STATS 201代做、代写Data Analysis、R编程语言调试、R代写代做

日期：2020-06-26 11:21

STATS 201/208

THE UNIVERSITY OF AUCKLAND

First SEMESTER 2020

Campus: City

STATISTICS

Data Analysis/Data Analysis for Commerce

INSTRUCTIONS

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after submitting it, to verify you have uploaded the correct document.

• Attempt all questions.

• There are 70 marks in total for this examination.

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1. This question refers to the Job Prestige Data in Appendix A. [Total 18 marks]

a. Using the pairs plot, comment on the most important features of the data.

[3 marks]

b. The model fit.prestige1 uses both the variables income and log10income

while the other models use only log10income. Give TWO reasons why income

was dropped instead of log10income. [2 marks]

c. What would happen to the P-value for income if we dropped log10income

from fit.prestige1 instead of dropping income? Briefly justify your answer.

[2 marks]

d. According to the model fit.prestige3, which job has greater prestige:

gov.administrators or physicists? Briefly justify your answer. [2 marks]

Reminder: The data contains information collected from a large number of

people averaged for each of 102 different professions.

e. Give a brief Executive Summary of the main conclusions of the

analysis of the Job Prestige Data in APPENDIX A. [9 marks]

Note: To describe the effect of log10income it will be extremely helpful

to note that a 0.30 increase in log10income is equivalent to doubling income.

2. This question refers to the m&m’s™ Data in Appendix B.

[Total 18 marks]

a. Comment on what the interaction plot at the start of Appendix B reveals.

[2 marks]

b. Write brief Methods and Assumption Checks of the analysis of the

m&m’s™ Data in Appendix B. [6 marks]

c. Give a brief Executive Summary of the main conclusions of the

m&m’s™ Data in Appendix B. [10 marks]

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3. This question refers to the Viral pock data in Appendix C.

[Total 10 marks]

a. Comment on what the plot at the start of Appendix C reveals. [2 marks]

b. Write down an equation of the final model fitted to the data, just as you would

for a Method and Assumption Checks section. [2 marks]

c. Give a brief Executive Summary of the main conclusions of the

analysis of the Viral pock data in Appendix C. [6 marks]

4. This question refers to the Turbines data in Appendix D.

[Total 10 marks]

a. Write brief Methods and Assumption Checks notes of the analysis of the

Turbines data in Appendix D. [4 marks]

b. Give a brief Executive Summary of the main conclusions of the

analysis of the Turbines data in Appendix D. [6 marks]

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5. This question refers to the Nitrous Oxide Data in Appendix E.

[Total 14 marks]

a. Comment on the time series plot at the start of Appendix E (Global atmospheric

concentration of Nitrous Oxide). [2 marks]

b. Comment on the Holt-Winters coefficients. [2 marks]

c. Is the Holt-Winters model a good model of the N2O data? Justify your answer.

[1 mark]

d. Using the Holt-Winters model, provide a 95% prediction interval for the

N2O level in May, 2020. [2 marks]

e. Comment on the diagnostic plots for the model SF.fit1. Why was this model

rejected? [2 marks]

f. Comment on the diagnostic plots for the model SF.fit2. Are the underlying

assumptions satisfied? Briefly explain. [2 marks]

g. Using model SF.fit2, write an equation to calculate the estimated N2O for

November, 2019 . This is the estimated equation, not the model equation, so

use the estimated values for the coefficients and substitute appropriate values

for variables. You DO NOT need to calculate the final value. [3 marks]

APPENDICES BOOKLET FOLLOWS

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APPENDICES BOOKLET

CONTENTS

Appendix Name Pages

A Job Prestige Data 8–13

B m&m’s™ Data 14–18

C Viral pock data 19–21

D Turbines data 22–25

E Nitrous Oxide Data 26–32

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Appendix A Job Prestige Data

Employment data was collected from a large number of people working in Canada in

1971 across 102 different professions. For each profession, the data was combined and

average education and income levels were calculated. The prestige score and percentage

of women was also recorded for each profession.

