MATH1725 Introduction to Statistics: Worked examples

Answer: 15
Question (lecture 8).
For values (x,y) as given below, obtain the sample correlation r.
Answer: 16
x i 1.1 2.2 3.4 4.5 5.0
y i 3.3 6.1 7.0 10.4 11.5
Question (lecture 10).
For values (x,y) as given below, obtain the line of regression for y given x. What does the residual
at the first data point x 1 = 1.1 equal If x = 4, what is the predicted value of y
Answer: 17
x i 1.1 2.2 3.4 4.5 5.0
y i 3.3 6.1 7.0 10.4 11.5
Question (lecture 10).
For values (x,y) as given below, a line of regression for y given x is fitted.
Test the hypothesis that the slope β equals zero.
Answer: 18
x i 1.1 2.2 3.4 4.5 5.0
y i 3.3 6.1 7.0 10.4 11.5
Question (lecture 11).
Suppose pr{X = x} = x 10
for x = 1,2,3,4. Check that the probability function is valid (is 0 ≤
pr{X = x} ≤ 1 for all x and does ∑ pr{X = x} = 1). Calculate E[X] and Var[X].
x
15 n = 4, ¯x = 4, s 2 = 3.333, µ 0 = 1, s 2 ¯x − µ0
/n = 0.8333. Test statistic is t =
σ/ √ n = √ 4 − 1 = 3.286. Test rule is
0.8333
reject H 0 if |t| > t 3(2.5%). As t 3(2.5%) = 3.182, reject H 0 at 5% level.
16 ¯x = 3.24, s 2 x = 1 X
(xi − ¯x) 2 = 1 “X
x
2
i − n¯x 2” = 2.593,
n − 1
n − 1
ȳ = 7.66, s 2 y = 1 X
(yi − ȳ) 2 = 1 “X
y
2
i − nȳ 2” = 11.033,
n − 1
n − 1
s xy = 1 X
(xi − ¯x)(y i − ȳ) = 1 “X
xiy i − n¯xȳ”
= 5.2645, r XY = s p xy/ s
n − 1
n − 1
2 xs 2 y = 0.984.
Check your answer using R!
x=c(1.1,2.2,3.4,4.5,5.0) # And setup y similarly.
cor(x,y)
17 ¯x = 3.24, ȳ = 7.66, s 2 x = 2.593, s 2 y = 11.033, s xy = 5.2645. Regression line is y = α + βx where ˆβ = s xy/s 2 x =
2.030, ˆα = ȳ − ˆβ¯x = 1.082 so fitted line is y = 1.082 + 2.030x. If x 1 = 1.1, predict ŷ 1 = 3.315. At x = 1.1, residual
is r 1 = y 1 − ŷ 1 = 3.3 − 3.315 = −0.015. If x = 4, predict y = 9.023. Check your answers using R!
x=c(1.1,2.2,3.4,4.5,5.0) # And setup y similarly.
lm(y∼x) # Gives parameter estimates.
model=lm(y∼x) # Stores regression model output as model.
model$residual[1]
r
# First residual value.
18 If H 0: β = 0, then ˆβ/ ˆσ
2
∼ t n−2, where S xx = P r
ˆσ
(x i − ¯x) 2 = (n − 1)s 2 2
x. Here = 0.2105 where
S xx S xx
S xx = (n − 1)s 2 x = 10.372. Thus t = 9.646. t 3(2.5%) = 3.182. As |t| > 3.182, reject H 0 at 5% level. Check
your answers using R!
x=c(1.1,2.2,3.4,4.5,5.0) # And setup y similarly.
model=lm(y∼x)
summary(model) # Can you find your answers in the R output
13