Data606 Lab 4a
Ritesh Lohiya
March 15, 2018
Foundations for statistical inference - Sampling distributions In this lab, we investigate the ways in which the statistics from a random sample of data can serve as point estimates for population parameters. We’re interested in formulating a sampling distribution of our estimate in order to learn about the properties of the estimate, such as its distribution.
The data We consider real estate data from the city of Ames, Iowa. The details of every real estate transaction in Ames is recorded by the City Assessor’s office. Our particular focus for this lab will be all residential home sales in Ames between 2006 and 2010. This collection represents our population of interest. In this lab we would like to learn about these home sales by taking smaller samples from the full population. Let’s load the data.
Set my working directory as C:-library.46064a
Load Data Set:
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionload("ames.RData")
head(ames)## Order PID MS.SubClass MS.Zoning Lot.Frontage Lot.Area Street Alley
## 1 1 526301100 20 RL 141 31770 Pave <NA>
## 2 2 526350040 20 RH 80 11622 Pave <NA>
## 3 3 526351010 20 RL 81 14267 Pave <NA>
## 4 4 526353030 20 RL 93 11160 Pave <NA>
## 5 5 527105010 60 RL 74 13830 Pave <NA>
## 6 6 527105030 60 RL 78 9978 Pave <NA>
## Lot.Shape Land.Contour Utilities Lot.Config Land.Slope Neighborhood
## 1 IR1 Lvl AllPub Corner Gtl NAmes
## 2 Reg Lvl AllPub Inside Gtl NAmes
## 3 IR1 Lvl AllPub Corner Gtl NAmes
## 4 Reg Lvl AllPub Corner Gtl NAmes
## 5 IR1 Lvl AllPub Inside Gtl Gilbert
## 6 IR1 Lvl AllPub Inside Gtl Gilbert
## Condition.1 Condition.2 Bldg.Type House.Style Overall.Qual Overall.Cond
## 1 Norm Norm 1Fam 1Story 6 5
## 2 Feedr Norm 1Fam 1Story 5 6
## 3 Norm Norm 1Fam 1Story 6 6
## 4 Norm Norm 1Fam 1Story 7 5
## 5 Norm Norm 1Fam 2Story 5 5
## 6 Norm Norm 1Fam 2Story 6 6
## Year.Built Year.Remod.Add Roof.Style Roof.Matl Exterior.1st Exterior.2nd
## 1 1960 1960 Hip CompShg BrkFace Plywood
## 2 1961 1961 Gable CompShg VinylSd VinylSd
## 3 1958 1958 Hip CompShg Wd Sdng Wd Sdng
## 4 1968 1968 Hip CompShg BrkFace BrkFace
## 5 1997 1998 Gable CompShg VinylSd VinylSd
## 6 1998 1998 Gable CompShg VinylSd VinylSd
## Mas.Vnr.Type Mas.Vnr.Area Exter.Qual Exter.Cond Foundation Bsmt.Qual
## 1 Stone 112 TA TA CBlock TA
## 2 None 0 TA TA CBlock TA
## 3 BrkFace 108 TA TA CBlock TA
## 4 None 0 Gd TA CBlock TA
## 5 None 0 TA TA PConc Gd
## 6 BrkFace 20 TA TA PConc TA
## Bsmt.Cond Bsmt.Exposure BsmtFin.Type.1 BsmtFin.SF.1 BsmtFin.Type.2
## 1 Gd Gd BLQ 639 Unf
## 2 TA No Rec 468 LwQ
## 3 TA No ALQ 923 Unf
## 4 TA No ALQ 1065 Unf
## 5 TA No GLQ 791 Unf
## 6 TA No GLQ 602 Unf
## BsmtFin.SF.2 Bsmt.Unf.SF Total.Bsmt.SF Heating Heating.QC Central.Air
## 1 0 441 1080 GasA Fa Y
## 2 144 270 882 GasA TA Y
## 3 0 406 1329 GasA TA Y
## 4 0 1045 2110 GasA Ex Y
## 5 0 137 928 GasA Gd Y
## 6 0 324 926 GasA Ex Y
## Electrical X1st.Flr.SF X2nd.Flr.SF Low.Qual.Fin.SF Gr.Liv.Area
## 1 SBrkr 1656 0 0 1656
## 2 SBrkr 896 0 0 896
## 3 SBrkr 1329 0 0 1329
## 4 SBrkr 2110 0 0 2110
## 5 SBrkr 928 701 0 1629
## 6 SBrkr 926 678 0 1604
## Bsmt.Full.Bath Bsmt.Half.Bath Full.Bath Half.Bath Bedroom.AbvGr
## 1 1 0 1 0 3
## 2 0 0 1 0 2
## 3 0 0 1 1 3
## 4 1 0 2 1 3
## 5 0 0 2 1 3
## 6 0 0 2 1 3
## Kitchen.AbvGr Kitchen.Qual TotRms.AbvGrd Functional Fireplaces
## 1 1 TA 7 Typ 2
## 2 1 TA 5 Typ 0
## 3 1 Gd 6 Typ 0
## 4 1 Ex 8 Typ 2
## 5 1 TA 6 Typ 1
## 6 1 Gd 7 Typ 1
## Fireplace.Qu Garage.Type Garage.Yr.Blt Garage.Finish Garage.Cars
## 1 Gd Attchd 1960 Fin 2
## 2 <NA> Attchd 1961 Unf 1
## 3 <NA> Attchd 1958 Unf 1
## 4 TA Attchd 1968 Fin 2
## 5 TA Attchd 1997 Fin 2
## 6 Gd Attchd 1998 Fin 2
## Garage.Area Garage.Qual Garage.Cond