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Tutorial 11

Scope of this tutorial:

• Discussion - scatter plots

• Regression Exercise

• Revision of earlier hypothesis tests

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X(determinant)

Y(outcome)

Relation

(a) Max. daily temperature

and soft drink sales

Max. daily

temperature

soft drink

sales

Positive

(b) Odometer reading and

sale price of used cars

Odometerreading

sale price ofused cars

Negative

(c) Annual income and

credit card balance of bank

clients

Annualincome

credit cardbalance

Positive

Worksheet 1: Q. 1 What sort of a relation is that?

333

Worksheet 1: Q.2: Lookingfor relationship

Rising trend, plusperiodic rise and fall.

What relation can you see in the

following scatter diagrams?

Negative linear relation betweentemperature and latitude:Higher latitude => lower temp. 444

Positive linear relation

between chest girth andweight for males.

Probably no relation betweenattendance of crowd at MCGand temperature.

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Astronomy: galaxyWhat is a galaxy? A galaxy is a collection of stars, ranging

from ten million (107) up to a hundred trillion (1014) stars.

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Group of galaxies

888

Expansion of the universe –

after the Big Bang creation

999

Worksheet 2: Relation between distance fromEarth and radial velocity of galaxies in the universe– Hubble’s law

There appears to be apositive relation betweenvelocity and distance.

V = -40.784+454.158*distance

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v = -40.784+454.158*distance

Meaning of the regression

equation (or meaning of the

slope):

For each increase of 1

megaparsec (Mpc) from Earth,

velocity increases by 454

km/sec, on average.

(1 Mpc=3.26 million light

years)

1 Mpc

454 km/sec

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Research Question: Is the distance from earth a

useful predictor of the radial velocity of galaxies?

H Ho: β = 0

A The relation appears reasonably linear.

The points seem to be fairly evenly spread roundthe line with no obvious outliers, indicating that

the residuals have constant standard deviation andresiduals may be normally distributed.

T t = 6.036, df=22

P p-value ≈ 0. Since p<0.05, reject Ho

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C:

There is a significant positive linear relationbetween distance and radial velocity (Hubble’slaw).

For each increase of a distance of 1 megaparsec(Mpc) from earth, a galaxy’s velocity increasesby 454 km/sec, on average.

We are 95% confident that the true increase is

between 298 and 610 km/sec.

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Predictions:Predict the radial velocity of a galaxy which is 1.25

Mpc from Earth.v=-40.784+454.158*distance=-40.784+454.158*1.25= 526.9 km/sec

Predict the radial velocity of a galaxy which is 2.25Mpc from Earth.

2.25 megaparsecs is out of range of data, hencenot valid to predict.

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Predictions:

Predict the distance from earth for a celestialobject which has a radial velocity of 400

km/sec.Not valid to predict independent variable (X)

from outcome (Y)

(For those curious:

If we really want to predict distance from velocity,we need to re-do the regression using velocity asx (independent variable) and distance as y(dependent variable). Then the new regression willbe Distance = a + b*velocity)

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Goodness-of-fit statistic r2

Interpret the goodness of fit statistic: r2 = 0.624.62.4% of the variation in radial velocity of galaxies

can be explained by the variation in distance fromEarth.

Calculate and interpret the correlation coefficient:

r=+√0.624 = 0.79, indicating there is a fairly strong positive linear relation between the two variables.

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Revision Questions

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Variable DescriptionID ID of male (1 – 252)

D Density determined from underwaterweighing

BF% Percent body fat from Siri's (1956) equation

Age Age (years)

W Weight (kg)

H Height (m)BMI Body Mass Index (kg/m2)

Nec Neck circumference (cm)

Che Chest circumference (cm)

Abd Abdomen circumference (cm)

Hip Hip circumference (cm)

Thi Thigh circumference (cm)

Kne Knee circumference (cm)

Ank Ankle circumference (cm)Bic Biceps (extended) circumference (cm)

Arm Forearm circumference (cm)

Wri Wrist circumference (cm)

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Question 1: Display

1. a) What type of graphical display should you provide tocompare the percentage body fat (BF%) of males aged lessthan 39 years and males aged 39 years or more?

BF%: numeric (continuous) variable

Less than or more than 39 years old: binary variable New variable

Hence comparative box plots

b) An obese person is said to have a body mass index (BMI)of more than 30. What type of graphical display should youprovide to compare the proportions of obese males aged

less than 39 years with those aged 39 or more years?

BMI above or below 30 (obese or not obese): binary variable New variable

Less than or more than 39 years old: binary variable New variable

Hence clustered bar chart.1919

Question 2: One-sample z-test

2. Research Question: Was the mean BMI of Australian males in 2008 the same as it was in the 1980s?

Assume the mean BMI of Australian men inthe 1980s was equal to 25 with a SD of 3.5.In 2008, a random sample of 20 Australianmales was selected and the BMI of eachmale was recorded.

