିఓ
ೞ ξ೙ ,df=n-1
df: degree of freedom
step4a: T ઼ ௗܶ௙,ఈ∄䖳

⌘᜿˖ ௡ܶିଵ,ఈᱟӾ㺘ѝ䈫ࠪⲴ
*step4b˖㇇ P value

A四 ⼆ 005
E = 1 . 64
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

ֻ仈 1
Georgianna claims that in a small city renowned for its music school, child takes less than 5
years of piano lessons. We have a random sample of 20 children from the average
city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years
(a) Evaluate Georgianna's claim (or that the opposite might be true) using a hypothesis test.

1-1 0 : M : 5
-
↳错的
H.int#5I=4.65=z.zx-MT=-=46誉 ⼆ - 0 -8 1 3 1
df = n - 1 = 1 9
7
1 9 , 005
⼆ 209

ITKTq.o.os-ifailtonef.ge Ho
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8. Error
єԦһ˖1. ᵜ䓛ᱟሩⲴ䘈ᱟ䭉Ⲵ 2. ⍻ࠪᶕ(Hypothesis)ᱟሩⲴ䘈ᱟ䭉Ⲵ
⚛䆖 vs ⵏⲴ⚛⚮
⚛䆖૽Ҷˈਁ⭏Ҷ⚛⚮˖ ሩⲴ
⚛䆖૽Ҷˈ⋑ਁ⭏⚛⚮˖Type I error (False positive)
⚛䆖⋑૽ˈ⋑ਁ⭏⚛⚮: ሩⲴ
⚛䆖⋑૽ˈਁ⭏Ҷ⚛⚮˖Type II error (False negative)
9. Two erros Type I error Reject the null hypothesis when the null hypothesis is true Happens with probability ߙ Type II error Fail to reject the null hypothesis when the null hypothesis is not true Happens with probabilityߚ 10. Power of test
ޜᔿ˖݌݋ݓ݁ݎ ൌ 1 െ ߚ
ਜ਼ѹ˖P (Reject ܪ଴| ܪ଴ is false) ,↓⺞Ⲵᤂ㔍ܪ଴Ⲵᾲ⦷
㤡᮷䀓䟺˖the probability of correctly rejecting the null hypothesis.






Accept˄Fail to Reject˅ܪ଴ Reject ܪ଴
ᇎ䱵ᱟሩⲴ Correct Type I error (ߙሻ ᤂⵏ
ᇎ䱵䈱䭉Ⲵ Type II error(ߚሻ᧕ٷ Correct
-
→ 更严重

-_-
→ 越⼤越好
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ


㓒㢢˖Type I error
㬍㢢˖ Type II error
㔯㢢˖Power of test 11. Increase power Ⲵᖡ૽
x ߙ໎࣐
x decrease variability in sampling distribution
x increase sample size
x “move the true” value away from the null

12. Confidence Interval








Ѯֻ˖ߙ ൌ 0.05,ܼഀ

ൌ 1.96

1. What is the purpose of a confidence interval?
A. None of these
B. It provides a range of the most plausible that a theoretical can take on
ܫ݂ ݌Ƹ ~ܰቌ݌,ඨ݌ሺ1 െ ݌ሻ
݊

Z ⡸ᵜ˖ܥܫ ൌ ሾ݌ െ ܼഀ

ට௣
ሺଵି௣ሻ

,݌ ൅ ܼഀ

ට௣
ሺଵି௣ሻ

]
T distribution ⡸ᵜ˖ܥܫ ൌ ሾ തܺ െ ݐഀ

,௡ିଵ ௦ඥሺ௡ሻ , തܺ ൅ ݐഀమ,௡ିଵ ௦ඥሺ௡ሻ]

B 1 - B
'
o
es 随 poweroftest 增加
☆ 好的影响
减少不确定性
增加样本数量
将 t.me wue移到 轗

⼆ critìcalvaw
( 1
0
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

C. It provides the complete range of values that a statistic (such as the
sample mean) can take on
D. It provides a range of values that repeated measurements of a statistic
(such as a sample mean) have taken on

2. We have a biased coin which has a probability p of landing heads when
flipped. Suppose we perform the following experiment 100 times: In each
experiment, we flip the coin 10 times and find the 95% confidence interval
for p based on the proportion of heads in the 10 flips. Which of the following is TRUE?
A. None of these
B. 95 of the intervals will contain p
C. The closer p is to 0.5, the more intervals will contain p
D. We expect that approximately 95 of the intervals will contain p

3. Suppose that after sampling a population, we estimate some proportion p
and compute a 95% confidence interval to be [0.05, 0.45]. What interpretation is correct?
A. With probability 0.95, the true proportion p lies between 0.05 and 0.45
B. With probability 0.95, the probability that our estimate is equal to the true
proportion p is between 0.05 and 0.45
C. We have sampled the same proportion repeatedly and estimate p each
time, and 95% of the time is p^ between 0.05 and 0.45
D. If we sampled the same population repeatedly and estimated p each time, 95% of the time p
should lie within the computed confidence interval


