Statistics Exam #2

Here as promised, word for word¹, is the second exam I gave my introductory statistics students. A prerequisite for the class was at least “Pre calculus”. I emphasize—in italics!—that I have taught this course before many times and have never had as much difficulty as I am having this time through.

This exam came after reviewing chapters three, four, and six for at least a week. These may be found in my typo-ridden class notes (a copy of which is linked on the far left of this page). These notes describe, in great detail, with plenty of in-class and out-of-class exercises, all about the binomial and normal distributions (where the parameters are assumed to be known; i.e., no estimation: this is pure probability). We spent a great deal on the problems of chapter six, which are like those below, but which are much harder.

At the end of the exam, I include a list of “Potentially useful formulas”, such as the binomial, the normal approximation to the binomial, and each necessary combinatoric equation. In other words, a cheat sheet. The students did not have to memorize any equations.

I am embarrassed to show this because I find it is shockingly easy. I made it so to bring the grade of the class up from its on-average failing (yes, failing) level. I do this because I am just a visitor and I do not want to cause a great disruption in the usual procedures.

Please, no sea lawyers arguing about ambiguity in the questions: the students (who came to class) have already had dozens of problems just like these and know what is expected of them. Like I said before, grades improved. The median score for this test was 76; the average was 70.

1. Each floor in one of Behemoth University’s multi-story dormitory houses 200 students. On Thursday nights, instead of remembering they are in college, and thus staying at home to study, history shows that about 70% of students “go out.”
   1. What is the probability that the third floor empties out on Thursday night?
   2. If the dormitory has 10 floors, what is the probability the entire building empties out?
2. In order for a professor to have a completely successful Statistics 101 class, each student who registers should pass the course. 35 students are registered for this class. The probability that any individual student passes is 90%. What is the probability of not having a completely successful Statistics 101 class?
3. 1. Sketch the normal distribution x ~ N(-25.0,12.5). Take your time and CLEARLY label your axes. Label all the usual points with their actual numbers, not their letters.²
   2. For this same normal, what is the value of w such that Pr(x>w|E_n)=0.025?
   3. What is the probability that Pr(x<-25.0|E_n)?
4. The FCC designates AM radio stations with either three or four letters, with the restriction that the first letter must be a K (for stations West of the Mississippi) or W (East of the Mississippi). Examples are WOR, KSLT and so on. How many unique AM radio stations are possible?
5. The amount of money that people spend on Christmas is of obvious interest to retailers. S-Mart’s top-of-the-line shotgun is the 12-gauge double-barreled Remington, retailing for $109.99. During the Christmas rush, each S-Mart branch sees about 5,000 paying customers. The chance any paying customer buys the Remington is 3%.
   1. What is the approximate probability any S-Mart branch sells more than 150 shotguns? Explain your answer for credit.
   2. We obviously do not know how much money an S-Mart store will make on selling Remingtons. But we can express our uncertainty in this number. Do so; and sketch a picture. Hint: If S-Mart sold just one Remington across all its stores, how much money would they make? If they sold just two? three? more?

The Answers are:

1. 1. 0.7²⁰⁰
   2. 0.7²⁰⁰⁰
2. 1 – 0.90³⁵
3. 1. The central parameter is -25, the two usual points are plus or minus 2 times the spread parameter³, i.e. -50 and 0. These points demarcate the approximate interval in which we expect 95% of the observables to lie. This sounds much harder than it is.
   2. 0
   3. 50% or 0.5
4. 2 x 26² x 27 (or any variation of this).
5. 1. Use the normal approximation to the binomial formula, where the central parameter is found to be 5000 x 0.03 = 150; thus the probability of selling more than 150 guns is 50%.
   2. Sketch the appropriate normal approximation to the binomial from the first part and multiply all the usual points (like the 150) by $109.99. The actual calculation was not required: just writing “150 x $109.99” was sufficient.

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¹The school’s name, which was part of the exam, has been changed; so has the name of the course.

²I.e., the “m” and “s” parameters of the normal. We had done dozens of these in and out of class by this point. Can you guess how many students wrote the letters and not numbers?

³2 is close enough to 1.96 for anybody, especially in this class.