Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Lab 6: Hypothesis Testing and CIs for:
Two Sample Problems
M. George Akritas
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Exercise 1: Section 9.3.7, p.356, #10
Use the commands
df=read.table( ”http://www.stat.psu.edu/~mga/401/Data/
Temp28DayStrength.txt”,header=TRUE)
x=df$Temp
y=df$Strength
out=lm(y∼x)
summary(out)
fit=aov(lm(y∼ x))
summary(fit)
to answer all questions of this problem.
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Exercise 2: The Notched Box Plot
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Use the data of n1 = 32 strength measurements of cold-rolled
and n2 = 35 measurements of two-sided galvanized steel.
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Read the data in R with the command: df=read.table(
”http://www.stat.psu.edu/~mga/401/Data/
SteelStrengthData.txt”, header = T)
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boxplot(df$Value∼ df$Sample,notch=TRUE)
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Turn in the boxplot and comment. (Note: As an informal
test, if notches do not overlap, population medians differ.)
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
The R command ”t.test”: How it is used
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The default is to assume σ1 6= σ2 , do 95% CI, and give the
p-value for H0 : µ1 − µ2 = 0 vs Ha : µ1 − µ2 6= 0:
t.test(y ∼ x) # For values in y and sample index in x
t.test(y[1:32],y[33:67]) # Specify the two samples separately
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or: x1=y[which(x==1)]; x2=y[which(x==2)]; t.test(x1,x2)
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For the pooled variance T test, and 99% CI do:
t.test(y ∼ x, var.equal = TRUE, conf.level = 0.99)
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To test H0 : µ1 − µ2 = 1.8 vs Ha : µ1 − µ2 < 1.8 do:
t.test(y ∼ x,mu=1.8, alternative = ”less”)
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Other options: alternative = ”greater” or the default
”two.sided”
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Exercise 3
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Use the data set from Exercise 2, to test the hypothesis
H0 : µ1 − µ2 = 0 vs Ha : µ1 − µ2 6= 0, at level of significance
0.9, without assuming σ1 6= σ2 . Report the outcome of the
test and also the 90% CI for µ1 − µ2 .
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Repeat the above using the assumption that σ1 = σ2 .
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems
Outline
Regression
Comparing Two Means: Independent Samples
Comparing Two Means: Paired Data
Exercise 4: Section 10.5.5, p.399, #2,a,b,c
Use the commands for the paired data t test for means:
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t.test(x, y, alternative = c(”two.sided”, ”less”, ”greater”), mu
= 0, paired = TRUE, conf.level = 0.95)
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x,y above can be replaced by y ∼ Sample
to answer parts a, b, and c of this problem.
M. George Akritas
Lab 6: Hypothesis Testing and CIs for: Two Sample Problems