We want to build a model to explain the perceived prestige of a profession in terms of

the other information collected. In particular, is there any evidence that professions with

a higher proportion of women are either more or less prestigious?

The variables are:

job The profession

prestige Pineo-Porter prestige score for the profession. A numeric measure of

prestige, the higher the score the more prestigious the profession.

education Average education (years) for people in the profession

income Average income ($) for people in the profession

log10income The logarithm to base 10 of income

women Percentage of women in the profession

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> pairs20x(Prestige.df[, c(1:5)])

> fit.prestige1 = lm(prestige ~ income + log10income + education

+ + women, data = Prestige.df)

> plot(fit.prestige1,which=1)

20 30 40 50 60 70 80

−20 −10

0 10 20

Fitted values

Residuals

Residuals vs Fitted

> summary(fit.prestige1)

Call:

lm(formula = prestige ~ income + log10income + education + women,

data = Prestige.df)

Residuals:

Min 1Q Median 3Q Max

-17.3357 -4.3855 -0.0915 4.3120 19.1733

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.097e+02 2.227e+01 -4.925 3.46e-06 ***

income 2.936e-05 3.736e-04 0.079 0.938

log10income 3.056e+01 6.557e+00 4.661 1.00e-05 ***

education 3.724e+00 3.669e-01 10.150 < 2e-16 ***

women 4.706e-02 3.012e-02 1.562 0.121

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.129 on 97 degrees of freedom

Multiple R-squared: 0.8351,Adjusted R-squared: 0.8283

F-statistic: 122.8 on 4 and 97 DF, p-value: < 2.2e-16

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> fit.prestige2 = lm(prestige ~ log10income + education + women,data=Prestige.df)

> summary(fit.prestige2)

Call:

lm(formula = prestige ~ log10income + education + women, data = Prestige.df)

Residuals:

Min 1Q Median 3Q Max

-17.364 -4.429 -0.101 4.316 19.179

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -110.9658 14.8429 -7.476 3.27e-11 ***

log10income 30.9427 4.4066 7.022 2.90e-10 ***

education 3.7305 0.3544 10.527 < 2e-16 ***

women 0.0469 0.0299 1.568 0.12

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.093 on 98 degrees of freedom

Multiple R-squared: 0.8351,Adjusted R-squared: 0.83

F-statistic: 165.4 on 3 and 98 DF, p-value: < 2.2e-16

> fit.prestige3 = lm(prestige ~ log10income + education, data = Prestige.df)

> summary(fit.prestige3)

Call:

lm(formula = prestige ~ log10income + education, data = Prestige.df)

Residuals:

Min 1Q Median 3Q Max

-17.0346 -4.5657 -0.1857 4.0577 18.1270

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -95.1940 10.9979 -8.656 9.27e-14 ***

log10income 26.3357 3.3090 7.959 2.94e-12 ***

education 4.0020 0.3115 12.846 < 2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.145 on 99 degrees of freedom

Multiple R-squared: 0.831,Adjusted R-squared: 0.8275

F-statistic: 243.3 on 2 and 99 DF, p-value: < 2.2e-16

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> plot(fit.prestige3,which=1)

20 30 40 50 60 70 80

−20 −10

0 10 20

Fitted values

Residuals

Residuals vs Fitted

67

46

82

> cooks20x(fit.prestige3)

observation number

0 20 40 60 80 100

0.00 0.05 0.10 0.15 0.20

Cook's Distance plot

Cook's distance

63

67 20

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> normcheck(fit.prestige3,main="histogram")

Theoretical Quantiles

Sample Quantiles

histogram

Residuals from lm(prestige ~ log10income + education)

> confint(fit.prestige3)

