Question

Homework 5, MAT 327/782, Fall 2018

For the R computations, submit the R commands you used and their output, either by taking a screenshot or by copying them into a text file. Submit your graph as a pdf or image file.

All graphs should have labeled axes and a title.

1. The R dataset nhtemp contains the mean annual temperature (in F) in New Haven, Connecticut from 1912 to 1971.

(a) In R, compute the mean, sample standard deviation, sample skewness, and sample kurtosis of nhtemp.

(b) What can you say about the spread and shape of the nhtemp data using the information from part (a)? Hint: Remember Chebychev’s rule and rules of thumb.

(c) Plot a histogram and boxplot of nhtemp. Are these plots what you expected from part (b)? Why or why not?

(d) nhtemp is a time series. Plot it as an index plot. Does this plot give any new information? Do the histogram and/or boxplot show anything about the data that is not seen or hard to see in the index plot?

2. The R dataset co2 contains 468 measurements of the amount of carbon dioxide (CO2) in the atmosphere. The measurements were taken monthly from 1959 to 1997, and are in parts per million (ppm).

(a) (b)

(c) (d)

3. (a)

(b)

In R, compute the mean, sample standard deviation, sample skewness, and sample kurtosis of co2.

What can you say about the spread and shape of the nhtemp data using the information from part (a)? Hint: Remember Chebychev’s rule and rules of thumb.

Plot a histogram and boxplot of co2. Are these plots what you expected from part (b)? Why or why not?

co2 is a time series. Plot it as an index plot. Does this plot give any new information? Do the histogram and/or boxplot show anything about the data that is not seen or hard to see in the index plot?

The R dataset islands contains the areas (in 1000s of square miles) of land- masses more than 10,000 square miles. Plot the boxplot. What are the outliers?

Classify the outliers in the islands dataset as potential or suspected.

4. For MAT 782 only. Prove that skewness is location and scale independent. That is, show that for data x1, x2, ..., xn and for any non-zero constants c, a ∈ R, if yi =cxi +aforall1≤i≤n,then

1 ?ni=1(xi − x ̄)3 1 ?ni=1(yi − y ̄)3 n s3 =n s3y

where y ̄ is the mean and sy is the standard deviation of y1, y2, ..., yn.

Answer 1

1) The R-Code to read the data:

year = c(1912:1971)
dt = data.frame(cbind(year,nhtemp))

1.a)  the mean, sample standard deviation, sample skewness, and sample kurtosis of nhtemp is obtained as:

mean(nhtemp)
51.16

sd(nhtemp)
1.265608

library(e1071)
skewness(nhtemp)
-0.07178758

kurtosis(nhtemp)
0.3832752

1.b) The skewness of mean annual temperature (in F) is -0.07178758 which indicates that the distribution of the mean annual temperature (in F) in New Haven, Connecticut is skewed towards the left. While, the kurtosis of mean annual temperature (in F) is 0.3832752, which indicates that the distribution of the mean annual temperature (in F) in New Haven, Connecticut is leptokurtic.

1.c) histogram and boxplot of nhtemp is given by:

hist(nhtemp, main = paste("Histogram of the Mean Annual Temperature (in F)
in New Haven, Connecticut"), xlab = "Mean Annual Temperature")

Thus, the histogram plot shows that the distribution is slightly skewed towards left and is leptokurtic.

boxplot(nhtemp, main = paste("Boxplot of the Mean Annual Temperature (in F)
in New Haven, Connecticut"))

From the boxplot we observe that there are 4 outliers in total, 2 of which are less than 3/2 times of the Inter Quartile Range and the other 2 is greater than 3/2 times of the Inter Quartile Range. The Inter Quartile Rangeis slightly shifted upwards indicating skewness towards left and leptokurtic.

1.d) The index Plot of nhtemp is given by:

plot(years, nhtemp, type = "l", main = paste("Index Plot of the Mean Annual Temperature (in F)
in New Haven, Connecticut"), xlab = "Years", ylab = "Mean Annual Temperature")

The index plot of the given data gives us the trend or pattern of the mean annual temperature where the boxplot and histogram where giving us the distribution of the mean annual. temperature.

2.a)  the mean, sample standard deviation, sample skewness, and sample kurtosis of co2 is obtained as:

mean(co2)
337.0535

sd(co2)
14.96622

library(e1071)
skewness(co2)
0.2419156

kurtosis(co2)
-1.223013

2.b) The skewness of the amount of carbon dioxide (CO2) in the atmosphere is 0.2419156 which indicates that the distribution of the amount of carbon dioxide (CO2) in the atmosphere is skewed towards the right. While, the kurtosis of amount of carbon dioxide (CO2) in the atmosphere is -1.223013, which indicates that the distribution of the amount of carbon dioxide (CO2) in the atmosphere is platykurtic.

1.c) histogram and boxplot of co2 is given by:

hist(co2, main = paste("Histogram of the Amount of Carbon Dioxide
(CO2) in the atmosphere"), xlab = "Amount of Carbon Dioxide")

Thus, the histogram plot shows that the distribution is slightly skewed towards right and is platykurtic.

boxplot(co2, main = paste("Boxplot of the Amount of Carbon Dioxide
(CO2) in the atmosphere"))

From the boxplot we observe that there are no outliers. The Inter Quartile Rangeis slightly shifted downwards indicating skewness towards right and platykurtic.

2.d) The index Plot of nhtemp is given by:

plot(co2, type = "l", main = paste("Index Plot of the Amount of Carbon Dioxide
(CO2) in the atmosphere"), xlab = "Years", ylab = "Amount of Carbon Dioxide (CO2)")

The index plot of the given data gives us the trend or pattern of the Amount of Carbon Dioxide (CO2) in the atmosphere where the boxplot and histogram where giving us the distribution of the Amount of Carbon Dioxide (CO2).

3.a) The boxplot of co2 is given by:

boxplot(islands, main = paste("Boxplot of the Areas in thousands of square miles of the landmasses"))

From the boxplot we observe that there are many outliers. The Inter Quartile Range is totally shifted downwards indicating skewness towards right and platykurtic.

The outliers are :

Africa : 11506, Antarctica: 5500, Asia : 16988, Australia : 2968, Europe : 3745, North America : 9390.