Download Introduction to Statistics, Lecture 5

Overview
Course 02402/02323 Introduction to Statistics
Lecture 5: Hypothesis test, power and model control - one
sample
1
Motivating example - sleeping medicine
2
One-sample t-test and p-value
p-values and hypothesis tests (COMPLETELY general)
Critical value and confidence interval
Per Bruun Brockhoff
3
Hypothesis test with alternatives
Hypothesis test - the general method
DTU Compute
Danish Technical University
2800 Lyngby – Denmark
4
Planning: Power and sample size
5
Checking the normality assumption
The Normal QQ plot
Transformation towards normality
e-mail: [email protected]
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Introduction to Statistics, Lecture 5
Fall 2015
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Motivating example - sleeping medicine
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Introduction to Statistics, Lecture 5
Example - sleeping medicine
Difference of sleeping medicines?
The hypothesis of no difference:
In a study the aim is to compare two kinds of sleeping medicine A and B.
10 test persons tried both kinds of medicine and the following 10
DIFFERENCES between the two medicine types were measured:
(For person 1, sleep medicine B was 1.2 sleep hour better than medicine A,
etc.):
Sample, n = 10:
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x = Beffect - Aeffect
1.2
2.4
1.3
1.3
0.9
1.0
1.8
0.8
4.6
1.4
Introduction to Statistics, Lecture 5
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Motivating example - sleeping medicine
Motivating example - sleeping medicine
person
1
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9
10
Fall 2015
H0 : µ = 0
Sample mean and standard
deviation:
x̄ = 1.670 = µ̂
Data: x̄ = 1.67, H0 : µ = 0
s = 1.13 = σ̂
NEW:p-value:
p − value = 0.00117
(Computed under the scenario,
that H0 is true)
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Is data in acoordance with the
null hypothesis H0 ?
NEW:Conclusion:
As the data is unlikely far away
from H0 , we reject H0 - we have
found a significant effect of
sleep medicine B as compared to
A.
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One-sample t-test and p-value
One-sample t-test and p-value
Metode 3.22: One-sample t-test and p-value
p-values and hypothesis tests (COMPLETELY general)
The definition and interpretation of the p-value
(COMPLETELY general)
How to compute the p-value?
For a (quantitative) one sample situation, the (non-directional) p-value is
given by:
p − value = 2 · P (T > |tobs |)
where T follows a t-distribution with (n − 1) degrees of freedom.
The observed value of the test statistics to be computed is
tobs =
x̄ − µ0
√
s/ n
Very strong evidence against H0
Strong evidence against H0
Some evidence against H0
Weak evidence against H0
Little or no evidence against H0
The p-value is the probability of obtaining a test statistic that is at least
as extreme as the test statistic that was actually observed. This probability
is calculated under the assumption that the null hypothesis is true.
H0 : µ = µ0
Introduction to Statistics, Lecture 5
One-sample t-test and p-value
p < 0.001
0.001 ≤ p < 0.01
0.01 ≤ p < 0.05
0.05 ≤ p < 0.1
p ≥ 0.1
Definition 3.12 of the p-value:
where µ0 is the value of µ under the null hypothesis:
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The p-value expresses the evidence against the null hypothesis – Table 3.1:
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p-values and hypothesis tests (COMPLETELY general)
Example - sleeping medicine
Per Bruun Brockhoff ([email protected])
Introduction to Statistics, Lecture 5
One-sample t-test and p-value
Fall 2015
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p-values and hypothesis tests (COMPLETELY general)
Example - sleeping medicine - in R - manually
The hypothesis of no difference:
## Enter data:
x <- c(1.2, 2.4, 1.3, 1.3, 0.9, 1.0, 1.8, 0.8, 4.6, 1.4)
n <- length(x)
## Compute the tobs - the observed test statistic:
tobs <- (mean(x) - 0) / (sd(x) / sqrt(n))
## Compute the p-value as a tail-probability in the t-distribution:
pvalue <- 2 * (1-pt(abs(tobs), df=n-1))
pvalue
H0 : µ = 0
Compute the p-value:
Compute the test-statistic:
1.67 − 0
√ = 4.67
tobs =
1.13/ 10
2P (T > 4.67) = 0.00117
2 * (1-pt(4.67, 9))
## [1] 0.0011659
Interpretation of the p-value in light of Table 3.1:
There is strong evidence agains the null hypothesis.
