• one-peso coin per student
• stop timer per group
DEVELOPMENT OF THE LESSON
A. Introduction/Motivation: The coin toss and breath-holding activities Look at the current one-peso coin in circulation. It has Jose Rizal on one side, which we will call Head (H), and the other side Tail (T). Ask learners to toss the one-peso coin three times and record on their Activity Sheet the results of the three tosses Use H for heads, and T for tails. If needed, define first the head side of the coin and the tail side of the coin. For example, a learner tosses heads, tails, heads. Then the learner should write HTH on his/her notebook. Ask them to count the number of heads that appeared and write it also on their Activity Sheets.
Next, have all the learners hold their breaths and record the time. This is best done if they time it as accurately as possible (if possible, use a cell phone timer and record up to the nearest hundredth of a second). If there is limited number of timers, do it one at a time with one learner holding the timer while the other one is holding his/her breath. Ask students to record the time on their Activity Sheets.
Then, record all the possible answers on the board for both activities. For the first activity, write all eight possible outcomes, and then list down which one had zero (0), 1, 2, or 3 heads. If you have time, tally the results. You can do this systematically so that you do not get confused later on. Start with the outcomes with zero (0) heads, then progress from there.
TTT TTH THT HTT THH HTH HHT HHH
The breath-holding activity is a little bit more challenging. Expect to have a lot of possible values. Just write about 10. Then, tell the learners if they have different values, they can raise their hands. Notice that the first one had only four possible values to take, while the second one is almost unique to each individual. What
could help may be getting the lowest value and the highest value recorded by students.
Emphasize the difference in the number of possible values in these two activities as this is important in the discussion.
B. Main Lesson
I. Experiments and Random Variables
Begin the discussion with the definition of a Statistical Experiment: An activity that will produce outcomes, or a process that will generate data. The outcomes have a corresponding chance of occurrence. Examples of which are (a) tossing three coins and counting the number of heads, (b) recording the time a person can hold his/her breath, (c) counting the number of students in the classroom who are present today, (d) obtaining the height of a student, etc.
Say that the two activities are examples of statistical experiments. Come up with several examples, such as recording the results of an examination, asking the weekly baon (or allowance) of students, identifying the waistline of students.
Emphasize that Statistical Experiments can have a few or a lot of possible outcomes. In the coin toss example, there are eight possible outcomes. In the breath example, there can be a lot of possible outcomes. However, they can indicate that the possible values are in the range of 10 seconds to 60 seconds (Ask the shortest and the longest times in class and use that as the limits for this
example).
Suppose you give a learner candy based on the number of heads that appear in the coin toss experiment (Remember: Giving a candy is optional). List down the
possible number of candies that can be given. Notice that it should only be zero (0), 1, 2, or 3. Then, you can list down all the outcomes of the experiment under each value:
Number of candies Outcomes
0 TTT
1 TTH, THT, HTT
2 THH, HTH, HHT
3 HHH
Next, define a Random Variable: It is a way to map outcomes of a statistical experiment determined by chance into number. It is typically denoted by a capital letter, usually X.
X: outcome ! number
Random variable is actually neither random nor a variable in the traditional sense that a variable is defined in an algebra class (where we solve for the value of a
variable). It is technically a function from the space of all possible events to the set ℝ of real numbers.
Tell students that a random variable must take exactly one value for each random outcome. Generally, as with functions, a number of possible outcomes may have the same value of the random variable, and in practice, this occurs frequently. For instance, three outcomes above for tossing a coin thrice would have 1 candy, and three outcomes would have 2 candies.
Learners need to understand that random variables are conceptually different from the mathematical variables that they have met before in math classes. A random variable is linked to observations in the real world, where uncertainty is involved.
Learners should be told that random variables are central to the use of probability in practice. They help model random phenomena, that is, random variables are relevant to a wide range of human activities and disciplines, including agriculture, biology, ecology, economics, medicine, meteorology, physics, psychology,
computer science, engineering, and others. They are used to model outcomes of random processes that cannot be predicted deterministically in advance (but the range of numerical outcomes may, however, be viewed).