Paved.Drive Wood.Deck.SF
## 1 528 TA TA P 210
## 2 730 TA TA Y 140
## 3 312 TA TA Y 393
## 4 522 TA TA Y 0
## 5 482 TA TA Y 212
## 6 470 TA TA Y 360
## Open.Porch.SF Enclosed.Porch X3Ssn.Porch Screen.Porch Pool.Area Pool.QC
## 1 62 0 0 0 0 <NA>
## 2 0 0 0 120 0 <NA>
## 3 36 0 0 0 0 <NA>
## 4 0 0 0 0 0 <NA>
## 5 34 0 0 0 0 <NA>
## 6 36 0 0 0 0 <NA>
## Fence Misc.Feature Misc.Val Mo.Sold Yr.Sold Sale.Type Sale.Condition
## 1 <NA> <NA> 0 5 2010 WD Normal
## 2 MnPrv <NA> 0 6 2010 WD Normal
## 3 <NA> Gar2 12500 6 2010 WD Normal
## 4 <NA> <NA> 0 4 2010 WD Normal
## 5 MnPrv <NA> 0 3 2010 WD Normal
## 6 <NA> <NA> 0 6 2010 WD Normal
## SalePrice
## 1 215000
## 2 105000
## 3 172000
## 4 244000
## 5 189900
## 6 195500We see that there are quite a few variables in the data set, enough to do a very in-depth analysis. For this lab, we’ll restrict our attention to just two of the variables: the above ground living area of the house in square feet (Gr.Liv.Area) and the sale price (SalePrice). To save some effort throughout the lab, create two variables with short names that represent these two variables.
area <- ames$Gr.Liv.Area
price <- ames$SalePriceLet’s look at the distribution of area in our population of home sales by calculating a few summary statistics and making a histogram.
summary(area)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 334 1126 1442 1500 1743 5642hist(area)
Exercise 1: Describe this population distribution.
Answer: The distribution is unimodal and right skewed.
The unknown sampling distribution In this lab we have access to the entire population, but this is rarely the case in real life. Gathering information on an entire population is often extremely costly or impossible. Because of this, we often take a sample of the population and use that to understand the properties of the population.
If we were interested in estimating the mean living area in Ames based on a sample, we can use the following command to survey the population.
samp1 <- sample(area, 50)This command collects a simple random sample of size 50 from the vector area, which is assigned to samp1. This is like going into the City Assessor’s database and pulling up the files on 50 random home sales. Working with these 50 files would be considerably simpler than working with all 2930 home sales.
Exercise 2: Describe the distribution of this sample. How does it compare to the distribution of the population?
Answer:
summary(samp1)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 540 1076 1335 1421 1608 3390hist(samp1)
The distribution is not perfectly normal. It is right skewed. Also its unimodal.
If we’re interested in estimating the average living area in homes in Ames using the sample, our best single guess is the sample mean.
mean(samp1)## [1] 1420.64Depending on which 50 homes you selected, your estimate could be a bit above or a bit below the true population mean of 1499.69 square feet. In general, though, the sample mean turns out to be a pretty good estimate of the average living area, and we were able to get it by sampling less than 3% of the population.
Exercise 3 : Take a second sample, also of size 50, and call it samp2. How does the mean of samp2 compare with the mean of samp1? Suppose we took two more samples, one of size 100 and one of size 1000. Which would you think would provide a more accurate estimate of the population mean?
mean(samp1)## [1] 1420.64samp2 <- sample(area, 50)
mean(samp2)## [1] 1594.36hist(samp2)
samp3 <- sample(area, 100)
mean(samp3)## [1] 1470.47hist(samp3)
samp4 <- sample(area, 1000)
mean(samp4)## [1] 1503.478hist(samp4)
Answer: With every sample the sample mean varies. The sample mean is close to each other(specially samp1, samp2 and samp3).
