#3) First I will begin by loading the data.

The Student Number data is integers from 1 to 20. This line of code assignes the values of 1 through 20 to the variable Student_Number.

Student_Number <- c(1:20)

The Credit hour data is a changing variable. For this I will enter the data on a single line, because 20 numbers isn’t too many. I could use a csv file or something but this will be fine.

Credit_Hours <- c(2,7,9,9,8,11,6,8,12,11,6,5,9,13,10,6,9,6,9,10)

Now I will combine the data into a data frame.

StudentDatabase <- data.frame(Student_Number, Credit_Hours)

The Data is in.

  1. Determine the mean, median, and mode for this sample of data. Write a sentence explaining what each means.

I will now present the mean Credit_Hours of the StudentDatabase data. R has a simple function for this. The function returns the mean of variable Credit_Hours as part of the StudentDatabase data frame.

mean(StudentDatabase$Credit_Hours)
[1] 8.3

8.3 is the mean number of Credit Hours.

Median:

median(StudentDatabase$Credit_Hours)
[1] 9

The median number of Credit Hours is 9.

The mode was really harder than it should be. first, the base package of R will perform mean, and median functions, but not mode. There were methods online to make it work but I really wanted the convenience of just typing mode. I found a package for descriptive statistics which had a function for mode with an uppercase ‘M’.

To find the mode I load the DescTools package then use it. Mode:

library(DescTools)
Mode(StudentDatabase$Credit_Hours)
[1] 9

the mode number of Credit Hours is 9.

b)It has been suggested that graduate students in business take fewer credits per quarter than the typical graduate student at this university. The mean for all graduate students is 9.1 credit hours per quarter, and the data are normally distributed. Set up the approperiate null and alternative hypothesis, and determine wheather the null hypothesis can be rejected at a 95% confidence level.

This function performs a t-test for the mu of 9.1, and the alternative hypothesis is that the mean number of credit hours taken by business students is less than that of others.

t.test(StudentDatabase$Credit_Hours, mu = 9.1,alternative = "less")

    One Sample t-test

data:  StudentDatabase$Credit_Hours
t = -1.3563, df = 19, p-value = 0.09545
alternative hypothesis: true mean is less than 9.1
95 percent confidence interval:
     -Inf 9.319888
sample estimates:
mean of x 
      8.3

the t-critical value is -1.729. The t-statistic is -1.3563. The test statistic falls within the Fail to Reject region. The test fails to reject the null hypothesis that business students take greater than or equal credit hours as the typical graduate at this university, at a 5% error level.

#10 Home Sales are often considered an important determinant of the future health of the economy. Thus, there is widespread interest in being able to forecast total houses sold (THS). Quarterly data for THS are shown in the following table in thousands of units:

First I started with a CSV file but thought just pasting the values for homes sold into one line of code was cooler, so I did that. This code segment loads the Total Houses Sold per Quarter into a variable THS, then puts the data into a data frame, THSDataFrame. Next I call the zoo library, it has a good function for quarterly data. I put the data frame of THS data into the variable z, starting the first quarter of 1989, and a quarterly frequency.

THS <- c(161,179,172,138,153,152,130,100,121,144,126,116,159,158,159,132,154,183,169,160,178,185,165,142,154,185,181,145,192,204,201,161,211,212,208,174,220,247,218,200,227,248,221,185,233,226,219,199,251,243,216,199,240,258,254,220,256,299,294,239,314,329,292,268,328,351,326,278,285,300,251,216,214,240)
THSDataFrame <- as.data.frame(THS)

library(zoo)

z <- zooreg(THSDataFrame$THS, start = as.yearqtr("1989-1"), frequency = 4)

The Data is Loaded in.

a)Prepare a time-series plot of THS. Describe what you see in this plot in terms of trend and seasonality.

This line plots the time series of the THS data, as the variable ‘z’. Including lables for the x and y axis. labeling the x axis “YEAR”, and the y axis “THS”.

plot.ts(z, xy.labels = "TRUE", xlab = "YEAR", ylab = "THS")

The trend appears positive, and there appears to be seasonality. Less homes are sold around the end or begining of the year.

  1. Calculate and plot the first twelve autocorrelation coefficients for PHS. What does this autocorrelation structure suggest about the trend?

This function calculates and plots the first 12 autocorrelation coefficients.

acf(THSDataFrame$THS,lag.max = 12, type = "correlation")

The trend appears to be positive.

c)De-trend the data by calculating first differences. Calculate and plot the first eight autocorrelation coefficients for DTHS. Is there a trend in DTHS?

This code calculates and plots the first 8 autocorrelation coefficients of the first difference of the THS data frame. I did it this way because the plots did not work well with the zoo library.


acf(diff(THSDataFrame$THS,differences = 1),lag.max = 8)

There still appears to be a trend.

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