In the coin example, we can define the random variable X to be the number of heads that appears from tossing a coin three times. While we do not know what the resulting specific outcome is, we know the possible values of X in this case are zero (0), 1, 2, or 3. You can also define another random variable Y to be the time a person can hold his/her breath. The possible values for this variable can be one of so many possible values.
In the second example, the possible values range between the lowest and the highest value recorded by students. Notice that it is really difficult to list down all the possible values. That is why in this example, it is better to state the possible values as an interval, such as
, if the lowest and highest values are 10 and 60, respectively.
II. Types of Random Variables:
Distinguish the two types of random variables, viz., discrete and continuous.
(a) Discrete Random Variables are random variables that can take on a finite (or countably infinite) number of distinct values. Examples are the number of heads obtained when tossing a coin thrice, the number of siblings a person has, the
number of students present in a classroom at a given time, the number of crushes a person has at a particular time, etc.
Categorical variables can be considered discrete variables. Example: whether a person has normal BMI or not, you can assign one (1) as the value for normal BMI and zero (0) for not normal BMI. You can also put numbers to represent certain categorical variables with more than two categories. You can also use ordinal variables, like how much they like adobo on a scale of 1 to 10 (where 1 means favorable and 10 unfavorable).
(b) Continuous Random Variables, on the other hand, are random variables that take an infinitely uncountable number of possible values, typically measurable quantities. Examples are the time a person can hold his/her breath, the height or weight or BMI of a person (if measured very accurately), the time a person takes for a person to bathe. The values that a continuous random variable can have lie on a continuum, such as intervals.
Extra Notes:
• You can modify the experiment to just tossing a coin twice instead of thrice to make things simpler. Here, the outcomes will be only four: HH, HT, TH, TT, and the possible values of X are 0, 1, and 2.
• You may use other examples of continuous variables such as height, weight, lengths, and age)
• Feel free to add more examples, or get examples from the seatwork that is in the next section.
C. Group Discussion
Group learners into threes. Given the following experiments and random variables, ask the groups to identify what the possible values of the random variables are.
Also, for each random variable, identify whether the variable is discrete or
continuous. (Answers in bold are Discrete, while answers in italics are Continuous) 1. Experiment: Roll a pair of dice
Random Variable: Sum of numbers that appears in the pair of dice 2. Experiment: Ask a friend about preparing for a quiz in statistics
Random Variable: How much time (in hours) he/she spends studying for this quiz
3. Experiment: Record the sex of family members in a family with four children Random Variable: The number of girls among the children
4. Experiment: Buy an egg from the grocery
Random Variable: The weight of the egg in grams
5. Experiment: Record the number of hours one watches TV from 7 pm to 11 pm for the past five nights.
Random Variable: The number of hours spent watching TV from 7 pm to 11 pm
D. Enrichment
In tossing a coin four times, how many outcomes correspond to each value of the random variable?
What if the coin would be tossed five times? six times? seven times? eight times?
Try to relate the outcomes to the numbers in Pascal’s triangle.
For tossing the coin four times, there will be five possible values, 0, 1, 2, 3, 4, with
1, 4, 6, 4, 1 outcomes, respectively.
For five coins there are six possible values,
0, 1, 2, 3, 4, and 5, with
1, 5, 10, 10, 5, 1 outcomes, respectively.
In general, for n tosses of a coin, there are n+1 possible values, 0, 1, 2, 3, …, n. If k is a possible value, then there are
nCx =
outcomes associated with x.
Next, possibly read on probability distributions, which will be covered in the next lesson.!
!
KEY POINTS
• A Random Variable may be viewed as a way to map outcomes of a statistical experiment determined by chance into number.
• There are two types of random variables:
o Discrete: takes on a finite (or countably infinite) number of values o Continuous: takes an infinitely uncountable number of possible values,
typically measurable quantities