2.5 % 97.5 %

(Intercept) -117.016298 -73.371676

log10income 19.769848 32.901591

education 3.383824 4.620081

> head(Prestige.df)

prestige education women income log10income job

1 68.8 13.11 11.16 12351 4.091702 gov.administrators

2 69.1 12.26 4.02 25879 4.412947 general.managers

3 63.4 12.77 15.70 9271 3.967127 accountants

4 56.8 11.42 9.11 8865 3.947679 purchasing.officers

5 73.5 14.62 11.68 8403 3.924434 chemists

6 77.6 15.64 5.13 11030 4.042576 physicists

> head(predict(fit.prestige3, interval = "confidence"))

fit lwr upr

1 65.02953 62.71844 67.34062

2 70.08810 65.99493 74.18127

3 60.38808 58.51342 62.26274

4 54.47327 52.71791 56.22863

5 66.66736 64.20611 69.12861

6 73.86069 70.95756 76.76381

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Appendix B m&m’s™ Data

In this study researchers were interested in whether low-fat nutrition labels increase the

actual consumption of chocolate candies (m&m’s™) for overweight and normal weight

consumers.

The researchers set up the experiment as follows. They asked forty adult family members

participating in a university open house to serve themselves unusual colours of m&m’s™

(gold, teal, purple, and white), which were clearly labelled either as “New Colors of

Regular m&m’s™” (regular-label) or as “New ‘Low-Fat’ m&m’s™ ” (low-fat-label).

They then measured how many calories of m&m’s™ the participants ate. The participant

was also assessed as being normal weight or overweight based on their body mass index.

The variables in the data set are:

Calories the number of calories consumed per helping by the participant,

BMI whether the participant had a BMI that was normal or ow for overweight,

MM whether m&m’s™ were labelled regular or lowfat.

The researchers were interested in whether the designations of ”low-fat” resulted in

greater calorific consumption and, if so, did the effect of the label on the m&m’s™depend

on whether the respondent was of normal weight or deemed overweight.

They also want the following estimated:

• the average number of calories consumed per helping by normal weight people eating

regular m&m’s™.

• the difference(s) in average number of calories consumed per helping between normal

and overweight people - for each type of m&m’s™.

• the difference(s) in average number of calories consumed per helping between when the

m&m’s™are labelled regular and lowfat - for both weight groups.

Reference: “Can “Low-Fat” Nutrition Labels Lead to Obesity?”, Wansink, B & Chandon,

P., Journal of Marketing Research, November 2006.

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We wish to order the factor MM so that regular is first.

> MM.df=within(MM.df,{MM=factor(MM,levels=c("regular","lowfat"))})

> interactionPlots(Calories~BMI+MM,ylab="Calories consumed",data =MM.df)

Plot of 'Calories consumed'

by levels of 'BMI' and 'MM'

BMI

Calories consumed

100 150 200 250 300 350

normal ow

lowfa

regul

> MM.fit1 = lm(Calories~BMI*MM, data = MM.df)

> eovcheck(MM.fit1)

200 220 240 260 280

−80 −40

0 20 60

Fitted values

Residuals

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> normcheck(MM.fit1)

Theoretical Quantiles

Sample Quantiles

Residuals from lm(Calories ~ BMI * MM)

> cooks20x(MM.fit1)

observation number

0 10 20 30 40

0.00 0.10 0.20

Cook's Distance plot

Cook's distance

28 26

34

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> anova(MM.fit1)

Analysis of Variance Table

Response: Calories

Df Sum Sq Mean Sq F value Pr(>F)

BMI 1 11465 11465 12.3800 0.001195 **

MM 1 35593 35593 38.4339 3.765e-07 ***

BMI:MM 1 9030 9030 9.7507 0.003528 **

Residuals 36 33339 926

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

> summary(MM.fit1)

Call:

lm(formula = Calories ~ BMI * MM, data = MM.df)

Residuals:

Min 1Q Median 3Q Max

-77.50 -14.82 2.56 15.10 56.89

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 187.790 9.623 19.514 < 2e-16 ***

