Lesson 5.5 – Continuous Random Variables
- Show that the probability distribution of a continuous random variable is valid and use the distribution to calculate probabilities.
- Determine the relative locations of the mean and median of a continuous random variable.
- Draw a normal probability distribution with a given mean and standard deviation.
Activity: How much do you get paid?
We started this Activity with a short story. Suppose that the teacher just opened up his/her own pizza shop and is looking for some employees. Since our stats classes are full of smart, hard-working students, we have decided to hire each student in class. But there is a catch. Each day the student works, the hourly rate for working at the pizza shop is determined by a random number generator and can be anywhere between $0 and $20 per hour. The random number generator has several different options for what type of distribution to use to pick a random number (uniform, right-skew, left-skew, bell-shaped). In the Activity, it is the students’ job to figure out which shaped distribution is to their advantage. WARNING: The website we are using mistakenly switched the definitions of right-skew and left-skew. We made our students aware of this mistake at the beginning of the Activity.
We let students work through the Activity, and then we provided a summary at the end.
For each of the distributions, we had students draw in a “curve” (density curve) that models the dot plot.
Uniform distribution: If we were to sample many more than 100 random numbers the dot plot would eventually have a similar height for each value. The density curve is therefore a horizontal line. Be sure to mention that the total area under the density curve is 1, and use that fact to calculate the height. Once you have the height, ask students how likely it is that they would make between $8 and $12 per hour. Use area to calculate this probability (see picture above). Also notice that the mean is similar to the median.
Left-skewed distribution: Total area under the density curve is still 1 and the mean is less than the median, which students already know from Lesson 1.6.
Right-skewed distribution: Total area under the density curve is still 1 and the mean is greater than the median, which students already know from Lesson 1.6.
Bell-shaped distribution: Also known as a Normal curve, this will become a ubiquitous idea for the remainder of the course. Notice that the mean is similar to the median. Ask students why the mean is similar to the median for Uniform and Bell-shaped distributions (they are both symmetric!). Some students may remember the 68-95-99.7 rule from another class, which we will be formally teaching tomorrow.
In the end, some students wanted to be paid with a random number from the bell-shaped curve (more consistent pay and lower the risk of a very low pay day), while most decided on a left-skewed distribution (as most pay days will be higher).