Question:

Q1. Use the below R command to randomly sample 15 data points from beaver1$temp and assign them to a vector, temp. (60 points - 6 points for each sub-question)

temp

A. Display the vector, temp. Since values in temp is randomly sampled, it is necessary that you report the data values in temp so that we can evaluate your answers for the downstream questions.

B. Compute the mean and median of temp. Convert the data in temp into integers and save them into a vector, temp_int. Compute the mode of temp_int.

C. Compute the variance and the standard deviation of temp.

D. Compute the five number summary, the interquartile range, and outliers, if any in temp.

E. Compute the standardized version (z-scores) of temp.

F. Identify values that are less than the first quartile of the data.

G. Using indexing, show the expression for accessing the first item and the last one of temp. The code should work for a vector of any size (in another word, no hard coding).

H. Create matrix of size 3 x 5 using the data in temp. The first five values belong to the first row of the matrix. Assign the result to the variable, temp.matrix, and display the result.

I. Show the code for displaying the first and the last columns of the matrix. The code should work for matrices of any size (in another word, no hard coding).

J. Assign row names for the temp.matrix as Row1, Row2, and Row3. Assign column names as Day1, Day2, .... The code should work for matrices of any size.

Q2. Using function data() to load R data set airquality. (40 points - 8 points for each sub-question)

A. How many variables are there? How many observations are there?

B. Are there missing values in the data set? If so, clean up the data by removing those observations with missing values and save the cleaned data set. Report the number of observations in the cleaned data set.

C. Show the mean, median, mode, 1st and 3rd quantiles of variables Temp and Wind in the cleaned data set.

D. In the cleaned data set, how many days the Temp is greater than 80 and how many days the Wind is less than 6?

E. The numbers in column Temp represent temperature in Fahrenheit. Modify the cleaned data set by adding a new column, TempC, showing temperature in Celsius. Display the new resulting data frame. The equation to convert Fahrenheit to Celsius: T(C) = (T(F) - 32) 5/9

Question Based on Ch.13 Non-parametric procedures are usually not our rst choice among statistical procedures because: \"1' they are less powerful than parametric procedures. A the probability of making a Type II error is too high. /"\\ the probability of making a Type I error is too high. A non-parametric procedures are not inferential statistics. Prove that if T is a normal transformation with spectral decomposition T = 2 ME\"; according to Spectral Theorem for Normal Transformations, then T' = 2 LEE\": is a spectral decomposition ofT'. *Spectral Theorem for Normal Transformation: Let T be a normal transformation on a 1rector space if over (I 1with a hermitian scalar product, and let {a1 , . . , as } be the distinct eigenvalues of T. Then there exist polynomials {flirt}, . .. fsnlfxl} in Clx] such that the transformation Ei = fitf}, l s: i c: s, are self-adjoint and satisfyr the conditions 1=E1+. . . + E5r and El-Ej = I] if i is not equal to i. More over T can he expresse din the form T = a1E1 +. . . + aSEE. Such a decomposition of T is called a spectral decomposition ofT. 1. What are the most common reasons you would select a non-parametric test over its parametric alternative? 2. Discuss the issue of statistical power of non-parametric methods as compared with their parametric counterparts? Why will parametric statistics always be more powerful than their non-parametric counterparts? 3. While the is a very useful non-parametric method, it is, by far, not the only non-parametric method. There are, in fact, a non-parametric counterpart of each parametric test. Do some research on your own and fill out the rest of the table. Then give a brief description of each non-parametric method. Parametric Statistic Non-Parametric Statistic What is it used for? Correlated Groups t-test Independent Groups t-test One Way ANOVA