4. In an experiment with two outcomes, let p be the unknown true probability
of getting a successful outcome on any one trial. If we repeat the trial n
times and let p^ = total number of successes / total number of trials, which of p and p^ is fixed
and which is constant?
A. p^ is random and p is fixed
B. p is random and p^ is fixed
C. Both p and p^ are fixed
D. Both p and p^ are random

5. What is the purpose of substituting in p^ for p in the equations for the
endpoints of a confidence interval for p?
A. p^ is a more conservative choice than p
B. p is a more conservative choice than p^
C. p^ is always unknown, but we can compute p following an experiment
D. p is always unknown, but we can compute p^ following an experiment if
we anticipate p^ Ĭ p

ADDAD


⽘ 错误说法
:⼀
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 1


⌘᜿˖ྲ᷌㘱ᐸ㔉Ҷ significance levelˈቡ⭘ significance level ᶕࡔᇊ reject 䘈ᱟ do not
rejectˈ⋑㔉Ⲵ䈍ቡ྇⭘㬍㢢Ⲵ㺘


















(a) Ho :P ⼆ 0 .2
H i i p I O
.
2
P = ⻮ ~~ o . 2 9 6 9
5 = 1- = 0䉁 = 00 5
Z = j = 1 9 4 ※本!作'
p (区 1 7 1 94 ) = 2 P ( Z 7 1 9 4)
= o.0 524
weakevideneeagaist.no
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㓳Ґ 1


⌘᜿˖ྲ᷌㘱ᐸ㔉Ҷ significance levelˈቡ⭘ significance level ᶕࡔᇊ reject 䘈ᱟ do not
rejectˈ⋑㔉Ⲵ䈍ቡ྇⭘㬍㢢Ⲵ㺘


















Ho
ˇ
(b) H 。 : p = 0 . 2
H . :P > 0 .
2
:⼀点
4
" ㄨ
l
P ( Z > 1 9 4) = o.oz.bz
⇒ moderateevidenceagainst.it

(C) 与 前⾯ ⼀样 Ho :P ⼆ 0 2 H , :P
ao.z.PL
Z < 1 9 4) = 097
⇒ P 701 , 所以noeuidenceagainst.lt。
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㓳Ґ 2

Given significance level is 0.05.


@ p = 0
0 5 4 P > 2
所mfh.to rejeu.tt 。
(b)
p = o.0262 P
⼈ 2
所以 t.lt H 。
4) P ⼆ 097 > ⼈
所攻 faivtoreject.no
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㓳Ґ 3





-
nri
(a) 5010
(b) bio ( 7 7 ,
o.05)
4) E ( x) = 77 x 0 -
0 5 = 37 5
3 . 7 5 7 2
No
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ


㓳Ґ 4





-
-
-
Ō
torejeuaiypeIerroringeutHowhileltistrueplrejectHoltt.is
true)
= P ( ⽕ 1 , ㄨ= 2 , ㄨ= 3 , ㄨ = 5 )
⼆ 012 to.lt 0.lt 0 - 1 = 0 . 5
(b) TypeElifil.tonejectttowhileltoisfnlsepliailtorejatH-IH.istlse)
= P (ㄨ = 4 , ㄨ = 6 )
= 0 3
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Test 2 㓳Ґ

1. If an experimental result is said to be "statistically significant", it means
that we conclude that the observed result contradicts the established
model and/or theory due to some systematic cause. TRUE

2. If the null hypothesis is rejected in favor of the alternative hypothesis, the
result is said to be statistically significant. TRUE

3. A p-value tells you how unlikely the observed value of the test statistic
(and more extreme values) is if the null hypothesis were true. TRUE

4. A p-value can be interpreted as how likely it is that the null hypothesis is
true. FALSE

5. The second step of statistical testing is to collect the data and calculate
the test statistic. TRUE

6. The test statistic in statistical testing is formulated under the assumption
that the alternative hypothesis is correct. FALSE

7. In statistical tests, the legal idea of innocent until proven guilty is
encapsulated by the alternative hypothesis. FALSE

8. The alternative hypothesis is usually what an experiment is trying to
establish. TRUE

9. TRUE or FALSE: Suppose we suspect that taking a certain medication
daily has an effect on reducing heart attacks. An appropriate null
hypothesis in this situation is "Taking the medication daily has no effect
on the incidence of heart attacks". TRUE

10. Suppose we have a coin that we suspect has been weighted so that it is
more likely to come up heads than tails and we will collect some data and
carry out a statistical test to see if this is the case. Let p denote the true
probability of heads for this coin. The appropriate null and alternative
hypotheses are
A. H0: p = 0.5 vs Ha: p Į 0.5
B. H0: p > 0.5 vs Ha: p = 0.5
C. H0: p = 0.5 vs Ha: p > 0.5
D. H0: p = 0.5 vs Ha: p < 0.5

11. Suppose we flip a coin many times and suspect that the coin is biased in
some way. We perform a statistical test to evaluate this hypothesis. If the
true probability of the coin landing heads is p, what is a reasonable
alternative hypothesis?

statisticalysignific.at
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nrnrz

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-
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