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Per Bruun Brockhoff ([email protected])
Introduction to Statistics, Lecture 5
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One-sample t-test and p-value
p-values and hypothesis tests (COMPLETELY general)
Example - sleeping medicine - in R - with inbuilt function
One-sample t-test and p-value
p-values and hypothesis tests (COMPLETELY general)
The definition of hypothesis test and significance (generally)
Definition 3.23. Hypothesis test:
t.test(x)
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We say that we carry out a hypothesis test when we decide against a null
hypothesis or not using the data.
One Sample t-test
A null hypothesis is rejected if the p-value, calculated after the data has
been observed, is less than some α, that is if the p-value < α, where α is
some pre-specifed (so-called) significance level. And if not, then the null
hypothesis is said to be accepted.
data: x
t = 4.67, df = 9, p-value = 0.0012
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
0.86133 2.47867
sample estimates:
mean of x
1.67
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Introduction to Statistics, Lecture 5
One-sample t-test and p-value
Fall 2015
Definition 3.28. Statistical significance:
An effect is said to be (statistically) significant if the p-value is less than
the significance level α.
(OFTEN we use α = 0.05)
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p-values and hypothesis tests (COMPLETELY general)
Example - sleeping medicine
Introduction to Statistics, Lecture 5
One-sample t-test and p-value
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Critical value and confidence interval
Critical value
Definition 3.30 - the critical values of the t-test:
The (1 − α)100% critical values for the (non-directional) one-sample t-test
are the (α/2)100% and (1 − α/2)100% quantiles of the t-distribution with
n − 1 degrees of freedom:
With α = 0.05 we can conclude:
Since the p-value is less than α so we reject the null hypothesis.
tα/2 and t1−α/2
And hence:
We have found a significant effect af medicine B as compared to A. (And
hence that B works better than A)
Metode 3.31: One-sample t-test by critical value:
A null hypothesis is rejected if the observed test-statistic is more extreme
than the critical values:
If |tobs | > t1−α/2 then reject
otherwise accept.
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One-sample t-test and p-value
Critical value and confidence interval
One-sample t-test and p-value
Critical value and confidence interval
Critical value and hypothesis test
Critical value and hypothesis test
The acceptance region are the values for µ not too far away from the data
- here on the standardized scale:
The acceptance region are the values for µ not too far away from the data
- now on the original scale:
Acceptance
Acceptance
Rejection
Rejection
t0.025
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0
Rejection
x − t0.975s
t0.975
Introduction to Statistics, Lecture 5
One-sample t-test and p-value
Rejection
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Critical value and confidence interval
n
x + t0.975s
Introduction to Statistics, Lecture 5
One-sample t-test and p-value
Critical value, confidence interval and hypothesis test
x
n
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Critical value and confidence interval
Proof:
Theorem 3.32: Critical value method = Confidence interval method
We consider a (1 − α) · 100% confidence interval for µ:
Remark 3.33
A µ0 inside the confidence interval will fullfill that
s
x̄ ± t1−α/2 · √
n
s
|x̄ − µ0 | < t1−α/2 · √
n
The confidence interval corresponds to the acceptance region for H0 when
testing the (non-directional) hypothesis
which is equivalent to
|x̄ − µ0 |
√s
n
H0 : µ = µ0
< t1−α/2
and again to
(New) interpretation of the confidence interval:
|tobs | < t1−α/2
The confidence interval covers those values of the parameter that we
believe in given the data.
Those values that we accept by the corresponding hypothesis test.
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which then exactly states that µ0 is accepted, since the tobs is within the
critical values.
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Hypothesis test with alternatives
Hypothesis test with alternatives
Hypothesis test with alternatives
Example - PC screens
So far - implied: (= non-directional)
Produkt specification
The alternative to H0 : µ = µ0 is : H1 : µ 6= µ0
A manufacturer of computer screens claims that screens in average uses
83W.
(And implied: below 83W would be just fine for the company then)
Furthermore we can assume, that the usage is normally distributed with
some variance σ 2 (W)2 . The underlying insight that makes this a one-sided
situation is that having a mean usage below 83W would be just fine for the
company.