BMIow 3.810 13.609 0.280 0.78112

MMlowfat 29.610 13.609 2.176 0.03622 *

BMIow:MMlowfat 60.100 19.247 3.123 0.00353 **

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 30.43 on 36 degrees of freedom

Multiple R-squared: 0.6272,Adjusted R-squared: 0.5961

F-statistic: 20.19 on 3 and 36 DF, p-value: 7.604e-08

> confint(MM.fit1)

2.5 % 97.5 %

(Intercept) 168.272946 207.30705

BMIow -23.791283 31.41128

MMlowfat 2.008717 57.21128

BMIow:MMlowfat 21.065891 99.13411

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> summary2way(MM.fit1,page="nointeraction")

Tukey multiple comparisons of means

95% family-wise confidence level

Fit: aov(formula = fit)

$BMI

diff lwr upr p adj

ow-normal 33.86 14.34295 53.37705 0.0011955

$MM

diff lwr upr p adj

lowfat-regular 59.66 40.14295 79.17705 4e-07

> summary2way(MM.fit1,page="interaction")

Tukey multiple comparisons of means

95% family-wise confidence level

Fit: aov(formula = fit)

$`Comparisons within BMI`

diff lwr upr p adj

normal:lowfat-normal:regular 29.61 -7.043392 66.26339 0.1494601

ow:lowfat-ow:regular 89.71 53.056608 126.36339 0.0000007

$`Comparisons between BMI`

diff lwr upr p adj

ow:regular-normal:regular 3.81 -32.84339 40.46339 0.9922076

ow:lowfat-normal:lowfat 63.91 27.25661 100.56339 0.0002140

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Appendix C Viral pock data

When creating vaccines and medications for treating viruses, researchers need to be able

to produce large amounts of samples of the virus in petri dishes. A set quantity of a viral

medium is placed in a petri dish and left for the virus to grow. One question researchers

had was what was the effect of diluting the viral medium on the viral Activity.

They set up an experiment as follows. A supply of viral medium was produced. This was

then divided into 5 batches, each with a different level of dilution applied to them. Petri

dishes were randomly assigned to the batches and filled. After a fixed amount of time,

the viral activity in each petri dish was assess by counting the number of pock marks

that appeared. The dilution levels are actually in increasing powers of 2, so each increase

of 1 in dilution level d

¯

oubles the dilution and therefore h

¯

alves the concentration of the

viral medium.

The variables are:

Count a count of the number of pock marks recorded in the petri dish

Dilution the dilution level used for viral medium in the petri dish

The researcher has two questions they want answered. What is the estimated number of

pock marks you would expect to see at dilution level 0 and what is the effect of doubling

the dilution level?

> plot(Count~Dilution, data=Pock.df)

0 1 2 3 4

0 50 100 150 200 250

Dilution

Count

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> fit.glm<- glm(Count ~ Dilution,family = poisson, data=Pock.df )

> summary(fit.glm)

Call:

glm(formula = Count ~ Dilution, family = poisson, data = Pock.df)

Deviance Residuals:

Min 1Q Median 3Q Max

-6.058 -2.031 -0.462 1.373 5.357

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 5.26793 0.02255 233.60 <2e-16 ***

Dilution -0.68094 0.01544 -44.09 <2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 2729.74 on 47 degrees of freedom

Residual deviance: 290.44 on 46 degrees of freedom

AIC: 562.42

Number of Fisher Scoring iterations: 4

> 1-pchisq(deviance(fit.glm), df.residual(fit.glm))

> fit.quasi=glm(Count ~ Dilution,family = quasipoisson, data=Pock.df )

> summary(fit.quasi)

Call:

glm(formula = Count ~ Dilution, family = quasipoisson, data = Pock.df)

Deviance Residuals:

