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Assignment3.pdf-Stat 241/251: Assignment 3 Due Mon...

Assignment3.pdf-Stat 241/251: Assignment 3 Due Monday

Assignment3.pdf-Stat 241/251: Assig...

Assignment3.pdf-Stat 241/251: Assignment 3 Due Monday

##### Page 1

Stat 241/251: Assignment 3

Due Monday March 21

st

by 8:50am

1. Suppose you work as a waiter/waitress at Mahoney & Sons Pub on UBC campus.

From your time working there you know that the average tip (per table) you receive is

$7.11 with a standard deviation of $5.90. The distribution for the amount of tip you

receive is highly skewed to the right.

(a) What is the probability that a table tips you more than $13.50? If this question

cannot be answered with the information given, state why.

(b) What is the probability that the average tip you receive from 5 randomly selected

tables you wait on is more than $13.50? If this question cannot be answered with

the information given, state why.

(c) On a typical weekend, you usually wait on 30 tables over the entire weekend.

What is the probability that on a given weekend (assuming you wait on exactly

30 tables) that you make more than $250 in tips?

If this question cannot be

answered with the information given, state why.

2. When using the Google search engine, one can see the amount of time that was taken

for the search engine to provide you with a list of pages.

Suppose that the amount

of time until a list of pages is provided is normally distributed with a mean of 0.33

seconds and a standard deviation of 0.08 seconds.

(a) What is the probability that a randomly selected search takes more than 0.4

seconds?

(b) What is the 95

th

percentile for search times?

(c) Now, consider a set of 70 searches using the Google search engine. What is the

probability that at most, 10 of the 70 searches take longer than 0.4 seconds? You

may use an approximate method.

(d) Again, consider a set of 70 searches.

What is the probability that the mean

amount of time for the 70 searches is more than 0.34 seconds?

1

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3. In this question we will examine the casino game Roulette.

Essentially, the game

involves randomly selecting a number, from 38 possible numbers.

There are 18 red

numbers, 18 black numbers and 2 green numbers. All numbers are equally likely to

show up on a given spin, and spins of the wheel are independent. Consider making

bets on the colour of the number coming up in Roulette (red or black). This bet pays

1:1, meaning if you bet $1, winning means you gain $1 and losing means you lose $1.

(a) If you make 50 bets, each of $1, what is the probability that you leave the casino

having made money? That is, you leave the casino with more money than you

started with. Use an approximate method.

(b) If you make 500 bets, each of $1, what is the probability you leave the casino

having made money? Use an approximate method.

(c) If you make 5000 bets, each of $1, what is the probability you leave the casino

having made money? Use an approximate method.

(d) Sketch out a plot representing the probability of making money (y) as a function

of the number of bets made (x). You don’t have to calculate any more than you

already have; you may approximate the shape of this curve.

Brieﬂy comment

on what happens to the probability of making money as the number of bets (n)

increases. Make sure to use statistical terms in your answer. Please limit your

answer to a few sentences.

(e) Consider again making 5000 bets, each of $1, on the colour of the number in

Roulette. Create an interval for the amount won (or lost) that you will fall into

95% of the time. Center this interval around the expected loss. Comment on this

interval.

4. Suppose that small airplanes arrive at a small airplane airport at a rate of 4 per hour,

and are assumed to follow a poisson process.

(a) What is the probability that at most 3 planes land in a 75 minute period?

(b) What is the probability that upon the airport opening for the day, it takes more

than 30 minutes for the ﬁrst plane to arrive?

(c) Consider an entire day. We will assume that a day consists of 16 hours of operation

for the airport.

What is the probability that at least 50 planes land over the

one day (16 hour) period? You may use an approximate method to answer this

question.

2

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