BUT there are other possible settings, e.g. one-sided (=directional), ”less”:
The alternative to H0 : µ ≥ µ0 is : H1 : µ < µ0
Or one-sided (=directional), ”greater”:
The alternative to H0 : µ ≤ µ0 is : H1 : µ > µ0
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Hypothesis test with alternatives
Hypothesis test with alternatives
Example - PC screens
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Hypothesis test - the general method
Generally a hypothesis test consists of the following steps:
Null hypothesis: H0 : µ ≥ 83. The alternative: H1 : µ < 83
With the purpose to be able to reject(=falsify) that the usage could be
greater.
1
Formulate the hypotheses and choose the level of significance α
(choose the "risk-level")
2
Calculate, using the data, the value of the test statistic
3
Calculate the p-value using the test statistic and the relevant sampling
distribution, and compare the p-value and the significance level α and
make a conclusion
4
(Alternatively, make a conclusion based on the relevant critical
value(s))
IF an external should document the claim wrong:
Null hypothesis: H0 : µ ≤ 83. The alternative: H1 : µ > 83
With the purpose to be able to reject(=falsify) that the usage is no larger
than 83.
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Metode 3.36. Steps by hypothesis tests - an overview
IF the company should document the claim:
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Hypothesis test with alternatives
Hypothesis test - the general method
Hypothesis test with alternatives
Hypothesis test - the general method
The two-sided (non-directional) one-sample t-test again
Det one-sided (directional) one-sample t-test
Method 3.37. The level α test is:
1 Compute t
obs as before
Methode 3.38. The level α one-sided (”less”) test is:
2
Compute the evidence against the null hypothesis H0 : µ = µ0 vs. the
alternative hypothesis H1 : µ 6= µ0 by the
1
Compute tobs as before
2
Compute the evidence against the null hypothesis H0 : µ ≥ µ0 vs. the
alternative hypothesis H1 : µ < µ0 by the
p–value = 2 · P (T > |tobs |)
p–value = P (T < tobs )
where the t-distribution with n − 1 degrees of freedom is used.
where the t-distribution with n − 1 degrees of freedom is used.
3
If p–value < α: We reject H0 , otherwise we accept H0 .
3
If p–value < α: We reject H0 , otherwise we accept H0 .
4
The rejection/acceptance conclusion could alternatively, but
equivalently, be made based on the critical value(s) ±t1−α/2 :
If |tobs | > t1−α/2 we reject H0 , otherwise we accept H0 .
4
The rejection/acceptance conclusion could alternatively, but
equivalently, be made based on the critical value(s) tα :
If tobs < tα we reject H0 , otherwise we accept H0 .
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Introduction to Statistics, Lecture 5
Hypothesis test with alternatives
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Hypothesis test - the general method
Per Bruun Brockhoff ([email protected])
Introduction to Statistics, Lecture 5
Hypothesis test with alternatives
The one-sided (directional) one-sample t-test
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Hypothesis test - the general method
Example - PC screens
Can we reject the claim of the producer?
Metode 3.39. The level α one-sided (”greater”) test is:
1
Compute tobs as before
2
Compute the evidence against the null hypothesis H0 : µ ≤ µ0 vs. the
alternative hypothesis H1 : µ > µ0 by the
p–value = P (T > tobs )
A group of consumers wants to test the manufacturer’s claim and 12
measurements of the usage are performed:
82 86 84 84 92 83 93 80 83 84 85 86
From this we estimate the mean usage at x̄ = 85.17 and s = 3.8099
Hence, one-sided ”greater” is the relevant test:
Null hypothesis: H0 : µ ≤ 83. The alternative: H1 : µ > 83
where the t-distribution with n − 1 degrees of freedom is used.
3
If p–value < α: We reject H0 , otherwise we accept H0 .
4
The rejection/acceptance conclusion could alternatively, but
equivalently, be made based on the critical value(s) t1−α :
If tobs > t1−α we reject H0 , otherwise we accept H0 .
Compute the test-statistic and the p-value:
tobs =
85.17 − 83
√ = 1.97
3.8099/ 12
p-value = P (T > 1.97) = 0.0373
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Hypothesis test with alternatives
Hypothesis test - the general method
Hypothesis test with alternatives
Example - PC screens
Hypothesis test - the general method
Errors in hypothesis testing
Conclusion using α = 0.05:
So we reject the null hypothesis. We have documented that the screen’s
mean usage is significantly greater than 83W.
x <- c(82, 86, 84, 84, 92, 83, 93, 80, 83, 84, 85, 86)
t.test(x, mu = 83, alt = "greater")
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Two kind of errors can occur (but only one at a time!)
