01
Normal approximation - Eating hot dogs
Frank is a competitive hot dog eater. He eats
in with . What is the probability that Frank manages to consume
in or less, in an upcoming competition? Use a normal approximation from the CLT to estimate this probability. State the reason that the normal approximation is applicable.
02
De Moivre-Laplace Continuity Correction
A fair die is rolled 300 times.
Use a normal approximation to estimate the probability that exactly 100 outcomes are either 3 or 6.
Do this with and without the continuity correction.
03
Normal approximation - Ventilator filters
A mechanical ventilator model uses air filters that last 100 hours on average with a standard deviation of 30 hours.
How many filters should be stocked so that the supply lasts 2,000 hours with probability at least 95%? Use a normal approximation to estimate the answer.
State the reason that the normal approximation is applicable.
04
Normal approximation - Grading many exams
An instructor has 50 exams to grade. The grading time for each exam follows a distribution with an average of 20 minutes and variance of 16. Assume the grading times per exam are independent.
Roughly what are the odds that after 450 minutes of grading, at least half the exams will be graded? Use a normal approximation to estimate the answer.
State the reason that the normal approximation is applicable.
05
Indicator method, exchangeability, summation rules
A class has 40 students: 24 women and 16 men. Each period the teacher selects a random student to present an exercise on the board from among those who have not presented already.
Let
count the number of times a man was chosen after 15 class periods. (a) Find
. (b) Find
. Hint: Is
independent of ? Do you know anyway?
06
Graphing convergence to a bell curve
Let
be independent RVs each having the following PMF: Notice that
and for each . Define
. So and , and therefore . By the CLT,
should converge to the standard normal distribution as . In this problem you explore the limit process by direct computation of the cases . (a) Compute the PMF of
in terms of . Hint steps:
is the number of sequences of outcomes of having a total sum of , times the probability of any particular such sequence. - Find the probability of any particular sequence of
outcomes of . - There are
ways to get outcomes of and outcomes of . Solve for in terms of . - Put 2. and 3. together in 1. to get your formula.
(b) Compute the PMF of
for using your formula from (a) together with a scaling (derived variable calculation). (c) Draw the graphs of the PMF of
and the PMF of for .
07
Normal approximation of the binomial - Double ones
Roll a pair of dice 10,000 times. Estimate the odds that the number of “snake eyes” (double ones) obtained is in the range
.
08
Normal approximation - Heads v. tails
Flip a coin 10,000 times. Let
measure the number of heads, and measure the number of tails. Estimate the probability that and are within 100 of each other. Hint: Write an inequality for the condition, then sub a formula for
in terms of .
09
Burning through light bulbs
A 100 Watt light bulb’s expected lifetime is 600 hours, with variance 360,000. An advertising board uses one of these light bulbs at a time, and when one burns out, it is immediately replaced with another. (The lifetime of each bulb is independent from the others.) Let the continuous random variable
be the total number of hours of advertising from 10 bulbs. (a) Find the expected value of
. (b) Find the variance of
. (c) Use the CLT to approximate the probability that
is less than 5,500 hours. (You should decide whether it is appropriate to use the continuity correction.)
10
Body weights
Assume that body weights of men are Normally distributed with a mean of 170 pounds and a standard deviation of 30 pounds.
What is the body weight threshold separating the lightest 90% from the heaviest 10%?
Solution
11
Community college ages
At a community college, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn.
What is the probability that the average age of the students in the random sample is less than 23 years?
12
Winning the lottery
Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.
Use the Central Limit Theorem and the continuity correction to approximate the probability of winning at least 20 times when you purchase 100 of these game cards.