27.84 30.44 29.86 31.04 30.81 24.93 23.57 21.23 30.98 26.25

24.84 27.03 31.02 23.54 25.49 29.38 24.52 27.62 32.94 22.46

Carry out a suitable hypothesis test to answerthe research question. Assume that thevariation in BMI has not changed.

2020

One-sample z-test, NOT t-test

20 22 24 26 28 30 32

0

1

2

3

4

5

6

BMIFreq.

s, NOT used

)82.28,76.25(20

5.396.12895.2796.1CI95% =×±=×±=

n y

σ 2121

Was there a difference between the average percentage bodyfat (BF%) of American males in 1985 aged less than 39 yearsand the average BF% of American males aged 39 years ormore? => 2-sample t-test

0 5 10 15 20 25 300

5

10

15

20

<39yrsFreq.

0 10 20 30 400

5

10

15

20

>39yrsFreq.

Question 3(a)

0.0003

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We are 95% confident that the BF% of males agedover 39 years between 1.87% and 6.26% higherthan the younger males on average.

CI/2 CI/2(-------------+------------)

4.066)26.6,87.1(198.2066.4

)(

forCI95%

21

1121

21

=

±=

+×±−=

−

nn pst y y ν

µ µ

2*2.198=4.396 is NOT CI.

It is the length of CI.

Double

this is

NOT the

CI.

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Question 3(b): Was the ankle circumference 5cmmore, on average, than the wrist circumference of American males in 1985? => paired t-test

2 3 4 5 6 7 8 9 1011 120

20

40

60

80

100

differenceFreq.

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Question 4: RegressionResearch Question: Was the BMI of American

males in 1985 a useful predictor of BF%?

Use the output to complete this question.

1. Which is the dependent/response variable?

2. Which is the independent/predictor variable?

3. Comment on the scatterplot.

4. Write down the equation of the regression line.

5. Test the statistical significance of the relation.

6. Predict, if appropriate, the expected % Body Fat for:

(a) a male with a BMI of 20; (b) a male with a BMI of 15

7. Predict, if appropriate, the expected BMI for a male with 20%Body Fat.

8. (a) Calculate r and interpret. (b) Calculate r2 and interpret.

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1. Which is the dependent/response variable? BF%

2. What is the independent/predictor variable? BMI?

3. Comment on the scatter plot.

• Positive linear relation: higher BMI => higher BF%

• Residual constant SD

• No outliers; symmetric on both sides

4. Regression equation

BMI F B *8186.19872.26%ˆ +−=

BF% vs BMI

0

10

20

30

40

50

15 20 25 30 35 40BMI

BF%

?

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5. Test the statistical significance of the relation.

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6. Predict, if appropriate, the expected % Body Fatfor: (a) a male with a BMI of 20; (b) a male with aBMI of 15

A male with a BMI of 20:BF% = -26.9872 + 1.8186*20 = 9.384

A male with a BMI of 15:Not valid to predict, since 15 is out of the range of

the data.

7. Predict, if appropriate, the expected BMI for a malewith 20% Body Fat.Not valid to predict the independent variable (predictor

or x) from the dependent variable (outcome or y)

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8. (a) Calculate r and interpret.r = √ 0.535 =0.73

There is a fairly strong positive linear relation between BMI and BF%.

(b) Calculate r2 and interpret.r2 = 0.535This indicates that about 53% of the

variation in BF% can be explained by thevariation in BMI.

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Question 5: Best predictorResearch Question: Which of the BMI, Neck Circumference or

Abdomen circumference is the best predictor of BF%?

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Best predictorFill in the table. Explain your answer.

Each of the predictors is a significant predictor of BF%;

the p-val for each of the predictors is 0.000.

Each regression equation satisfies the assumptions of linearity,constant spread and normality of the residuals.

However, the abdomen circumference (Abd) provides the best fitting as r 2 = 67% is much higher than the others.

Note: 1. NEVER compare values of b. 2. It is easier, and better, to compare r 2 instead of p-vals. 3. Discard (cross out)variables if they break assumptions or if p-val>0.05.

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Practice Exercises: Question 1

Consider the computer output which shows the relationbetween students’ ideal weights and their actualweights for females. Note the dotted line represents the

cases when the ideal weight is the same as the actualweight.

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Question 1

(a) By comparing the regression line (solid) with the

line y=x (ie ideal weight_f = weight_f) (dotted),

comment on the scatter plot.

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Question 1

(b) From the partial EcStat output above, perform an

appropriate hypothesis test to see if there is alinear relation between Ideal weight_f (Y) and

Weight_f (X).

Partial ans: t=21.29

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Question 1 (answers)

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Question 1 (continued)

(c) What is the value of goodness-of-fit statistic?

Interpret its meaning.

Ans: 70.8% Meaning: ……………….

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Question 2

The table on the right shows Accounting and

Statistics marks for 12 students.

Research question: Can Accounting marks

(X) be used to predict Statistics marks (Y)?

Use the partial EcStat output below to answer

the research question.