A. p = 0.5
B. p < 0.5 or p > 0.5
C. p < 0.5 and p > 0.5
D. None are correct

12. The mean result from some experiment with 50 observations was 1.5 and
the standard deviation was 1. Let ȝ denote the true mean result. We wish
to to test H0: ȝ=1 vs Ha: ȝ>1. The p-value is
A. P(t50 ı 3.5)
B. P(t49 İ 3.5)
C. P(t49 ı 3.5)
D. P(t50 İ 3.5)


13. We wish to test H0: p = 0.5 vs Ha: p > 0.5 where p denotes the true
probability of heads for a particular coin. We flip the coin 1000 times and
observe 512 heads. Without doing any calculations, the p-value and test
conclusion are most likely
A. Close to 0 so we reject the null hypothesis.
B. Close to 1 so we reject the null hypothesis.
C. Close to 1 so we fail to reject the null hypothesis.
D. Close to 0 so we fail to reject the null hypothesis.

14. We wish to test H0: p = 0.5 vs Ha: p Į 0.5 where p denotes the true
probability of heads for a particular coin. We flip the coin 1000 times and
observe 510 heads. Without doing any calculations, the p-value and test
conclusion are most likely
A. Close to 0 so we reject the null hypothesis.
B. Close to 1 so we reject the null hypothesis.
C. Close to 1 so we fail to reject the null hypothesis.
D. Close to 0 so we fail to reject the null hypothesis.

15. The mean result of some observations was 50. Let ȝ denote the true mean
result. We wish to test the hypothesis H0: ȝ = 40 vs Ha: ȝ < 40. Without
doing any calculations, the p-value and test conclusion are most likely
A. Close to 0 so fail to reject the null hypothesis
B. Close to 1 so fail to reject the null hypothesis
C. Close to 0 so reject the null hypothesis
D. Close to 1 so reject the null hypothesis


16. Suppose we flip a coin many times and observe a suspiciously large
number of heads. How does a statistical test help us assess whether or not the coin is biased?
A. We decide based on how likely the coin is to be unbiased given our
observations
B. We decide based on how unlikely our observations would be if the coin
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

were unbiased
C. We decide based on how likely our observations would be if the coin were
biased
D. We decide based on how unlikely the coin is to be unbiased given our
observations

17. Why can't we calculate the test statistic as the first step of a statistical
test?
A. Because it is calculated from the p-value, which we haven't calculated yet
B. Because it appears in the statement of the null hypothesis, which we
haven't formulated yet
C. Because it is calculated from the sample data, which we haven't obtained
yet
D. Because it can only be calculated if the null hypothesis is true

18. Suppose we suspect that the average means, ȝ1 and ȝ2, of two
populations are different. What is an appropriate alternative hypothesis for
this situation?
A. HA: ­1 ı ­2
B. HA: ȝ1 = ȝ2
C. HA: ­1Į ­2
D. None are correct


19. Suppose we calculate sample means from two populations. While the
sample means are not identical, the difference between them seems to be
small. How can a statistical test help here?
A. It can determine whether the population means are different
B. It can determine whether the difference is statistically reasonable if the
population means are different
C. It can determine whether the difference is statistically reasonable if the
population means are identical
D. It can determine whether the population means are identical

20. What is the purpose of a test statistic?
A. All are correct
B. It serves as a numerical summary of the data under the null hypothesis
C. It allows us to assess how extreme our observed data is if our suspicions
are false
D. It allows us to calculate the p-value

21. Which of the following is TRUE about the p-value?
A. It indicates the likelihood of obtaining data at least as extreme as the
observed data if the null hypothesis were true
B. A large p-value indicates that the null hypothesis is false
C. A large p-value indicates that the null hypothesis is true
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

D. It indicates the likelihood of obtaining data at least as extreme as the
observed data if the null hypothesis were false


22. Which of the following hypotheses would be suitable for testing using a
statistical test?
A. The average age of undergraduate students is 20
B. 4 is the only solution to the function x^2í16=0
C. Every undergraduate student is at least 16 years old
D. The average employee is healthy

23. Suppose we suspect that University A tends to graduate a significantly
different proportion of students than University B does. If we assessed
this using a statistical test, how would the test statistic be formulated?
A. Under the assumption that the two proportions are equal
B. Under the assumption that the two proportions are different
C. Under the assumption that both universities graduate the same number of
students
D. Under the assumption that each universities graduate a different number
of students

24. You want to investigate the relationship between exercise and weight loss
among university students, but you do not know whether weight gain due
to increased muscle mass might be offset by weight loss due to
decreased fat stores. You should FIRST...
A. Design your study to distinguish between the two influences on weight
B. Define precisely your null hypothesis and alternative hypothesis
C. Use a two-tailed test to evaluate the significance of your results
D. Use a one-tailed test to evaluate the significance of your results


