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.
Solution
04
(1) The normal approximation in this case is applicable since:
Assumptions:
- Frank eats a large number of hot dogs
the sample size, or , is sufficiently large - We assume that the amount of time Frank spends on each hot dog does not depend on how many he has had previously
the times to consume each hot dog are independent and identically distributed
(2) Let
be the time taken to eat the -th hot dog. Let be the time taken to eat hot dogs. Then seconds with seconds. Since
minutes is seconds, by the CLT we have: Link to original
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.
Solution
06
In this case the normal approximation is applicable since we have a large sample size (need a large number of filters to last
hours) and they follow follow independent but identical distributions. Let
be how long the -th filter lasts. Let where we want to find such that . By normal approximation and the Central Limit Theorem, we have and thus
Link to originalfilters are required.
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?
Solution
08
(a)
Let
be the indicator that a man was chosen in the -th period. Then for each , and the are independent for each . Let be the total number of times a man was chosen. We can use a similar argument to Problems 2 or 3, or we can simply use linearity of expectation: (b)
(1) Using the standard formula for the variance of a sum of random variables, we have:
Since the variables are identically distributed, their variances are equal. Thus,
(2) Now, the sum has
terms, and since each is identically distributed, each term is identical. Thus, for some fixed
.
(3) We then have
Finally,
Link to original, and thus Plugging these values in, we have .
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 .
Solution
07
Let
and be the number of heads and tails respectively. Then we have the following two conditions: Thus,
. Let be the event that the -th flip is a head. Then for each and . Thus, by CLT,
. By the normal approximation, using the continuity correction, we have:
Link to original
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.)
Solution
09
(a)
(b)
(c)
The standardized variable is:
Then:
Link to original
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
10
Let
. From the CDF table,
: Link to original
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?
Solution
11
Sample mean RV:
has and . Standardize this and approximate with a normal distribution:
Link to original
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.
Solution