Acc Sta t

74 81

93 86

55 67

41 35

23 30

92 100

64 55

40 52

71 76

33 24

30 48

71 87

df: 10

coeff SE t p-value

7.0194 7.971 0.8806 0.399 -10.741 24.779

0.9560 0.129

r-sq: 0.845 Resid SS: 1046.876 s: 10.232

outcome:

predictor

constant

Acc

Stat

95% C.I.

20

30

40

50

60

70

80

90

100

110

20 30 40 50 60 70 80 90 100Acc

Stat

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(Partial Ans: t=7.411)

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(b) What is the value of goodness-of-fit statistic?


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Question 3

Research question: Can Weight (X) be used to predictHeight (Y)?

Using the partial EcStat output below to answer theresearch question.

150

155

160

165

170

175

180

185

190

195

40 50 60 70 80 90 100Weight

Height

df: 82coeff SE t p-value

130.1702 4.041 32.2109 0.000 122.131 138.209

0.6699 0.061

r-sq: 0.595 Resid SS: 2855.483 s: 5.901

outcome: predictor

constant

Weight

Height95% C.I.

4040



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Question 4

(Swap X and Y in Question 3.)

Research question: Can Height (X) be used to predictWight (Y)?

Using the partial EcStat output below to answer theresearch question.

40

50

60

70

80

90

100

150 160 170 180 190Height

Weight

df: 82

coeff SE t p-value

-89.1380 14.096 -6.3238 0.000 -117.179 -61.097

0.8881 0.081 10.9760 0.000 0.727 1.049

r-sq: 0.595 Resid SS: 3785.467 s: 6.794

outcome:

predictor

constant

Height

Weight

95% C.I.

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(c) Explain why the value of r2 is the same as that in

Question 3.

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Question 5

For each of the following given regression equations,

interpret (i) the equation and (ii) r2.

(a) X=time a bee spends on a flow

Y = % pollen removed,

r2 = 0.384

Interpretation of equation (slope):

Interpretation of r2:

x05.213y+=

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(b) X = students’ high school results

Y = STAT170 exam results

r2 = 6.2%

x54.023.29y +=

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(c) X = number of cans of beer drank

Y = blood alcohol content

r2 = 82.1% x y 0203.00217.0ˆ +−=

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Computer (EcStat) Exercises

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Question 1(Q.2 of previous exercise)

Research question: Can Accountingmarks (X) be used to predictStatistics marks (Y)?

1. Enter the 2 columns as shown.

2. Optional but recommended:

Pre-highlight Y (Account), thenpress Ctrl key and highlight X(Stat).

3. Click “Relationship” (4th icon).

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Make sure the X(Account) and Y

(Stat) are chosencorrectly,otherwise you willhave the wronggraph, and wrongregression results.

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df: 10

coeff SE t p-value

7.0194 7.971 0.8806 0.399 -10.741 24.779

0.9560 0.129 7.3927 0.000 0.668 1.244

r-sq: 0.845 Resid SS: 1046.876 s: 10.232

Fitted line: Stat (Y) = 7.0194 + 0.956 Account (X)

outcome:

predictor

constant

Account (X)

Stat (Y)

95% C.I.

20

30

40

50

60

70

80

90

100

110

20 40 60 80 100Account (X)

Stat (Y)

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Question 1(continued)

Fill in the following answers:

(a) Ho: ___________________

(b) Write down the regression equation:

______________________________

(c) What is the value of test statistic? (Include symbol z/t)___________________

(d) What is the value of p-val? ________

(e) Do you reject or not reject Ho? _________

(f) What is a 95% CI for β ? ____________________(g) Does the 95% CI for β include the null value? ______

(h) What is the value of goodness-of-fit statistic? _______53

Question 2 (Pract 8 Exercises)

Load the file “pulse.xls” (used in Pract/WASP 8)

Research question: Can Height (X) be used to predictWeight (Y) ?

Perform the hypothesis test using EcStat. Then answerthe questions on the next slide.

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(a) Ho: ___________________

(b) Write down the regression equation:______________________________




(f) What is a 95% CI for β ? ____________________

(g) Does the 95% CI for β include the null value? ______



Load the file “Storks.xls” (used in Pract/WASP 8)

Research question: Can the number of storks (Stork)be used to predict the number of babies born (Birth)?


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(a) Ho: ___________________


______________________________




(f) What is a 95% CI for β ? ____________________(g) Does the 95% CI for β include the null value? ______



Load the file “Peru.xls” (used in Pract/WASP 8)

Research question: Can the number of years (Years)since migration be used to predict the systolic bloodpressure (Systol)?


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(a) Ho: ___________________

(b) Write down the regression equation:______________________________




(f) What is a 95% CI for β ? ____________________

(g) Does the 95% CI for β include the null value? ______



Continue with the file “Peru.xls”.

Research question: Can Forearm (X) be used topredict Weight (Y)?


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(a) Ho: ___________________


______________________________




(f) What is a 95% CI forβ

? ____________________(g) Does the 95% CI for β include the null value? ______

(h) What is the value of goodness-of-fit statistic? _______

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