25. Consider the following scenario: we have a coin which we suspect to be
biased. We get to flip the coin as many times as we want to perform a
statistical test. If we incorrectly conclude that the coin is biased at the end
of the experiment, we must pay $100; otherwise, no transfer of money
takes place. Which of the following is TRUE about Type I and Type II
errors?
A. We should aim to reduce Type II error, because the consequences of a
Type I error are not particularly costly
B. We should aim to reduce Type I and Type II error equally, because the
consequences of either of them are particularly costly
C. We should aim to reduce Type I error, because the consequences of a
Type II error are not particularly costly
D. We should aim to reduce Type I error, because the consequences of a
Type I error are particularly costly
1-5 TTTFT 6-10 FFTTC 11-15 BCCCB 16-20 BCCCA 21-25 AAAAD















































































































b)=1-A(b) P(cb)= P(Zb)=A(a)+1- A(b) A(Z)˖㺘ѝ䈫ࠪᶕⲴᮠˈᡆ⭘ excel/R 䇑㇇
ܲሺܺ ൌ ݔሻ ൌ
ߣ௫݁ିఒ
ݔ!
ܧሺܺሻ ൌ ܸܽݎሺܺሻ ൌ ߣ
P (⽕ 2) = P (X : 0) t p N = 1 )
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

(I) Sampling distributions
1. Population vs sample
ᴬ㒆 Population (Census) Sample
⵲♆㑯 ߤ ݔҧ
㒸Ⳮ ߪଶ ݏଶ
㜆⃅Ⳮ ߪ ݏ
㜶㘫㑯 N n

2. Central limit Theorem
ਜ਼ѹ˖ᖃањṧᵜ䏣ཏབྷᰦˈsample mean ઼ variance 䘁լҾ normal
䏣ཏབྷḷ߶˖ а㡜ḷ߶˖n>30
㘱ᐸḷ߶˖݊݌ ൒ 10 ݊ሺ1 െ ݌ሻ ൒ 10


↕僔
1. 傼䇱ᱟ੖ਟԕ⭘ central limit theorem
2. ≲ࠪ E(X)઼ SD
3. ྲ᷌㔉Ⲵᱟᾲ⦷˖⭘ normal
ྲ᷌㔉ࠪ standard deviation: ⭘ t distribution

As n approaches λ
ܧሺ തܺሻ ൌ ߤ
ߪ௫ҧ ൌ
ߪ
ξ݊

݌~ܰሺ݌,݌ሺ1 െ ݌ሻ
݊

=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

ֻ仈 1˖ A repair shop specializes in repairing swibbles. Swibbles are big business (everyone has one!) so under normal circumstances, 200 swibbles are repaired by the shop each day. One out of every six swibbles can be repaired by replacing a broken widget. If the shop has 47 widgets in stock, what is, approximately, the probability that they run out of widgets? Ans˖0.0048

o np = 200 ㄨ t > 1 0 n ( 1 - p) ⼆ zoox 号 > 1 0 - ✓
µ = 亡 ⼦ : (⼀是 = 0 02 63
1
5
= ⿏ 。 235
P ( P > 思) = P (
中寺 > 2,6
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

ֻ仈 2 A certain machine produces computer chips. The probability that the machine produces a defective chip is equal to 0.05. If the machine produces 1000 computer chips, then what is the probability that more than 900 will be non-defective? Ans˖1































P ( P < 0 - 1)
⼏ P > lo
h ( 1- 1》 > 1 0
µ = o . 05 5
=酒 ⼆ 命器
P ( P < 0 1) = p l
中寻 < ""等 )
= P ( Z <7 25) ~~ 1
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ


Module 5: Data collection
⸕䇶⛩ 1˖Observationalstudy
1. Observational study vs experiment
ޡ਼⛩˖⹄ウ x(explanatory variable)઼ y(response variable)Ⲵޣ㌫ Recall: explanatory variable=predictor=covariate=independent variable response variable=dependent variable=outcome Observational study Experiment
ᮠᦞᱟ⧠ᴹᆈ൘Ⲵ Data are measurements of existing characteristics ᮠᦞᱟڊᇎ傼ᗇࠪⲴˈᱟڊ㔃䇪Ⲵᴰྭᯩ⌅The best way we have to make causal conclusions
⹄ウӪઈਚ㿲⍻ᮠᦞˈн᧗ࡦ Investigator/Scientist has no control over what might have caused the data to have certain relationships
⹄ウӪઈ᧗ࡦᮠᦞ Investigator/Scientist has applied a “treatment” to see how it affects the response variable



2. Mechanism Causation: x ઼ y ᱟഐ᷌ޣ㌫ ഐѪ x ᡰԕ y Reverse causation˖x ઼ y ᱟ৽ഐ᷌ޣ㌫ ഐѪ y ᡰԕ x Association˖ᐗਸˈx ઼ y ⋑ޣ㌫ Common cause˖ሬ㠤 x ઼ y Ⲵޡ਼৏ഐ Confounding variable˖ᒢᢠഐ㍐ˈн㜭⺞ᇊ y ⲴਈॆᱟഐѪ x 䘈ᱟഐѪ confounding variable Lurking variable˖՚⴨ޣˈྲ࡫ᶯঠ䊑˖ྣਨᵪнՊᔰ䖖