Min 1Q Median 3Q Max

-6.058 -2.031 -0.462 1.373 5.357

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 5.26793 0.05678 92.78 <2e-16 ***

Dilution -0.68094 0.03888 -17.51 <2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for quasipoisson family taken to be 6.338981)

Null deviance: 2729.74 on 47 degrees of freedom

Residual deviance: 290.44 on 46 degrees of freedom

AIC: NA

Number of Fisher Scoring iterations: 4

> confint(fit.quasi)

2.5 % 97.5 %

(Intercept) 5.1548909 5.3775069

Dilution -0.7583492 -0.6058599

> exp(confint(fit.quasi))

2.5 % 97.5 %

(Intercept) 173.2769014 216.4819000

Dilution 0.4684391 0.5456051

> 100*(exp(confint(fit.quasi))-1)

2.5 % 97.5 %

(Intercept) 17227.69014 21548.19000

Dilution -53.15609 -45.43949

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Appendix D Turbines data

An experiment was set up to test the life of turbine wheels. Of interest was the likelihood

of the turbine wheels developing fissures (narrow cracks) that would require maintenance

to prevent more serious damage.

In an experiment, turbine wheels were randomly allocated to be run for a number of hours.

Each wheel was then examined to see whether or not it had developed any fissures.

The data collected contains the variables:

Hours the number of hours the turbines were run, a numeric vector,

Turbines the number of turbine wheels that where tested at that number of hours,

Fissures the number of turbine wheels that developed fissures at that number of hours.

The researchers wanted to know the effects of time on the likelihood of turbine wheels

developing fissures. They also wanted to estimate how many hours the turbine could run

before the probability of the wheel developing fissures exceeded 25%.

> Turbines.df

Hours Turbines Fissures

> plot(Fissures/Turbines~Hours,ylab="probability",data=Turbines.df,

+ main="Probability of having a fissure by number of hours")

1000 2000 3000 4000

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Probability of having a fissure by number of hours

Hours

probability

> turbines.glm <- glm( Fissures/Turbines ~ Hours, family=binomial,

+ weights=Turbines, data=Turbines.df)

> plot(turbines.glm,which=1)

−3 −2 −1 0

−1

0 1 2

Predicted values

Residuals

Residuals vs Fitted

9

1

7

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> summary(turbines.glm)

Call:

glm(formula = Fissures/Turbines ~ Hours, family = binomial, data = Turbines.df,

weights = Turbines)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.5055 -0.7647 -0.3036 0.4901 2.0943

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -3.9235966 0.3779589 -10.381 <2e-16 ***

Hours 0.0009992 0.0001142 8.754 <2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 112.670 on 10 degrees of freedom

Residual deviance: 10.331 on 9 degrees of freedom

AIC: 49.808

Number of Fisher Scoring iterations: 4

> 1-pchisq(deviance(turbines.glm), df.residual(turbines.glm))

[1] 0.3243236

> confint(turbines.glm)

2.5 % 97.5 %

(Intercept) -4.704479229 -3.219091395

Hours 0.000783381 0.001231879

> exp(100*confint(turbines.glm))

2.5 % 97.5 %

(Intercept) 4.864778e-205 1.572668e-140

Hours 1.081488e+00 1.131097e+00

> 100*(exp(100*confint(turbines.glm))-1)

2.5 % 97.5 %

(Intercept) -100.000000 -100.00000

Hours 8.148825 13.10969

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> Pred.data = data.frame(Hours = seq(400,5000,by=100))

> Turbine.pred = predict(turbines.glm, Pred.data)

> Prob=round(exp(Turbine.pred) / (1 + exp(Turbine.pred)),3)

> names(Prob)=seq(400,5000,by=100)

> plot(Fissures/Turbines~Hours,ylab="probability", main="Probability of

+ turbine having a fissure by number of hour",data=Turbines.df)

> lines(seq(400,5000,by=100),Prob)

