Research studies are conducted in order to answer some kind of research question(s). For example, the researchers in the Vegan Health Study define at least eight primary questions that they would like answered about the health of people who eat an entirely animal-free diet (no meat, no dairy, no eggs). Another research study was recently conducted to determine whether people who take the pain medications Vioxx or Celebrex are at a higher risk for heart attacks than people who don't take them. The list goes on. Researchers are working every day to answer their research questions.

What do you think about these research questions?

1. What percentage of college students feel sleep-deprived?
2. What is the probability that a randomly selected PSU student gets more than seven hours of sleep each night?
3. Do women typically cry more than men?
4. What is the typical number of credit cards owned by Stat 414 students?

Assuming that the above questions don't float your boat, can you formulate a few research questions that do interest you?

If we were to attempt to answer our research questions, we would soon learn that we couldn't ask every person in the population if they feel sleep-deprived, how often they cry, or the number of credit cards they have.

In trying to answer each of our research questions, whether yours or mine, we unfortunately can't ask every person in the population. Instead, we take a random sample from the population, and use the resulting sample to learn something, or make an inference, about the population:

For the research question "what percentage of college students feel sleep-deprived?", the population of interest is all college students. Therefore, assuming we are restricting the population to be U.S. college students, a random sample might consist of 1300 randomly selected students from all of the possible colleges in the United States. For the research question, "what is the probability that a randomly selected Penn State student gets more than 7 hours of sleep each night?", the population of interest is a little narrower, namely only Penn State students. In this case, a random sample might consist of, say, 300 randomly selected Penn State students. For the research question "what is the typical number of credit cards owned by Stat 414 students?", the population of interest is even more narrow, namely only the students enrolled in Stat 414. Ahhhh! If we are only interested in students currently enrolled in Stat 414, we have no need for taking a random sample. Instead, we can conduct a census, in which all of the students are polled.

Now, for each of the research questions you previously defined, identify the population of interest and describe a potential random sample.

The answers (or data) we get to our research questions of course depend on who ends up in our random sample. We can't possibly predict the possible outcomes with certainty, but we can at least create a list of possible outcomes.

In order to answer my first research question, we would need to take a random sample of U.S. college students, and ask each one "Do you feel sleep-deprived?" Each student should reply either "yes" or "no." Therefore, we would write the sample space as:

\(\mathbf{S} = \{\text{yes}, \text{no}\}\)

In order to answer my second research question, we would need to know how many hours of sleep a random sample of college students gets each night. One way of getting this information is to ask each selected student to record the number of hours of sleep they had last night. In this case, if we let h denote the number of hours slept, we would write the sample space as:

\(\mathbf{S} = \{h: h \ge 0 \text{ hours}\}\)

Hmmm, if we conducted a random study to answer my third research question, how would we define our sample space? Well, of course, it depends on how we went about trying to answer the question. If we asked a random sample of men and women "on how many days did you cry last month?", we would write the sample space as:

\(\mathbf{S} = \{0, 1, 2, ..., 31\}\)

Finally, if we were interested in learning about students who took Stat 414 in the past decade when trying to answer my fourth research question, we might ask all current Stat 414 students "how many credit cards do you have?" In that case, we would write our sample space as:

\(\mathbf{S} = \{0, 1, 2, ...\}\)

There is not always just one way of obtaining an answer to a research question. For my second research question, how would we define the sample space if we instead asked a random sample of college students "did you get more than seven hours of sleep last night?"

For each of the research questions you created:

1. Formulate the question you would ask (or describe the measurement technique you would use).
2. Define the resulting sample space.

Once we collect sample data, we need to do something with it. Like summarizing it would be good! How we summarize data depends on the type of data we collect.

Now, for their definitions!

1. Determine the number, \(n\), in the sample.
2. Determine the frequency, \(f_i\), of each outcome \(i\).
3. Center a rectangle with base of length 1 at each observed outcome \(i\) and make the height of the rectangle equal to the frequency.

For our crying (out loud) data, we would first tally the frequency of each outcome:

and then we'd use the first column for the horizontal-axis and the third column for the vertical-axis to draw our frequency histogram:

Well, of course, in practice, we'll not need to create histograms by hand. Instead, we'll just let statistical software (such as Minitab) create histograms for us.