⸕䇶⛩ 2: Experiment
1.TermExperiment ᇊѹ˖⹄ウӪઈᯭ࣐ᖡ૽Ⲵ䈅傼 Researcher manipulates or applies a treatment to the explanatory variables, then observes the response variable. Experimental unit: 䈅傼অս
2.FactorandlevelFactor: ഐ㍐Ⲵ⿽㊫ˈྲᙗ࡛ˈ㦟Ⲵ㊫࡛ Level: Factor Ⲵ٬ˈྲ⭧ᙗྣᙗˈ㦟⢙ Aˈ㦟⢙ Bˈᆹហࡲ˄Placebo˅
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

3.Extraneousfactor
ѝ᮷˖ཆ⭼ഐ㍐
x ਟ㜭Պᖡ૽ yˈնᱟнᱟ⹄ウሩ䊑 not of interest in the current study, but may affect the response
x Extraneous factor 䴰㾱㻛᧗ࡦˈ⇿㓴Ⲵਈ䟿ᓄ䈕аṧ(Blocking)
4.Blocking
ਜ਼ѹ˖֯ཆ⭼ഐ㍐؍ᤱа㠤
ֻྲ˖⍻䈅 bell ઼ rogers Ⲵ㖁䙏ˈࡉ䴰㾱᧗ࡦ⇿њ㖁㔌лᴹ⴨਼ⲴӪᮠ֯⭘㖁㔌ˈфڊⲴһᛵ⴨

5.Experiment ৏ࡉ
᧗ࡦᡰᴹⲴ extraneous factor
䲿ᵪ࠶䝽ᇎ傼ሩ䊑(No selection bias)
䟽༽䈅傼
㤡᮷䀓䟺˖ Control any extraneous factors Randomly assign experimental units to treatment group Replicate, by applying each treatment to many experimental units
6.CausalStatementsandExperimentalDesign


7. ᾲᘥ(1) Blinding Single blind:ᇎ傼ሩ䊑н⸕ᛵDŽ∄ྲ⌘ሴ⯛㤇ˈн੺䇹༷⌘ሴⲴӪ⌘ሴⲴᱟ⯛㤇䘈ᱟ⭏⨶ⴀ≤
-
>
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Double blind: ᇎ傼ሩ䊑઼ᇎ傼㘵䜭н⸕ᛵˈ⌘ሴ⯛㤇⌘ሴⲴӪ઼५⭏䜭н⸕䚃ᱟ⯛㤇䘈ᱟ⭏
⨶ⴀ≤ (2) Placebo Effect
ѝ᮷˖ᆹហࡲ᭸ᓄ
Ѯֻ˖⌘ሴ⭏⨶ⴀ≤Ⲵ䛓а㓴Ӫҏ㿹ᗇ㠚ᐡᣥᣇ࣋ਈᕪҶ
(3)Controlgroup
ѝ᮷˖᧗ࡦ㓴ˈሩ➗㓴
ਜ਼ѹ˖᧕ਇᆹហࡲⲴ䛓а㓴ˈ⭘ᶕоⵏ↓ᇎ傼ⲴӪ֌ሩ∄

8.MeanvsproportionMean: ᒣ൷ᮠˈᱟ quantitative variable Proportion˖∄ֻᡆ categorical variable
㓳Ґ 1˖





施加影响
-
experimentobservationdsudyi-treatmenti.ua
riay
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ




Blocked-competey.amdoniud
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 3





















nr-iexperiment.explanatorgaplacebo.iq
vc ,3gVC.3gVCtonsetofawldrespohsedumtionof.twwmencd
Yes
No
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ
































-
experimentiblockchiaseedundplaeeboYes.su/No.
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ





























ipositivehnearrdationsh.ir
随⼼ y T
N , 没控制变量 ,otherpossibleconfundmgvarneshhe.IQadfamìy income
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ





O-e-rnnnsimplemndomsamphngfstrat.fied䶈村庄
=++19:'bb


ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Module 8 &9
⸕䇶⛩ 1˖Hypothesis
1. ⍱〻
ᨀࠪٷ䇮-傼䇱ٷ䇮-ᗇࠪ㔃䇪(ᤂ㔍䘈ᱟ᧕ਇ)

2. ᨀࠪٷ䇮
Null hypothesis
ㅖਧ˖ܪ଴
⢩⛩˖ᑖㅹਧ
Alternative hypothesis
ㅖਧ˖ HୟᡆHଵ
⢩⛩˖нᑖㅹਧ
Ѯֻ˖
Two sided example:
We would like to test the average year income of a Canadian is whether $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ് 50000

One sided example:
We would like to test the average year income of a Canadian is above $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ൐ 50000

We would like to test the average year income of a Canadian is below $50,000 or not. Write the null
hypothesis and alternative hypothesis.
ܪ଴: ܫ ൌ 50000
ܪ௔: ܫ ൏ 50000