1000 2000 3000 4000

0.0 0.2 0.4 0.6

Probability of

turbine having a fissure by number of hour

Hours

probability

> Prob

400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

0.029 0.032 0.035 0.038 0.042 0.046 0.051 0.056 0.062 0.068 0.074 0.081 0.089

1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900

0.098 0.107 0.117 0.127 0.139 0.151 0.164 0.179 0.194 0.210 0.227 0.245 0.264

3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200

0.284 0.304 0.326 0.348 0.371 0.395 0.419 0.444 0.468 0.493 0.518 0.543 0.568

4300 4400 4500 4600 4700 4800 4900 5000

0.592 0.616 0.639 0.662 0.684 0.705 0.726 0.745

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Appendix E Nitrous Oxide Data

The following data are the monthly average global atmospheric concentrations of N2O

from January 2008 to October 2019 in parts per billion (ppb).

> N2O.ts = ts(N2O.df$N2O,start=2008,frequency=12)

> N2O.ts

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2008 321.3 321.4 321.4 321.4 321.4 321.5 321.4 321.4 321.5 321.6 321.8 322.0

2009 322.2 322.3 322.3 322.1 322.0 322.0 322.0 322.1 322.3 322.5 322.7 322.8

2010 322.9 322.9 322.9 323.0 323.0 323.0 323.0 323.1 323.3 323.5 323.7 323.9

2011 324.0 324.1 324.1 324.1 324.1 324.0 324.1 324.1 324.2 324.4 324.6 324.8

2012 324.9 325.0 325.0 324.9 324.9 324.9 324.9 325.0 325.1 325.3 325.4 325.5

2013 325.6 325.6 325.7 325.7 325.8 325.9 326.0 326.0 326.1 326.2 326.4 326.5

2014 326.6 326.7 326.7 326.8 326.8 326.9 327.0 327.2 327.3 327.5 327.7 327.9

2015 328.0 328.1 328.0 328.0 327.9 328.0 328.0 328.1 328.2 328.4 328.6 328.8

2016 328.9 328.9 328.9 328.9 328.9 328.9 328.9 328.8 328.9 329.0 329.2 329.4

2017 329.5 329.5 329.5 329.5 329.5 329.6 329.7 329.8 329.9 330.0 330.2 330.3

2018 330.4 330.6 330.7 330.7 330.7 330.7 330.7 330.9 331.1 331.3 331.5 331.7

2019 331.8 331.7 331.7 331.6 331.6 331.7 331.9 331.9 331.9 332.1

> plot(N2O.ts,xlab="month",ylab="ppb",main="Global atmospheric concentration

+ of Nitrous Oxide - Jan 2008 to Oct 2019")

Global atmospheric concentration

of Nitrous Oxide − Jan 2008 to Oct 2019

month

ppb

2008 2010 2012 2014 2016 2018 2020

322 324 326 328 330 332

Page 26 of 32

STATS 201/208

> HW.fit = HoltWinters(N2O.ts)

> HW.fit

Holt-Winters exponential smoothing with trend and additive seasonal component.

Call:

HoltWinters(x = N2O.ts)

Smoothing parameters:

alpha: 0.8527811

beta : 0.01102914

gamma: 1

Coefficients:

[,1]

a 332.17880227

b 0.07663861

s1 0.01925176

s2 0.10884812

s3 0.18119490

s4 0.22432467

s5 0.23304180

s6 0.14142845

s7 0.01506802

s8 -0.08650202

s9 -0.14807969

s10 -0.18645879

s11 -0.16506783

s12 -0.07880227

Page 27 of 32

STATS 201/208

> HW.pred = predict(HW.fit,n.ahead=12,prediction.interval=T)

> plot(HW.fit,HW.pred,main="Global atmospheric concentration of Nitrous Oxide")

