## Fox Module 1 Statistical models HW

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NEAS
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Regression analysis, Module 1, “Statistical models”

(The attached PDF file has better formatting.)

Homework assignment: probabilities

Fox uses the data in Table 1.1 on page 5 to infer that judges grant leave at different rates.

A.    If all judges grant leave in 25% of cases, and the differences among judges are random fluctuations, what is the probability a judge (Desjardins) grants leave in 49% or more of cases?

B.    If all judges grant leave in 25% of cases, and the differences among judges are random fluctuations, what is the probability that a judge (Pratte) grants leave in 9% or fewer of cases?

Write an algebraic expression for the solution. You need not compute a numerical solution.

Note: Judge Desjardins heard 47 cases and granted leave in 49% × 47 = 23 cases.

Write the expression for 23 successes in 47 cases with a probability of 25%.

This is a binomial probability with ð = 25%.

Write the summation for 23 through 47 successes. You need not evaluate the sum.

The sum goes from 23 successes to 47 successes.

Judge Pratte heard 57 cases and granted leave in 9% × 57 = 5 cases.

Write the expression for 5 successes in 57 cases with a probability of 25%.

Write the summation for 0 through 5 successes. You need not evaluate the sum.

The sum goes from 0 successes to 5 successes.

Note: The PMF of the binomial distribution is

where n is the number of trials and p is the probability of success on each trial.

You do not have to compute any figures for this homework assignment.

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Fox Module 1.HW.pdf (3.6K views, 43.00 KB)
noturbizniss
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Wow...it's been so long since I did stats in college or did basic probability...is the sum question just one of those with i = x to y and then the binomial expression after, or am I missing something?

P.S. Any tips on how to type math expressions would be great!

[NEAS: Correct. For math expresssions, you can sometimes copy from Word to the discussion forum.]

Chas
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No help on the equations but the sum i = x to y sounds right to me and then just use the binomial expression.

[NEAS: Fox’s textbook, like much modern statistical analysis, emphasizes the conditional distribution of the response variable (the dependent variable). Yes, this is a binomial distribution, which is used again in later modules. When the response variable is a probability, the proper conditional distribution is binomial, not normal. In practice, if the number of observations is large enough, the probability is close to 50%, and the observed values are near the mean, the normal distribution is a good approximation. The exact solution uses the binomial distribution. You should be familiar with the binomial distribution for the actuarial exams. If you want to check a formula for any statistical distribution, check the distribution in Wikipedia, which gives all the formulas and expressions you might need.]

jordanp
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Are the first section of questions (below) meant to be an algebraic expression or a numeric solution?

If all judges grant leave in 25% of cases, and the differences among judges are random fluctuations, what is the probability a judge (Desjardins) grants leave in 49% or more of cases?

If all judges grant leave in 25% of cases, and the differences among judges are random fluctuations, what is the probability that a judge (Pratte) grants leave in 9% or fewer of cases?

FrequentlySevere
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I believe that we are only intended to provide the algebraic expression. I think that these are supposed to be *like* significance tests which are covered later on.

I agree with others, this question was reasonable, but also totally out of left field.

dclevel
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When the overview instructs me to know a certain formula is it implying that I need to memorize said formula?
dwscott07
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For those who are like me and have to take that extra step:

Part A) Prob(N>=23) ~ 0.03389%

Part B) Prob(N<=5) ~ 0.1768%

I did this using excel to calculate each probability for N from 23 through 47:

=FACT(M)/(FACT(M-N)*FACT(N))*(.25)^N*(.75)^(M-N)

and just sum the probabilities together. If you spread the range from 0-47, you will see that the total probability is equal to 1.00. To get part B, repeat with M=57.

chichiri7
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Hey there,

I'm not sure whether I understand the first set of two questions correctly. Am I suppose to use a normal approximation or a binomial expression? If it is that I have to use a normal approx, then I'm assuming 25% is the average and i have to calculate the standard deviation from the table? Or if it's binomial, then would I have to assume a random number of trials and then use 25% as the probability of success and calculate the probability from there?

Any help would be appreciated, thanks!

[NEAS: Use a binomial distribution. Table 1.1 gives the number of cases for the two judges (Desjardins and Pratte).]

Edited 8 Years Ago by NEAS
Brodon
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Edited 8 Years Ago by Brodon
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