3. P value
ᇊѹ˖ٷ䇮 H0 ↓⺞ˈࠪ⧠䈕һԦⲴᾲ⦷DŽ
㔃䇪 1˖ᾲ⦷䎺վˈ䈤᰾ H0 ↓⺞Ⲵᾲ⦷䎺վˈࡉᤂ㔍 H0 (⌘᜿˖н㜭䈤 H1 ᱟሩⲴʽ)
Ѯֻ˖ྲ᷌䇱᰾⧠൘⭧৻ᱟ⑓⭧ˈнԓ㺘лањ⭧ᴻ৻нᱟ⑓⭧

㔃䇪 2˖ᾲ⦷䎺儈ˈ䈤᰾ H0 ↓⺞Ⲵᾲ⦷䎺儈ˈࡉ᧕ਇ(Fail to reject) H0
⌘᜿˖㘳䈅བྷ仈н㜭߉ accept 㾱߉ fail to reject


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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ







4. Level of significance
ㅖਧ˖ߙ
ਜ਼ѹ:ањѤ⭼٬ˈབྷҾԆࡉ᧕ਇ˄do not reject˅ܪ଴ˈሿҾ 0 ࡉᤂ㔍ሺܴ݆݁݁ܿݐሻܪ଴




5. Estimating a proportion ݌Ƹ doing a hypothesis test
݌Ƹ˖㿲⍻ࠪⲴᾲ⦷
p: assume Ⲵᾲ⦷
step1: ߉ࠪ hypothesis
step2: Ự⍻ྲ᷌┑䏣 central limit theorem:
Recall:֯⭘ central limit theorem ᶑԦ˖np>10 and np(1-p)൒10
step3:
݌Ƹ ൌ
ݕ
݊

ܼ ൌ
݌Ƹ െ ݌
ට݌ሺ1 െ ݌ሻ
݊

step4a: Z ઼ܼ௖௥௜௧௜௖௔௟∄䖳
step4b˖㇇ P value ઼ significant level ∄䖳

⼘。 错误可能性
0
test 5
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ



6. Critical value ܼ௖௥௜௧௜௖௔௟ᡆ Z*
⭡ᶕ˖Significance level ᡰሩᓄⲴ٬

઼㇇ࠪᶕⲴ Z ᰐޣʽ
઼㇇ࠪᶕⲴ Z ᰐޣʽ
઼㇇ࠪᶕⲴ Z ᰐޣʽ

ᑨ⭘ Critical value ٬
݂ܵ݅݃݊݅݅ܿܽ݊ܿ݁ ݈݁ݒ݈݁ ሺߙሻ One sided Two sided
0.1 1.28 1.645
0.05 1.645 1.96
0.01 2.33 2.576

7. Student t distribution
䘲⭘ᶑԦ˖仈ⴞѝ᰾⺞ᨀࡠ standard deviation

䀓仈↕僔
㊫լ Z test

step1: ࡇٷ䇮
step2: ܶ ൌ ௑
തିఓ
ೞ ξ೙ ,df=n-1
df: degree of freedom
step4a: T ઼ ௗܶ௙,ఈ∄䖳

⌘᜿˖ ௡ܶିଵ,ఈᱟӾ㺘ѝ䈫ࠪⲴ
*step4b˖㇇ P value

A四 ⼆ 005
E = 1 . 64
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

ֻ仈 1
Georgianna claims that in a small city renowned for its music school, child takes less than 5
years of piano lessons. We have a random sample of 20 children from the average
city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years
(a) Evaluate Georgianna's claim (or that the opposite might be true) using a hypothesis test.

1-1 0 : M : 5
-
↳错的
H.int#5I=4.65=z.zx-MT=-=46誉 ⼆ - 0 -8 1 3 1
df = n - 1 = 1 9
7
1 9 , 005
⼆ 209

ITKTq.o.os-ifailtonef.ge Ho
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

8. Error
єԦһ˖1. ᵜ䓛ᱟሩⲴ䘈ᱟ䭉Ⲵ 2. ⍻ࠪᶕ(Hypothesis)ᱟሩⲴ䘈ᱟ䭉Ⲵ
⚛䆖 vs ⵏⲴ⚛⚮
⚛䆖૽Ҷˈਁ⭏Ҷ⚛⚮˖ ሩⲴ
⚛䆖૽Ҷˈ⋑ਁ⭏⚛⚮˖Type I error (False positive)
⚛䆖⋑૽ˈ⋑ਁ⭏⚛⚮: ሩⲴ
⚛䆖⋑૽ˈਁ⭏Ҷ⚛⚮˖Type II error (False negative)
9. Two erros Type I error Reject the null hypothesis when the null hypothesis is true Happens with probability ߙ Type II error Fail to reject the null hypothesis when the null hypothesis is not true Happens with probabilityߚ 10. Power of test
ޜᔿ˖݌݋ݓ݁ݎ ൌ 1 െ ߚ
ਜ਼ѹ˖P (Reject ܪ଴| ܪ଴ is false) ,↓⺞Ⲵᤂ㔍ܪ଴Ⲵᾲ⦷
㤡᮷䀓䟺˖the probability of correctly rejecting the null hypothesis.






