STA 111 Lab 4

Goal

In this lab, we are going to explore the normal distribution, which is also called the Gaussian distribution. We are going to see a few examples of things that are normally distributed, as well as seeing how we can use properties of the normal distribution to work with data.

Loading the Data

We are going to work with data from a study interested in determining information about pregnant women and their newborn children. To load the data, put the following in a chunk of code and press play.

babyData <- read.csv('http://people.hsc.edu/faculty-staff/blins/classes/spring17/math222/data/babies.csv')

The data provide information on \(n = 1236\) individuals. The variables contained in the data are:

• bwt - birth weight of the baby (in ounces)
• gestation - length of the pregnancy (in days)
• parity - 1 if baby was the parent’s first born child, 0 otherwise
• age - mother’s age (in years)
• height - mother’s height (in inches)
• weight - mother’s weight (in lbs.)
• smoke - 1 if the mother is a smoker, 0 otherwise

Exploring the mother’s height

The first variable we are going to explore is the height of the mother, in inches. We are interested in the probability that a mother is less than 69 inches (5 foot 9) in height. To answer questions like this, it is first helpful to look at the distribution of heights matches a named distribution, like the normal distribution.

Once we have looked at the distribution, we start to consider whether the distribution we are seeing matches a named distribution. A named distribution is just a distribution of data that has been given a name and whose properties are well known. Some examples are the Normal, the F, the T, the Chi-Square, and the Gamma distributions. In this lab, we are focusing on the normal distribution.

One thing that we know about the normal distribution is that it is symmetric. When we have a symmetric distribution, we know that when we look at the mean and the median of the distribution, these two measures of center should be roughly the same.

Defining the Parameters

If we have decided that a normal distribution might be a reasonable way to describe the distribution of the data, the next step is to find the value of the two parameters that describe the distribution. These two parameters are:

• \(\mu\), which is the mean.
• \(\sigma\), which is the standard deviation.

Okay, so at this point, we have a good idea that a normal distribution is a plausible choice for describing the distribution of heights, and we also have an idea of the center (\(\mu\)) and spread (\(\sigma\)) of the distribution. The last step is to actually draw the normal distribution you specified in Question 6 on top of our histogram to make sure that the curve is a good representation of the bars in the histogram.

We can use R to draw our normal distribution on top of the data, but this takes a few steps. The first step is to tell R the range of values that are on the X axis. In other words, what is the smallest value of height (from =) and what is the largest value of height (to =) in our data? This information can be found in the summary we did earlier. Copy the code below, put it in a chunk, and press play.

# Step 1: Specify the range of your x axis
range <- seq(from = 53, to = 72, length = 40)

The next step is to tell R the parameters for the normal distribution you want to draw. We already know what those values are, but now we need to tell R what we have chosen. To do this, copy the code below and put it in a chunk. Then, replace MU with the numeric value for \(\mu\) you chose in Question 4, and SIGMA with the numeric value for \(\sigma\) that you chose in Question 5.

# Step 2
fun <- dnorm(range, mean = MU, sd = SIGMA)

The final step is to actually draw the curve on the histogram!

# Step 3: Histogram
hist(babyData$height, prob = TRUE, col = "white",
     ylim = c(0, max(fun)),
     main = "Histogram with normal curve",
     xlab = "Height (in inches)")
lines(range, fun, col = 2, lwd = 2)

You will notice that this looks a little different from our standard histogram code. The additions are to allow us to draw the normal distribution on top of the histogram.

Using the Normal Distribution

Once we have decided that a normal distribution is a reasonable choice, we can use that to answer questions about the data.

Another Example

We have just seen one example of where the normal distribution occurs in nature - human heights. It turns out that it is not just height that tends to be normally distributed!