Not surprisingly, every time we take another random sample, we get a different sample mean. It’s useful to get a sense of just how much variability we should expect when estimating the population mean this way. The distribution of sample means, called the sampling distribution, can help us understand this variability. In this lab, because we have access to the population, we can build up the sampling distribution for the sample mean by repeating the above steps many times. Here we will generate 5000 samples and compute the sample mean of each.
sample_means50 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 50)
sample_means50[i] <- mean(samp)
}
hist(sample_means50)
If you would like to adjust the bin width of your histogram to show a little more detail, you can do so by changing the breaks argument.
hist(sample_means50, breaks = 25)
Here we use R to take 5000 samples of size 50 from the population, calculate the mean of each sample, and store each result in a vector called sample_means50. On the next page, we’ll review how this set of code works.
Exercise 4: How many elements are there in sample_means50? Describe the sampling distribution, and be sure to specifically note its center. Would you expect the distribution to change if we instead collected 50,000 sample means?
length(sample_means50)## [1] 5000mean(sample_means50)## [1] 1500.369sample_means50_1 <- rep(NA, 50000)
for(i in 1:50000){
samp <- sample(area, 50)
sample_means50_1[i] <- mean(samp)
}
hist(sample_means50_1)
length(sample_means50_1)## [1] 50000mean(sample_means50_1)## [1] 1499.996There are 5000 elements in sample_means50. The sampling distribution looks like a very good normal distribution, with mean mean very close to population mean. For 50,000 samples, the sampling distribution looks similar, and the mean would also be very close.
Interlude: The for loop Let’s take a break from the statistics for a moment to let that last block of code sink in. You have just run your first for loop, a cornerstone of computer programming. The idea behind the for loop is iteration: it allows you to execute code as many times as you want without having to type out every iteration. In the case above, we wanted to iterate the two lines of code inside the curly braces that take a random sample of size 50 from area then save the mean of that sample into the sample_means50 vector. Without the for loop, this would be painful:
sample_means50 <- rep(NA, 5000)
samp <- sample(area, 50)
sample_means50[1] <- mean(samp)
samp <- sample(area, 50)
sample_means50[2] <- mean(samp)
samp <- sample(area, 50)
sample_means50[3] <- mean(samp)
samp <- sample(area, 50)
sample_means50[4] <- mean(samp)and so on.
With the for loop, these thousands of lines of code are compressed into a handful of lines. We’ve added one extra line to the code below, which prints the variable i during each iteration of the for loop. Run this code.
sample_means50 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 50)
sample_means50[i] <- mean(samp)
print(i)
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## [1] 5000Let’s consider this code line by line to figure out what it does. In the first line we initialized a vector. In this case, we created a vector of 5000 zeros called sample_means50. This vector will will store values generated within the for loop.
The second line calls the for loop itself. The syntax can be loosely read as, “for every element i from 1 to 5000, run the following lines of code”. You can think of i as the counter that keeps track of which loop you’re on. Therefore, more precisely, the loop will run once when i = 1, then once when i = 2, and so on up to i = 5000.
The body of the for loop is the part inside the curly braces, and this set of code is run for each value of i. Here, on every loop, we take a random sample of size 50 from area, take its mean, and store it as the ith element of sample_means50.
In order to display that this is really happening, we asked R to print i at each iteration. This line of code is optional and is only used for displaying what’s going on while the for loop is running.
The for loop allows us to not just run the code 5000 times, but to neatly package the results, element by element, into the empty vector that we initialized at the outset.
Exercise 5: To make sure you understand what you’ve done in this loop, try running a smaller version. Initialize a vector of 100 zeros called sample_means_small. Run a loop that takes a sample of size 50 from area and stores the sample mean in sample_means_small, but only iterate from 1 to 100. Print the output to your screen (type sample_means_small into the console and press enter). How many elements are there in this object called sample_means_small? What does each element represent?