Accept˄Fail to Reject˅ܪ଴ Reject ܪ଴
ᇎ䱵ᱟሩⲴ Correct Type I error (ߙሻ ᤂⵏ
ᇎ䱵䈱䭉Ⲵ Type II error(ߚሻ᧕ٷ Correct
-
→ 更严重

-_-
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ


㓒㢢˖Type I error
㬍㢢˖ Type II error
㔯㢢˖Power of test 11. Increase power Ⲵᖡ૽
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x decrease variability in sampling distribution
x increase sample size
x “move the true” value away from the null

12. Confidence Interval








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1. What is the purpose of a confidence interval?
A. None of these
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

C. It provides the complete range of values that a statistic (such as the
sample mean) can take on
D. It provides a range of values that repeated measurements of a statistic
(such as a sample mean) have taken on

2. We have a biased coin which has a probability p of landing heads when
flipped. Suppose we perform the following experiment 100 times: In each
experiment, we flip the coin 10 times and find the 95% confidence interval
for p based on the proportion of heads in the 10 flips. Which of the following is TRUE?
A. None of these
B. 95 of the intervals will contain p
C. The closer p is to 0.5, the more intervals will contain p
D. We expect that approximately 95 of the intervals will contain p

3. Suppose that after sampling a population, we estimate some proportion p
and compute a 95% confidence interval to be [0.05, 0.45]. What interpretation is correct?
A. With probability 0.95, the true proportion p lies between 0.05 and 0.45
B. With probability 0.95, the probability that our estimate is equal to the true
proportion p is between 0.05 and 0.45
C. We have sampled the same proportion repeatedly and estimate p each
time, and 95% of the time is p^ between 0.05 and 0.45
D. If we sampled the same population repeatedly and estimated p each time, 95% of the time p
should lie within the computed confidence interval


4. In an experiment with two outcomes, let p be the unknown true probability
of getting a successful outcome on any one trial. If we repeat the trial n
times and let p^ = total number of successes / total number of trials, which of p and p^ is fixed
and which is constant?
A. p^ is random and p is fixed
B. p is random and p^ is fixed
C. Both p and p^ are fixed
D. Both p and p^ are random

5. What is the purpose of substituting in p^ for p in the equations for the
endpoints of a confidence interval for p?
A. p^ is a more conservative choice than p
B. p is a more conservative choice than p^
C. p^ is always unknown, but we can compute p following an experiment
D. p is always unknown, but we can compute p^ following an experiment if
we anticipate p^ Ĭ p

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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 1


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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 1


⌘᜿˖ྲ᷌㘱ᐸ㔉Ҷ significance levelˈቡ⭘ significance level ᶕࡔᇊ reject 䘈ᱟ do not
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 2

Given significance level is 0.05.


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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

㓳Ґ 3





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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ


㓳Ґ 4





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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

Test 2 㓳Ґ

1. If an experimental result is said to be "statistically significant", it means
that we conclude that the observed result contradicts the established
model and/or theory due to some systematic cause. TRUE

2. If the null hypothesis is rejected in favor of the alternative hypothesis, the
result is said to be statistically significant. TRUE

3. A p-value tells you how unlikely the observed value of the test statistic
(and more extreme values) is if the null hypothesis were true. TRUE

4. A p-value can be interpreted as how likely it is that the null hypothesis is
true. FALSE

5. The second step of statistical testing is to collect the data and calculate
the test statistic. TRUE

6. The test statistic in statistical testing is formulated under the assumption
that the alternative hypothesis is correct. FALSE

7. In statistical tests, the legal idea of innocent until proven guilty is
encapsulated by the alternative hypothesis. FALSE

8. The alternative hypothesis is usually what an experiment is trying to
establish. TRUE

9. TRUE or FALSE: Suppose we suspect that taking a certain medication
daily has an effect on reducing heart attacks. An appropriate null
hypothesis in this situation is "Taking the medication daily has no effect
on the incidence of heart attacks". TRUE

10. Suppose we have a coin that we suspect has been weighted so that it is
more likely to come up heads than tails and we will collect some data and
carry out a statistical test to see if this is the case. Let p denote the true
probability of heads for this coin. The appropriate null and alternative
hypotheses are
A. H0: p = 0.5 vs Ha: p Į 0.5
B. H0: p > 0.5 vs Ha: p = 0.5
C. H0: p = 0.5 vs Ha: p > 0.5
D. H0: p = 0.5 vs Ha: p < 0.5

11. Suppose we flip a coin many times and suspect that the coin is biased in
some way. We perform a statistical test to evaluate this hypothesis. If the
true probability of the coin landing heads is p, what is a reasonable
alternative hypothesis?

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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

A. p = 0.5
B. p < 0.5 or p > 0.5
C. p < 0.5 and p > 0.5
D. None are correct

12. The mean result from some experiment with 50 observations was 1.5 and
the standard deviation was 1. Let ȝ denote the true mean result. We wish
to to test H0: ȝ=1 vs Ha: ȝ>1. The p-value is
A. P(t50 ı 3.5)
B. P(t49 İ 3.5)
C. P(t49 ı 3.5)
D. P(t50 İ 3.5)


13. We wish to test H0: p = 0.5 vs Ha: p > 0.5 where p denotes the true
probability of heads for a particular coin. We flip the coin 1000 times and
observe 512 heads. Without doing any calculations, the p-value and test
conclusion are most likely
A. Close to 0 so we reject the null hypothesis.
B. Close to 1 so we reject the null hypothesis.
C. Close to 1 so we fail to reject the null hypothesis.
D. Close to 0 so we fail to reject the null hypothesis.

