Fox Module 4: Bivariate Displays HW


Fox Module 4: Bivariate Displays HW

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NEAS
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Module 4: Bivariate Displays

 

(The attached PDF file have better formatting.)

 

Homework Assignment: quantile comparison plots

 

Quantile comparison plots are discussed in Module 3 and are used later in the text. This homework assignment discusses quantile comparison plots, not bivariate displays

 

We compare quantile comparison plots for two distributions:

 


           Figure 3.9 on page 37: A t-distribution with 3 degrees of freedom.

           Figure 3.8 on page 37: A ÷-squared distribution with 2 degrees of freedom.


 

 

Below is a quantile comparison plot for 1,000 random draws from a t-distribution with 3 degrees of freedom.

 

The quantile comparison plot for a t-distribution with 2 degrees of freedom is shaped like an S-curve.

 


 

A.     At the upper tail, are values more or less extreme than in a normal distribution?

B.     At the lower tail, are values more or less extreme than in a normal distribution?

C.    Is the t-distribution with 2 degrees of freedom (i) symmetric thin-tailed, (ii) symmetric thick-tailed, (iii) positively skewed, or (iv) negatively skewed?

 

 


 

Below is a quantile comparison plot for 1,000 random draws from a χ-squared distribution with 2 degrees of freedom.

 

 

The quantile comparison plot for a χ-squared distribution with 2 degrees of freedom is shaped like a convex banana.

 


 

A.     At the upper tail, are values more or less extreme than in a normal distribution?

B.     At the lower tail, are values more or less extreme than in a normal distribution?

C.    Is a ÷-squared distribution with df = 2 (i) symmetric thin-tailed, (ii) symmetric thick-tailed, (iii) positively skewed, or (iv) negatively skewed?

 

 


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CalLadyQED
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Is the t-distribution plot 2df like in the textbook or 3df?

[NEAS: A t-distribution with 2 degrees of freedom is thicker-tailed than a t-distribution with 3 degrees of freedom, but the shape of the distribution is similar. As the degrees of freedom approaches infinity, the t-distribution becomes a normal distribution. For quantile comparison graphs of symmetric thick-tailed distributions, final exam problems use a t-distribution with 2 or 3 degrees of freedom.]


CalLadyQED
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What is meant by "extreme"? I don't recall that term being used in any of my stats courses or in the textbook. Does "more extreme" mean a steeper slope (i.e., the derivative is larger at that point)?

[NEAS: Extreme is used in its lay sense, as the distance from the mean in units of standard deviation. An extreme event is many standard deviations away from the mean. Consider a normal distribution vs a t-distribution with 3 degrees of freedom, both of which have a mean of zero and a standard deviation of one. For which distribution is the 99th percentile higher? The t-distribution is thick tailed, so its extreme events (the 99th percentile and the 1st precentile) are further from the mean.]

 

 


jgorab17
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I'm curious as to the definition of extreme here as well. I was figuring the steeper slope as well.

Also, I don't recall seeing symmetric thin/thick tailed being defined either...would the first plot be symmetric thin tailed and the second be positively skewed?

[NEAS: Thin and thick tailed are relative to the normal distribution. For example, a t-distribution is symmetric thick tailed.]


wangxy
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Please define extreme. I would really like to know it as well. I am guessing if the points lie above the comparison line, it is more extreme, otherwise it would be less extreme. Am I correct? Text book Page 37 Figure 3.9 defines heavy tail, I am guessing this is also symmetric thick tail? then should the symmetric thin tail be opposite of what is draw in figure 3.9???? (below comparison line at higher values and above comparison line at lower values????)

[NEAS: Thick tailed distributions (heavy tailed, long tailed, fat tailed) have many definitions. This course refers to thick tailed as any distribution with thicker tails than the normal distribution. A normal distribution has a straight line for a quantile comparison plot. A thin or thick tailed distribution has points that deviate from the straight line, but in opposite directions. Consider the lognormal distribution, which is thick tailed on the right but thin tailed on the left (since values are bounded by zero). Examine the quantile comparison plot: what does it mean that the points are above or below the straight line?]


NJS26
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I've seen answers about the "thick tail", but none about the meaning of extreme.  Any help?  Thanks.
CalLadyQED
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Yeah, I guess the question we all want answered is this: Is thick-tailed more or less "extreme" than normal?

[NEAS: Thick tailed is more extreme than normal; thin-tailed is less extreme.]

CalLadyQED: Oh, good. That's what I was guessing. Thank you for clarifying that for us all.


NessaT23
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How does the definition of extreme work for the lower (negative) end of the graphs?

For example, the t distribution graph. There's a point on the graph that looks like it's approximately (-3, -7). This is (theoretical, sample), so this says that theoretical > sample. I would interpret this to mean that it's not extreme, but I think the correct answer is that it's extreme (just in a negative direction.) Is this correct (do we use absolute value when making this determination?) Thanks.

[NEAS: Use absolute value.]


dclevel
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There is a typo in the original HW question. Figure 3.9 on page 37 is a t-distribution with 2 degrees of freedom.
smh1021
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I am unsure if I am interpreting the Chi-Square plot correctly. I believe that the upper tail is more extreme because it continues to increase at a much faster rate than the normal distribution, indicating that it will continue to deviate farther from the mean of the normal distribution. For the lower tail, it appears that it is leveling off near 0 while the normal distribution would continue to get more and more negative if you expanded the graph; I would think this indicates that the Chi-Square is less extreme if we are to look at things on a absolute value basis. Is the accurate?

[NEAS: Yes, that is a good description.]


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