normality tests - amazon s3

Normality testsP R A C T I C I N G S TAT I S T I C S I N T E R V I E W Q U E S T I O N S I N R

Zuzanna ChmielewskaActuary

PRACTICING STATISTICS INTERVIEW QUESTIONS IN R


Testing normalityStatistical tests

Shapiro-Wilk test

Kolmogorov-Smirnov test

Visual measure

Q-Q plot


Shapiro-Wilk test


Shapiro-Wilk test

Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors andAnderson–Darling tests"

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P-value


Shapiro-Wilk test


Transforming data for normality


Checking normality in R

Method Function

Shapiro-Wilk test shapiro.test(x)

Kolmogorov-Smirnov test ks.test(x, y = "pnorm")

Q-Q plot qqnorm(x); qqline(x)


Summarynormality tests

Shapiro-Wilk test


p-value

Q-Q plot

data transformation

checking normality in R

Let's practice!P R A C T I C I N G S TAT I S T I C S I N T E R V I E W Q U E S T I O N S I N R

Inference for a meanP R A C T I C I N G S TAT I S T I C S I N T E R V I E W Q U E S T I O N S I N R



Inference for a mean


Inference for a meancon�dence interval

one-sample mean


AssumptionsThe t-test's assumptions:

normally distributed underlying data (recall CLT!)

random sample

independent observations


Con�dence interval


95% con�dence interval


One-sample t-testH : μ = μ

H : μ ≠ μ

where:

μ - the population's mean

μ - the hypothesized mean

0 0

1 0

0


t-test in R

t.test(x)


t-test in R

t.test(x, mu = 2)


t-test in R

t.test(x, mu = 2, conf.level = 0.9)


Summaryt-test's assumptions

con�dence interval

one-sample t-test

t.test() in R

Comparing twomeans

P R A C T I C I N G S TAT I S T I C S I N T E R V I E W Q U E S T I O N S I N R



Comparing two means


HypothesesTwo-tailed test:

H : μ = μ

H : μ ≠ μ

where:

μ - the �rst population's mean

μ - the second population's mean

0 1 2

1 1 2

1

2


AssumptionsThe two-sample t-test's assumptions:

normally distributed underlying data

random samples


homogeneity of variances


AssumptionsThe two-sample t-test's assumptions:

normally distributed underlying data

random sample


homogeneity of variances - e.g. bartlett.test()


Teaching methods


t-test in RTwo-sample t-test

t.test(value ~ group, data = df, var.equal = TRUE)


t-test in RTwo-sample t-test

t.test(value ~ group, data = df, var.equal = TRUE)

Paired t-test

t.test(value ~ group, data = df, paired = TRUE)


Summarytwo-sample t-test

hypotheses

assumptions

paired t-test

t.test() in R

ANOVAP R A C T I C I N G S TAT I S T I C S I N T E R V I E W Q U E S T I O N S I N R



Hypotheses

H : μ = μ = ... = μ

H : ∃ μ ≠ μ

0 1 2 n

1 (i,j) i j


Hypotheses

H : μ = μ = μ0 1 2 3


HypothesesTwo-sample t-test:

H : μ = μ0 1 2


Hypotheses

H : μ = μ = ... = μ0 1 2 n


Why not multiple t-tests?


Assumptionsindependence of cases

normal distributions

homogeneity of variances


ANOVA in R

oneway.test(value ~ group, data, var.equal = TRUE)


Box plot


Summaryhypotheses of ANOVA

type I and II errors

assumptions of ANOVA

oneway.test() in R

box plot

normality tests - amazon s3

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