Introduction to Inferential Statistics and Hypothesis Testing
발제자: 박재현
Normal Curve
Mean, median, mode: center
Symmetrical ball shaped curve
(fig. 3-1)
Eg. Intelligence, attitude, personality
1Z=1SD
95% -> +- 1.96Z
99% -> +- 2.58Z
Percentiles
Relative score of a given score
Compare scores on tests that have different means and SDs
(No. of scores less than a given score/total No. of score)*100
25th percentile=first quartile
50th percentile=second quartile (median)
75th percentile=third quartileIt’s point, not range (table 3-1)
If z score >0 then 50+ ( ) %
If z score <0 then 50- ( ) % (Appendix A)
mean에 가까울수록 등수 올리기가 쉽다.
Standard scores
Relative distance from the mean
600점=1SD -> Z score=1
Page 56
Z score -> mean=0, SD=1
Can compare different distribution
소수나 음수가 생김
Transformed standard scoreT-scores: mean 50, SD 10
다른 score 도 가능
Correcting failures in normality through data transformations
Data that do not meet normality, linearity, homoscedasticity(등분산성)
Recalulate the measure of skewness
Moderately positive skewed -> square root
Substantially positive skewed -> log severely positive skewed -> inverse
Negative skewed -> reverse score (make it positive) or recode -> transform
If not successful -> make it categorical
Central limit theorem
Sample mean = population mean?
Sample No increase -> sample mean: normally distributed
SD of sample mean = Standard error of the mean = SD of sample/square root of n(in one sample: calculated)
SD of sample increase -> need more sample to estimate
Probability
Eg. Mammography(assume that Patient X is on the line, similar with group)
Fig. 3-2, Table 3-2
Probability of event = 100% -probability of opposite of event
Definition of probability
Frequency probabilityThrough empirical observation
Page 62 공식
assume that Patient X is on the line, similar with group
Must be random process(in mathematical theory)
Equal chance of being chosen
Each choice mus be independent
If No. of sample increase -> sample prob. = pop. prob.
위 연구의 한계
Determined logically 이 또한 objective
Priori probability
Table 3-3
Subjective probabilityRational assessment not arbitrary beliefs
17.9%: Is it close to 0 or not?
Partly intuitive and historical
4 heads in a row: 6%, 5 heads in a row: 3%
5% cutoff(Fisher , 1926): convenient
연구의 성격에 따라 1%, 10%, 20% 등 다양하게쓸 수 있음.
Probability rules
Conditional probabilityIf events are dependent
Table 3-4
Multiplication ruleIf events are independent
Independent means that given knowledge of event A does not change of prob of event B
Additional ruleIf events are mutually exclusive
Hypothesis testing
One study can not prove anything!
Null Hypothesis: H0There is no difference -> accepted
There is difference -> reject
Types of errorType I, Type II (p 69) – trade-off 관계
Probability of making type I error = alpha
Alpha ↓-> power of test ↓, risk of type II error ↑Alpha ↑-> power of test ↑, risk of type II error ↓Sample size ↑ , SD ↓ , effect size ↑ -> risk of type II error ↓ (effect size = group 간 차이 / SD
Genetic defect test : type II error 줄이는 것이 중요
Preschool preparation: type I error 줄이는 것이 중요
Power of a test
Powerful test: more likely to reject H0 when the difference exist
One tailed testmore powerful
Must have sound theoretical basis
Fig. 3-3, fig. 3-4
cf> Degrees of freedomRelated to No. of scores, items or other units in data set to the idea of freedom to vary
Sample mean known -> df=n-1
Statistical Inference & statistical Estimate
Statistical InferenceSample mean to be representative -> random selection (every member have same prob to be selected)
Statistical EstimatesSample mean = point estimate
CI = range or interval if value (infer the true value of an unknown population parameter of pop.)
Confidence Intervals
Page 74
Sampling error = the difference between sample mean and pop mean (CI tells us)
Question 1
Question 2
Question 3It happens because we calculate using sample not pop
It differ from sample to sampleWe may not conclude that probability is 95% that “mean of pop is between mean+-CI”95% refers to the average accuracy of the procedure
The relationship between confidence intervals and significance test
Significance test: Page 76 table
(t-test)
CI: Fig. 3-5
Consistency checks for evaluating research reports: cross check!
Value of CIsIt contains more information because it is equivalent to performing a sig. Test for all values of the parameter, not just a single value
Cautionrandom sample > sample from explicitly defined pop >non-representative sample
Sample size
Sample size is related to power, effect size, significance level
Power = the likelihood of rejecting the null hypothesis (80%: adequate level)
Effect sizeCohen(1987)
0.2 SD = small effect size
0.5 SD = moderate effect size
0.8 SD = large effect size
Significance levelProb of rejecting a true H0 (type I error) = alpha
Small sample size -> Non-significant result
등분산성
등분산성
선형회귀모형의 가정독립변수 X값들은 고정되어 있음(비확률 변수, 수리적 변수)
변수 X의 추정치는 오차(error)를 갖지 않음
각 X값에 따라 Y값들의 하부모집단이 존재함. 이하부모집단은 정규분포를 따른다고 가정함.
Y의 하부모집단의 분산들은 모두 동일함(등분산성)
Y의 하부모집단들의 평균은 일직선상에 놓임(선형성의 가정)
Y값들은 통계적으로 독립임.
단순선형 회귀모형의 도식
등분산성, 선형성 분석
• 분석시작 전에 반드시 산점도를 그려보고 문제점을 파악할 것.
• 이 경우에는 선형성과 등분산성에 문제가 있음.
이분산성 해결
이분산성독립변수가 증가함에 따라 분산도 증가하는 경우가 흔함
예X: 개별가구 연간소득
Y: 연간 소비지출
고소득 가구보다는 저소득 가구의 소비 변동량이 일반적으로 작음.
이분산성의 해결독립변수에 로그, ln, root등을 취해transformation을 함.
표본수 계산과 검정력
단순임의추출에서평균비교 시 표본수 계산
단순임의추출에서비율비교 시 표본수 계산
검정력(power)와 effect size
Hand out 참조
신뢰구간의 의미 해석
박재현
평균 추정
가정모집단의평균: μ 분산: σ (보통잘모름)표본집단의 평균 x, 분산: s, 표본크기: n
모집단의 평균을 추정X은 여러 개 있을 수 있음(표본크기가 n인추출가능한 모든 표본을 생각해볼 수 있음)
즉, X1 , X2 , X3 , X4 . . . . 이 있을 수 있음.
이때 표본의 크기가 n인 X1 , X2 , X3 ,X4 . . . 의 평균은 표본수가 많을 수록 μ에근사함
평균의 표준오차(standard error)
평균의표준오차(표준오차) = X1 , X2 , X3 ,
X4 . . . . 들의 표준편차 = σ/ = s/ (σ를 아는경우는매우드뭄)왜냐하면수많은 S2들의평균은 σ2임
구간 추정
σ 를알고있는경우 μ에대한 95% 신뢰구간은 (X – 1.96 σ / , X + 1.96 σ/ )
신뢰구간의의미해석모집단에서표본의크기가 n인표본을반복추출하여이들표본으로부터얻은신뢰구간의 95%에서모수 μ를포함한다는것임.
틀린해석모수 가특정 95%신뢰구간에포함될확률
구간 추정