Okay, so let's use the above frequency histogram to answer a few more questions:

• What percentage of the surveyed women reported not crying at all in the month?
• What percentage of the surveyed women reported crying two times in the month? and three times?

Clearly, the frequency histogram is not a 100%-user friendly. To answer these types of questions, it would be better to use a relative frequency histogram:

Now, the answers to the questions are a little more obvious — about 12% reported not crying at all; about 28% reported crying two times; and about 18% reported crying three times.

1. Determine the number, \(n\), in the sample.
2. Determine the frequency, \(f_i\), of each outcome \(i\).
3. Calculate the relative frequency (proportion) of each outcome \(i\) by dividing the frequency of outcome \(i\) by the total number in the sample \(n\) — that is, calculate \(\frac{f_i}{n}\) for each outcome \(i\).
4. Center a rectangle with base of length 1 at each observed outcome i and make the height of the rectangle equal to the relative frequency.

While using a relative frequency histogram to summarize discrete data is a worthwhile pursuit in and of itself, my primary motive here in addressing such histograms is to motivate the material of the course. In our example, if we

• let X = the number of times (days) a randomly selected student cried in the last month, and
• let x = 0, 1, 2, ..., 31 be the possible values

Then \(h_0=\frac{f_0}{n}\) is the relative frequency (or proportion) of students, in a sample of size \(n\), crying \(x_0\) times. You can imagine that for really small samples \(\frac{f_0}{n}\) is quite unstable (think \(n = 5\), for example). However, as the sample size \(n\) increases, \(\frac{f_0}{n}\) tends to stabilize and approach some limiting probability \(p_0=f(x_0)\) (think \(n = 1000\), for example). You can think of the relative frequency histogram serving as a sample estimate of the true probabilities of the population.

It is this \(f(x_0)\), called a (discrete) probability mass function, that will be the focus of our attention in Section 2 of this course.

The major difference is that you first have to group the data into a set of classes, typically of equal length. There are many, many sets of rules for defining the classes. For our purposes, we'll just rely on our common sense — having too few classes is as bad as having too many.

1. Determine the number, \(n\), in the sample.
2. Define \(k\) class intervals \((c_0, c_1], (c_1, c_2], ..., (c_{k-1}, c_k]\).
3. Determine the frequency, \(f_i\), of each class \(i\).
4. Calculate the relative frequency (proportion) of each class by dividing the class frequency by the total number in the sample — that is, \(\frac{f_i}{n}\).
5. For a frequency histogram: draw a rectangle for each class with the class interval as the base and the height equal to the frequency of the class.
6. For a relative frequency histogram: draw a rectangle for each class with the class interval as the base and the height equal to the relative frequency of the class.
7. For a density histogram: draw a rectangle for each class with the class interval as the base and the height equal to \(h(x)=\dfrac{f_i}{n(c_i-c_{i-1})}\) for \(c_{i-1}<x \leq c_i\), \(i = 1, 2,..., k\).

Here's what the work would like for our nose length example if we used 5 mm classes centered at 30, 35, ... 70:

For example, the relative frequency for the first class (27.5 to 32.5) is 2/60 or 0.033, whereas the height of the rectangle for the first class in a density histogram is 0.033/5 or 0.0066. Here is what the density histogram would like in its entirety:

Note that a density histogram is just a modified relative frequency histogram. That is, a density histogram is defined so that:

• the area of each rectangle equals the relative frequency of the corresponding class, and
• the area of the entire histogram equals 1.

Again, while using a density histogram to summarize continuous data is a worthwhile pursuit in and of itself, my primary motive here in addressing such histograms is to motivate the material of the course. As the sample size \(n\) increases, we can imagine our density histogram approaching some limiting continuous function \(f(x)\), say. It is this continuous curve \(f(x)\) that we will come to know in Section 3 as a (continuous) probability density function.

So, in Section 2, we'll learn about discrete probability mass functions (p.m.f.s). In Section 3, we'll learn about continuous probability density functions (p.d.f.s). In Section 4, we'll learn about p.m.f.s and p.d.f.s for two (random) variables (instead of one). In Section 5, we'll learn how to find the probability distribution for functions of two or more (random) variables. Wow! That's a lot of work. Before we can take it on, however, we will first spend some time in this Section 1 filling up our probability toolbox with some basic probability rules and tools.