sample_means_small <- rep(NA, 100)
for(i in 1:100){
samp <- sample(area, 50)
sample_means_small[i] <- mean(samp)
}
sample_means_small## [1] 1543.32 1489.00 1531.78 1535.80 1403.66 1523.44 1479.86 1340.84
## [9] 1431.24 1525.22 1515.88 1520.82 1498.96 1505.90 1462.80 1543.80
## [17] 1547.42 1470.16 1449.22 1536.94 1665.68 1439.20 1647.12 1508.60
## [25] 1685.18 1471.28 1561.76 1442.20 1550.56 1535.10 1485.76 1404.76
## [33] 1436.92 1564.02 1591.62 1497.62 1567.60 1394.70 1500.54 1602.32
## [41] 1361.36 1492.16 1507.30 1385.20 1478.90 1582.02 1615.78 1473.02
## [49] 1432.46 1523.12 1385.84 1407.56 1578.46 1494.72 1366.80 1437.42
## [57] 1531.44 1477.92 1543.04 1606.44 1550.48 1427.82 1604.50 1604.68
## [65] 1498.20 1455.20 1544.98 1558.46 1575.00 1493.04 1477.56 1602.62
## [73] 1481.80 1548.08 1557.38 1432.56 1408.12 1548.64 1558.40 1555.98
## [81] 1503.14 1473.80 1514.62 1475.08 1471.78 1674.98 1405.40 1532.04
## [89] 1519.64 1457.96 1388.88 1565.90 1524.38 1557.44 1581.04 1489.02
## [97] 1542.60 1474.08 1500.00 1556.56There are 100 elements in sample_means_small, representing the mean from each 100 samples
Sample size and the sampling distribution
Mechanics aside, let’s return to the reason we used a for loop: to compute a sampling distribution, specifically, this one.
hist(sample_means50)
The sampling distribution that we computed tells us much about estimating the average living area in homes in Ames. Because the sample mean is an unbiased estimator, the sampling distribution is centered at the true average living area of the the population, and the spread of the distribution indicates how much variability is induced by sampling only 50 home sales.
To get a sense of the effect that sample size has on our distribution, let’s build up two more sampling distributions: one based on a sample size of 10 and another based on a sample size of 100.
sample_means10 <- rep(NA, 5000)
sample_means100 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 10)
sample_means10[i] <- mean(samp)
samp <- sample(area, 100)
sample_means100[i] <- mean(samp)
}Here we’re able to use a single for loop to build two distributions by adding additional lines inside the curly braces. Don’t worry about the fact that samp is used for the name of two different objects. In the second command of the for loop, the mean of samp is saved to the relevant place in the vector sample_means10. With the mean saved, we’re now free to overwrite the object samp with a new sample, this time of size 100. In general, anytime you create an object using a name that is already in use, the old object will get replaced with the new one.
To see the effect that different sample sizes have on the sampling distribution, plot the three distributions on top of one another.
par(mfrow = c(3, 1))
xlimits <- range(sample_means10)
hist(sample_means10, breaks = 20, xlim = xlimits)
hist(sample_means50, breaks = 20, xlim = xlimits)
hist(sample_means100, breaks = 20, xlim = xlimits)
The first command specifies that you’d like to divide the plotting area into 3 rows and 1 column of plots (to return to the default setting of plotting one at a time, use par(mfrow = c(1, 1))). The breaks argument specifies the number of bins used in constructing the histogram. The xlim argument specifies the range of the x-axis of the histogram, and by setting it equal to xlimits for each histogram, we ensure that all three histograms will be plotted with the same limits on the x-axis.
Exercise 6: When the sample size is larger, what happens to the center? What about the spread?
Answer: When the sample size is large the center is closer to the mean of the population and and the spread gets narrower.
On your own So far, we have only focused on estimating the mean living area in homes in Ames. Now you’ll try to estimate the mean home price.
- Take a random sample of size 50 from price. Using this sample, what is your best point estimate of the population mean?
Answer:
sample_meansprice50 <- sample(price, 50)
mean(sample_meansprice50)## [1] 175862.9- Since you have access to the population, simulate the sampling distribution for x¯price by taking 5000 samples from the population of size 50 and computing 5000 sample means. Store these means in a vector called sample_means50. Plot the data, then describe the shape of this sampling distribution. Based on this sampling distribution, what would you guess the mean home price of the population to be? Finally, calculate and report the population mean.
sample_means50 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(price, 50)
sample_means50[i] <- mean(samp)
}
hist(sample_means50, breaks = 25)
summary(sample_means50)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 147539 173031 180402 180787 188097 220572mean(sample_means50)## [1] 180786.8mean(price)## [1] 180796.1- Change your sample size from 50 to 150, then compute the sampling distribution using the same method as above, and store these means in a new vector called sample_means150. Describe the shape of this sampling distribution, and compare it to the sampling distribution for a sample size of 50. Based on this sampling distribution, what would you guess to be the mean sale price of homes in Ames?
Answer:
sample_means150 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(price, 50)
sample_means150[i] <- mean(samp)
}
hist(sample_means150, breaks = 25)
summary(sample_means150)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 145795 173015 180465 180896 188271 230722mean(sample_means150)## [1] 180895.7mean(price)## [1] 180796.1- Of the sampling distributions from 2 and 3, which has a smaller spread? If we’re concerned with making estimates that are more often close to the true value, would we prefer a distribution with a large or small spread?
Answer: The one with larger sample size has the smaller spread. In this case the second (sample_means150) has smaller spread. It is preferable to have a distribution with a smaller spread when trying to make estimates.
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.