Question

Instructions

First, download real estate data from the city of Ames, Iowa: download.file("http://www.openintro.org/stat/data/ames.RData", destfile = "ames.RData") load("ames.RData") Now write code in R to answer the following questions. Make sure the script that you turn in includes the code that you write, the output that you get from that code (in a comment), and a sentence or more answering the question if there was one (also in a comment). Consider the variable Gr.Liv.Area of the ames data frame, which represents the above ground living area of the house in square feet. Calculate the mean of the values. Now assume that you'd like to get an idea of the average living area, but you have time to check only 50 of the 2930 houses sold in Ames. Take a sample of size 50 from Gr.Liv.Area without replacement and save it in the variable ames.gr.liv.area.sample. Calculate the mean of the sample. Is the mean for the sample different from the mean for the entire population? Why or why not? Use the par function with the mfrow argument to set up the plotting space with 2 rows and 1 column. Then calculate the range of Gr.Liv.Area in the data and save it in a variable named area.xlim. Plot a histogram of Gr.Liv.Area in the first row, setting its x axis limits to match area.xlim. Then use abline to draw a red vertical line representing the mean of the distribution on top of the histogram. Plot a histogram of the sample of size 50 in the second row, and draw a red vertical line for its mean. Make sure that the x axis limits are the same as those of the other histogram. How different is the plot of the sample from the plot of the full population? Each time we take a sample, we'll get a different mean. Use replicate to repeat the following process 5000 times: Take a sample of Gr.Liv.Area of size 10 Take the mean. Save these 5000 means in a variable named area.means.10. Plot a histogram of these means and describe the shape of the distribution. Use replicate two more times, once taking 5000 samples of Gr.Liv.Area of size 50, and once taking 5000 samples of size 100. Save your results in two variables area.means.50 and area.means.100. Set up the plotting space with 3 rows and 1 column. Then calculate the range of area.means.10 and save it in a variable named area.means.10.xlim. Plot histograms for area.means.10, area.means.50, and area.means.100 in the three rows. Set all of their x axis limits to match area.means.10.xlim, and set all of their breaks to 20. If you wanted to estimate a mean, and you had to choose between a sample size of 10, 50, and 100, which one would you choose? Explain how the histograms here support your choice. What would the distribution of means look like if you took samples of size 1? What would it look like if you took samples of size 2930?

Answer 1

download.file("http://www.openintro.org/stat/data/ames.RData", destfile = "ames.RData")
load("ames.RData")

1)

#MEan of the variable Gr.Liv.Area
mean(ames$Gr.Liv.Area) # Output: 1499.69 #

2)

#MEan of the sample of 50 for variable variable Gr.Liv.Area
ames.gr.liv.area.sample<-sample(ames$Gr.Liv.Area,50, replace = FALSE)

mean(ames$Gr.Liv.Area[ames.gr.liv.area.sample]) # Output: 1515.64 #
#Means of the population and sample are not very different. The sample is representative of the population

3)

# Plot set plotting space
par(mfrow=c(2,1))

#Save range of variable Gr.Liv.Area
area.xlim = range(ames$Gr.Liv.Area)

4)

#Plot histogram for variable Gr.Liv.Area
hist(ames$Gr.Liv.Area, xlim=area.xlim, main = "Histogram for Living Area")
abline(v=mean(ames$Gr.Liv.Area),col="red", lwd="4")

5)

#Plot histogram for sample of 50 for variable variable Gr.Liv.Area
hist(ames$Gr.Liv.Area[ames.gr.liv.area.sample], xlim=area.xlim, main = "Histogram for Living Area (Sample of 50)")
abline(v=mean(ames$Gr.Liv.Area[ames.gr.liv.area.sample]),col="red", lwd="4")

#The histograms for the variable Gr.Liv.Area for the population and sample are similar

6)

replicate(5000,mean(ames$Gr.Liv.Area[sample(ames$Gr.Liv.Area,10)]))

7)

area.means.10<-replicate(5000,mean(ames$Gr.Liv.Area[sample(ames$Gr.Liv.Area,10)]))
hist(area.means.10)
#The shape of the histogram is that of a normal distribution