Homework 5, MAT 327/782, Fall 2018
For the R computations, submit the R commands you used andtheir output, either by taking a screenshot or by copying them intoa 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, sampleskewness, and sample kurtosis of nhtemp.
(b) What can you say about the spread and shape of the nhtempdata using the information from part (a)? Hint: RememberChebychev’s rule and rules of thumb.
(c) Plot a histogram and boxplot of nhtemp. Are these plotswhat you expected from part (b)? Why or why not?
(d) nhtemp is a time series. Plot it as an index plot. Doesthis plot give any new information? Do the histogram and/or boxplotshow anything about the data that is not seen or hard to see in theindex plot?
2. The R dataset co2 contains 468 measurements of the amountof carbon dioxide (CO2) in the atmosphere. The measurements weretaken 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, sampleskewness, and sample kurtosis of co2.
What can you say about the spread and shape of the nhtemp datausing the information from part (a)? Hint: Remember Chebychev’srule and rules of thumb.
Plot a histogram and boxplot of co2. Are these plots what youexpected from part (b)? Why or why not?
co2 is a time series. Plot it as an index plot. Does this plotgive any new information? Do the histogram and/or boxplot showanything about the data that is not seen or hard to see in theindex plot?
The R dataset islands contains the areas (in 1000s of squaremiles) of land- masses more than 10,000 square miles. Plot theboxplot. What are the outliers?
Classify the outliers in the islands dataset as potential orsuspected.
4. For MAT 782 only. Prove that skewness is location and scaleindependent. That is, show that for data x1, x2, ..., xn and forany 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.
