6  Probability

Before we can talk about inferential statistics we have to understand probability.

We all know, that if I toss a quarter in the air. There are only two possible options. Either the coin lands on a head, or it lands on a tail.

By using the sample() function, we can tell R to randomly sample an event for us.

We ‘tossed’ a coin, and it landed on heads. What is the probability of this happening?

Probability is written as:

\[P(A) = \frac{1}{2} = .5000\]

In other words, the number of possible outcomes favorable to A divided by the total number of possible outcomes.

Let’s think about this in terms of playing cards:

In a standard card deck, there are 52 total cards:

Two Colors: Red and Black

Four Suits: Clubs, Hearts, Spades, Diamonds

2,3,4,5,6,7,8,9,10,J,Q,K,A

What is the probability of picking a King of Hearts

\[P(King of Hearts) = \frac{1}{52}= .0192\]

What is the probability of picking a 10 of Hearts and a Nine of Hearts with replacement?

\[P(10 of Hearts) - P(9 of Hearts) \frac{1}{52}X \frac{1}{52} = .01923X .01923 = .00037\] What is the probability of picking a 10 of Hearts and a Nine of Hearts without replacement?

\[P(10 of Hearts) - P(9 of Hearts) \frac{1}{52}X \frac{1}{51} = .01923X .01961 = .00037\]

In order to do work with some datasets, we need to know what they are composed of. So far we have many worked with numbers. If we were to look at string data. Also know as, character data, we can use a new command called which(). Additionally we can also use a workaround using something new: ==. Without getting into the details, there is a difference between = and ==.

There are two different ways we can break-up our data.

We can use a command such as:

6.1 Proportioning Data

The below code will allow you to take a set of data and see what proportion is accounted for given a certain criteria.

For example: If I have a data set of 1’s and 0’s, how can I know how many I have of each. Additionally, how can I tell the proportion of each?

To do this we need to use the which()command as well as how much of the data is made up of 1’s and 0’s.

We can see here that the amount of 1’s is pretty close to the amount of 0’s. Now let us try with a more involved example.

6.1.1 The rnorm Function

In most, if not all of my slides, you have seen me use the function rnorm(). In class, I’ve used this as a way to generate large sets of data with one line of text. There are a few arguments that are necessary in order to take this function and use it for the purposes of sampling.

rnorm() stands for “Random Normal”. Essentially, what rnorm() does it create a random set of numbers (specified by you) and generates data.

Here are the arguments rnorm() accepts:

• n: Specifies the number of data points you want to create.

• mean: Specifies the mean you want your sample to have.

• sd: Specifies the standard deviation you want your sample to have.

Although not essential, a nice function to use is round( digits = )

• This function makes your data look a little bit neater, you will see that when we create a sample from rnorm() the numbers are a little messy.

So, let’s make a sample of 20 numbers with a mean of 12 and a standard deviation of 3.

See how the numbers came out a little unwieldy? Let’s fix that:

Notice how the digits= command is preceded by a comma. This is because it is a part of the round() function but not a part of the rnorm() function.

We can specify the mean and the standard deviation of the dataset, but does introducing the variability of the standard deviation make it harder for the mean to be exactly what we asked for? Let’s see!

6.1.2 Sampling

We can have our dataset of 100 points, but if we take a sample of that data, how representative will it be if the true mean?

Let’s take a look:

Do you see how it becomes difficult to tell a certain property of a random sample when you only have small sample to use?