Exploration Appendix
Chapter 3 Explorations
We can visualize this with an exploration.
Chapter 4 Explorations
We can visualize this with an exploration.
We can visualize this with an exploration.
Chapter 5 Explorations
In this exploration, we will be looking at the interaction of sum squared errors and the least squares regression line.
We can visualize this with an exploration.
Chapter 6 Explorations
To understand the density of a continuous random variable, we can use the following Desmos exploration.
If you change the values of \(a\) and \(b\) by dragging the points along the \(x\)-axis, we get a new probability of form \(P(a < X\leq b)\).
We can explore a generic distribution function using the Desmos exploration below which you can interact with.
Chapter 7 Explorations
The following Desmos exploration gives a geometric demonstration of what \(\mu\) and \(\sigma\) represent in a Normal distribution.
What you see is a graph of the “nasty” function in Definition 7.1. The graph is what is known as the Bell Curve and resembles a fairly simple roller coaster track. To help in our discussion of the distribution, we have included a “Coaster Car” which you can drag along the Normal curve. The red line it drags along with it represents the “level” of the car as it moves along the track.
Dragging the Coaster Car to the top of the track where the red line is parallel to the ground below, i.e. the \(x\)-axis, we find that this maximum value occurs when \(x = \mu\).
If you drag the Coaster Car to the right of \(x = \mu\) and start heading “down” the track, then you will see the red line get steeper and steeper… to a point. A red dot is included on the \(x\)-axis to aid in visualization. There is a point when the track is as steep as it ever gets. If you carefully drag the car around, this should occur precisely \(x = \mu + \sigma\). Similarly, we could have found the point on the ascent up the track to the left of \(x = \mu\) and we would find the steepest point of the graph here is at \(x = \mu - \sigma\). The fancy term for the values \(x = \mu \pm \sigma\) are the Inflection Points of the graph. Inflection points represent the “steepest points” of graphs, in some sense.
This exploration shows that if \(X\) is a Normal random variable modeling IQ scores, that \(P(124 < X \leq 148) \approx 0.0541\). It also shows that the \(P(X \leq 124) \approx 0.9452\) and that \(P(X>148)\approx 0.0007\).
Play around with the tool and answer the following questions.
What is the probability that a random person has an IQ less than 100? [Note: This one should be “obvious.”]
What is the probability that a random person has an IQ more than 120?
What is the probability that a random person has an IQ between 60 and 90?
The exploration below can be used to find the probabilities/proportions associated to the Standard Normal Distribution.
Chapter 8 Explorations
In the introduction of Chapter 8 we discussed probabilities associated to flipping a coin 50 times. The following exploration simulates just that where \(X\) is the number of “Heads” found after 50 flips.
Estimate \(P(X = 25)\) for a fair coin.
Estimate \(P( 22 \leq X \leq 28)\) for a fair coin.
Estimate \(P( 20 \leq X \leq 30)\) for a fair coin.
Shifting the coin to “Variable” and adjusting its probability with the purple slider, how high does the coin’s probability need to be for \(P( X \geq 48)\) to appear to be at least 1%?
This exploration allows you to look at how often a sample of BADRs were as unlucky as Melody.
You can change the sample size by dragging the black dot along the horizontal line and get a new sample by dragging the Randomizer slider.