14. We wish to test H0: p = 0.5 vs Ha: p Į 0.5 where p denotes the true
probability of heads for a particular coin. We flip the coin 1000 times and
observe 510 heads. Without doing any calculations, the p-value and test
conclusion are most likely
A. Close to 0 so we reject the null hypothesis.
B. Close to 1 so we reject the null hypothesis.
C. Close to 1 so we fail to reject the null hypothesis.
D. Close to 0 so we fail to reject the null hypothesis.

15. The mean result of some observations was 50. Let ȝ denote the true mean
result. We wish to test the hypothesis H0: ȝ = 40 vs Ha: ȝ < 40. Without
doing any calculations, the p-value and test conclusion are most likely
A. Close to 0 so fail to reject the null hypothesis
B. Close to 1 so fail to reject the null hypothesis
C. Close to 0 so reject the null hypothesis
D. Close to 1 so reject the null hypothesis


16. Suppose we flip a coin many times and observe a suspiciously large
number of heads. How does a statistical test help us assess whether or not the coin is biased?
A. We decide based on how likely the coin is to be unbiased given our
observations
B. We decide based on how unlikely our observations would be if the coin
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

were unbiased
C. We decide based on how likely our observations would be if the coin were
biased
D. We decide based on how unlikely the coin is to be unbiased given our
observations

17. Why can't we calculate the test statistic as the first step of a statistical
test?
A. Because it is calculated from the p-value, which we haven't calculated yet
B. Because it appears in the statement of the null hypothesis, which we
haven't formulated yet
C. Because it is calculated from the sample data, which we haven't obtained
yet
D. Because it can only be calculated if the null hypothesis is true

18. Suppose we suspect that the average means, ȝ1 and ȝ2, of two
populations are different. What is an appropriate alternative hypothesis for
this situation?
A. HA: ­1 ı ­2
B. HA: ȝ1 = ȝ2
C. HA: ­1Į ­2
D. None are correct


19. Suppose we calculate sample means from two populations. While the
sample means are not identical, the difference between them seems to be
small. How can a statistical test help here?
A. It can determine whether the population means are different
B. It can determine whether the difference is statistically reasonable if the
population means are different
C. It can determine whether the difference is statistically reasonable if the
population means are identical
D. It can determine whether the population means are identical

20. What is the purpose of a test statistic?
A. All are correct
B. It serves as a numerical summary of the data under the null hypothesis
C. It allows us to assess how extreme our observed data is if our suspicions
are false
D. It allows us to calculate the p-value

21. Which of the following is TRUE about the p-value?
A. It indicates the likelihood of obtaining data at least as extreme as the
observed data if the null hypothesis were true
B. A large p-value indicates that the null hypothesis is false
C. A large p-value indicates that the null hypothesis is true
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ح٤͹. G R H K X J b ; : 9 - ߻ Ԛ

D. It indicates the likelihood of obtaining data at least as extreme as the
observed data if the null hypothesis were false


22. Which of the following hypotheses would be suitable for testing using a
statistical test?
A. The average age of undergraduate students is 20
B. 4 is the only solution to the function x^2í16=0
C. Every undergraduate student is at least 16 years old
D. The average employee is healthy

23. Suppose we suspect that University A tends to graduate a significantly
different proportion of students than University B does. If we assessed
this using a statistical test, how would the test statistic be formulated?
A. Under the assumption that the two proportions are equal
B. Under the assumption that the two proportions are different
C. Under the assumption that both universities graduate the same number of
students
D. Under the assumption that each universities graduate a different number
of students

24. You want to investigate the relationship between exercise and weight loss
among university students, but you do not know whether weight gain due
to increased muscle mass might be offset by weight loss due to
decreased fat stores. You should FIRST...
A. Design your study to distinguish between the two influences on weight
B. Define precisely your null hypothesis and alternative hypothesis
C. Use a two-tailed test to evaluate the significance of your results
D. Use a one-tailed test to evaluate the significance of your results


25. Consider the following scenario: we have a coin which we suspect to be
biased. We get to flip the coin as many times as we want to perform a
statistical test. If we incorrectly conclude that the coin is biased at the end
of the experiment, we must pay $100; otherwise, no transfer of money
takes place. Which of the following is TRUE about Type I and Type II
errors?
A. We should aim to reduce Type II error, because the consequences of a
Type I error are not particularly costly
B. We should aim to reduce Type I and Type II error equally, because the
consequences of either of them are particularly costly
C. We should aim to reduce Type I error, because the consequences of a
Type II error are not particularly costly
D. We should aim to reduce Type I error, because the consequences of a
Type I error are particularly costly
1-5 TTTFT 6-10 FFTTC 11-15 BCCCB 16-20 BCCCA 21-25 AAAAD

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