Type I: Rejection of H0 when H0 is true
Type II: Non-rejection (acceptance) of H0 when H1 is true
One Sample t-test
The risks of the two types or errors:
data: x
t = 1.97, df = 11, p-value = 0.037
alternative hypothesis: true mean is greater than 83
95 percent confidence interval:
83.192
Inf
sample estimates:
mean of x
85.167
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Hypothesis test with alternatives
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P (Type I error) = α
P (Type II error) = β
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Introduction to Statistics, Lecture 5
Hypothesis test with alternatives
Court of law analogy
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Hypothesis test - the general method
Errors in hypothesis testing
A man is standing in a court of law:
Theorem 3.43: Significance level = The risk of a Type I error
A man is standing in a court of law accused of criminal activity.
The null- and the the alternative hypotheses are:
H0 :
The man is not guilty
H1 :
The man is guilty
The significance level α in hypothesis testing is the overall Type I risk:
P (Type I error) = P (Rejection of H0 when H0 is true) = α
Two possible truths vs. two possible conclusions:
That you cannot be proved guilty is not the same as being proved innocent:
H0 is true
H0 is false
Or differently put:
Accepting a null hypothesis is NOT a statistical proof of the null
hypothesis being true!
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Reject H0
Type I error (α)
Correct rejection of H0 (Power)
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Fail to reject H0
Correct acceptance of H0
Type II error (β)
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Planning: Power and sample size
Planning: Power and sample size
Planning, Power
Planning and power
What is the power of a future study/experiment:
When the test to use has been set:
If you know (or set/guess) four out of the following five pieces of
information, you can find the fifth:
The probability of detecting an (assumed) effect
P (Reject H0 ) when H1 is true
Probability of correct rejection of H0
The sample size n
Challenge: The null hypothesis can be wrong in many ways!
Practically: Scenario based approach
Significance level α of the test.
A change in mean that you would want to detect (effect size) µ0 − µ1 .
E.g. "What if µ = 86, how good will my study be to detect this?"
E.g. "What if µ = 84, how good will my study be to detect this?"
etc
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The population standard deviation, σ.
The power (1 − β).
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Planning: Power and sample size
0.5
Under H 0
0.4
β
Probability density
Probability density
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Under H 0
Under H 1
α
Power
0.3
0.2
0.1
α
β
Power
0.3
0.2
0.1
β
Power
Power
α
0.0
µ0
µ1
Introduction to Statistics, Lecture 5
βα
0.0
µ0
Low power
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High power example
Under H 1
0.4
Fall 2015
Planning: Power and sample size
Low power example
0.5
Introduction to Statistics, Lecture 5
µ1
High power
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Introduction to Statistics, Lecture 5
Planning: Power and sample size
Planning: Power and sample size
Planning, Sample size n
Example - The power for n = 40
The big practical question: What should n be?
power.t.test(n = 40, delta = 4, sd = 12.21,
type = "one.sample", alternative= "one.sided")
The experiment should be large enough to detect a relevant effect with
high power (usually at least 80%):
Metode 3.48: The one-sample, one-sided sample size formula:
For the one-sided, one-sample t-test for given α, β and σ:
z1−β + z1−α 2
n= σ
(µ0 − µ1 )
Where µ0 − µ1 is the change in means that we would want to detect and
z1−β , z1−α are quantiles of the standard normal distribution.
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Planning: Power and sample size
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One-sample t test power calculation
n
delta
sd
sig.level
power
alternative
=
=
=
=
=
=
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40
4
12.21
0.05
0.65207
one.sided
Introduction to Statistics, Lecture 5
Checking the normality assumption
Fall 2015
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The Normal QQ plot
Example - The sample size for power= 0.80
Example - student heights - are they normally distributed?