Global atmospheric concentration of Nitrous Oxide

Time

Observed / Fitted

2010 2012 2014 2016 2018 2020

322 326 330 334

> HW.pred

fit upr lwr

Nov 2019 332.2747 332.4768 332.0726

Dec 2019 332.4409 332.7077 332.1741

Jan 2020 332.5899 332.9096 332.2702

Feb 2020 332.7097 333.0756 332.3437

Mar 2020 332.7950 333.2028 332.3873

Apr 2020 332.7801 333.2266 332.3336

May 2020 332.7303 333.2132 332.2475

Jun 2020 332.7054 333.2227 332.1881

Jul 2020 332.7205 333.2707 332.1703

Aug 2020 332.7587 333.3406 332.1768

Sep 2020 332.8568 333.4693 332.2442

Oct 2020 333.0197 333.6620 332.3773

Page 28 of 32

STATS 201/208

Note: Define Month as a factor for month of the year coded as 1 for January, 2 for

February, etc. Define Time as a numeric variable with values 1 for January 2008, 2 for

February 1980, ..., 142 for October 2019.

> Time = 1:142

> Month = factor(c(rep(1:12,11),(1:10)))

> SF.fit1 = lm(N2O.ts~Time+Month)

> plot.ts(residuals(SF.fit1),main="Residual Series")

Residual Series

Time

residuals(SF.fit1)

0 20 40 60 80 100 120 140

−0.3 −0.1 0.1 0.2 0.3

Page 29 of 32

STATS 201/208

> acf(residuals(SF.fit1))

0 5 10 15 20

−0.5 0.0 0.5 1.0

Lag

ACF

Series residuals(SF.fit1)

> SF.fit2 = lm(N2O.ts[-1]~Time[-1]+Month[-1]+N2O.ts[-142])

> plot.ts(residuals(SF.fit2),main="Residual Series")

Residual Series

Time

residuals(SF.fit2)

0 20 40 60 80 100 120 140

−0.15 −0.05 0.05 0.15

Page 30 of 32

STATS 201/208

> acf(residuals(SF.fit2))

0 5 10 15 20

−0.2 0.2 0.6 1.0

Lag

ACF

Series residuals(SF.fit2)

> normcheck(SF.fit2, main = "")

Theoretical Quantiles

Sample Quantiles

Residuals from lm(N2O.ts[−1] ~ Time[−1] + Month[−1] + N2O.ts[−142])

Page 31 of 32

STATS 201/208

> summary(SF.fit2)

Call:

lm(formula = N2O.ts[-1] ~ Time[-1] + Month[-1] + N2O.ts[-142])

Residuals:

Min 1Q Median 3Q Max

-0.163659 -0.042180 0.005527 0.041432 0.157619

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 30.135431 11.598702 2.598 0.010480 *

Time[-1] 0.007490 0.002867 2.613 0.010067 *

Month[-1]2 -0.045989 0.025966 -1.771 0.078939 .

Month[-1]3 -0.098020 0.025925 -3.781 0.000239 ***

Month[-1]4 -0.129731 0.025956 -4.998 1.88e-06 ***

Month[-1]5 -0.130448 0.026398 -4.941 2.40e-06 ***

Month[-1]6 -0.088718 0.027186 -3.263 0.001414 **

Month[-1]7 -0.092309 0.027628 -3.341 0.001096 **

Month[-1]8 -0.070900 0.028129 -2.521 0.012958 *

Month[-1]9 -0.022152 0.028311 -0.782 0.435404

Month[-1]10 0.031276 0.027794 1.125 0.262593

Month[-1]11 0.063619 0.027393 2.322 0.021800 *

Month[-1]12 0.046722 0.026635 1.754 0.081815 .

N2O.ts[-142] 0.906416 0.036144 25.078 < 2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.06205 on 127 degrees of freedom

Multiple R-squared: 0.9997,Adjusted R-squared: 0.9996

F-statistic: 2.938e+04 on 13 and 127 DF, p-value: < 2.2e-16

> N2O.ts[142]

[1] 332.1

Page 32 of 32