power.t.test(power = .80, delta = 4, sd = 12.21,
type = "one.sample", alternative= "one.sided")
x <- c(168,161,167,179,184,166,198,187,191,179)
hist(x, xlab="Height", main="", freq = FALSE)
lines(seq(160, 200, 1), dnorm(seq(160, 200, 1), mean(x), sd(x)))
0.02
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58.984
4
12.21
0.05
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one.sided
Density
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delta
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One-sample t test power calculation
0.00
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Height
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Checking the normality assumption
The Normal QQ plot
Checking the normality assumption
The Normal QQ plot
Example - 100 observations from a normal distribution:
Example - student heights - ecdf
xr <- rnorm(100, mean(x), sd(x))
hist(xr, xlab="Height", main="", freq = FALSE)
lines(seq(130, 230, 1), dnorm(seq(130, 230, 1), mean(x), sd(x)))
plot(ecdf(x), verticals = TRUE)
xp <- seq(0.9*min(x), 1.1*max(x), length.out = 100)
lines(xp, pnorm(xp, mean(x), sd(x)))
0.030
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160
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x
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Checking the normality assumption
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Per Bruun Brockhoff ([email protected])
The Normal QQ plot
Introduction to Statistics, Lecture 5
Checking the normality assumption
Example - student heights - Normal Q-Q plot
xr <- rnorm(100, mean(x), sd(x))
plot(ecdf(xr), verticals = TRUE)
xp <- seq(0.9*min(xr), 1.1*max(xr), length.out = 100)
lines(xp, pnorm(xp, mean(xr), sd(xr)))
qqnorm(x)
qqline(x)
Per Bruun Brockhoff ([email protected])
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Normal Q−Q Plot
ecdf(xr)
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The Normal QQ plot
Example - 100 observations from a normal distribution, ecdf:
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Introduction to Statistics, Lecture 5
Checking the normality assumption
The Normal QQ plot
Checking the normality assumption
Example - student heights - Normal Q-Q plot - compare
with other simulated normally distributed data
The Normal QQ plot
Normal Q-Q plot
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Method 3.52 - The formal definition (And remark 3.53)
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The ordered observations x(1) , . . . , x(n) are plotted versus a set of expected
normal quantiles zp1 , . . . , zpn . Different definitions of p1 , . . . , pn exist:
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pi =
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pi =
i − 3/8
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In R, when n ≤ 10:
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Theoretical Quantiles
Quantiles
Per Bruun Brockhoff ([email protected])
Introduction to Theoretical
Statistics,
Lecture 5
Checking the normality assumption
0.5 1.0 1.5
Theoretical Quantiles
Fall 2015
46 / 50
Transformation towards normality
Per Bruun Brockhoff ([email protected])
Introduction to Statistics, Lecture 5
Checking the normality assumption
Example - Radon data
Fall 2015
Transformation towards normality
Example - Radon data - log-transformed are closer to a
normal distribution
## READING IN THE DATA
radon<-c(2.4, 4.2, 1.8, 2.5, 5.4, 2.2, 4.0, 1.1, 1.5, 5.4, 6.3,
1.9, 1.7, 1.1, 6.6, 3.1, 2.3, 1.4, 2.9, 2.9)
##A HISTOGRAM AND A QQ-PLOT
par(mfrow=c(1,2))
hist(radon)
qqnorm(radon,ylab = 'Sample quantiles',xlab = "Normal quantiles")
qqline(radon)
Histogram of radon
##TRANSFORM USING NATURAL LOGARITHM
logRadon<-log(radon)
hist(logRadon)
qqnorm(logRadon,ylab = 'Sample quantiles',xlab = "Normal quantiles")
qqline(logRadon)
Normal Q−Q Plot
Histogram of logRadon
Normal Q−Q Plot
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47 / 50
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radon
Per Bruun Brockhoff ([email protected])
5
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1
2
0.0
Normal quantiles
Introduction to Statistics, Lecture 5
0.5
1.0
logRadon
Fall 2015
48 / 50
Per Bruun Brockhoff ([email protected])
1.5
2.0
−2
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0
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2
Normal quantiles
Introduction to Statistics, Lecture 5
Fall 2015
49 / 50
Checking the normality assumption
Transformation towards normality
Overview
1
Motivating example - sleeping medicine
2
One-sample t-test and p-value
p-values and hypothesis tests (COMPLETELY general)
Critical value and confidence interval
3
Hypothesis test with alternatives
Hypothesis test - the general method
4
Planning: Power and sample size
5
Checking the normality assumption
The Normal QQ plot
Transformation towards normality
Per Bruun Brockhoff ([email protected])
Introduction to Statistics, Lecture 5
Fall 2015
50 / 50