Homework

Due Wed

1)  When I was 16 I drove a Dodge Colt. The gas gauge was empty, but I knew the distance it could go on a tank of gas is normally distributed with a mean of 320 miles and a standard deviation of 15 miles.

  1. What is the probability that the next tank will go for 310 miles? 0
  2. What is the probability that the next tank will go for less than 301.4 miles? 0.107488
  3. There’s no way to guarantee I’ll never run out of gas, but I want to make sure it only happens 5% of the time. How many miles do I dare drive before filling up again? 295.3

2)  If the Engineering Building catches fire the amount of heat the structural support beams will be subjected to is N(1500, 2002) degrees.

  1. What is the probability a random fire will subject the beam to more than 2128 degrees? 0.000845
  2. What is the probability that a random fire will subject the beam to temperatures between 1868 and 1944 degrees? 0.019675
  3. The beams are supposed to be built so that they only fail in 2% of the fires. What temperature do they need to be able to withstand? 1910

3)  The height of wind turbines in South Dakota is N(80, 152) meters.

The height of wind turbines in Wyoming is N(120, 222) meters.

A planning committee wants to build a 112 meter wind turbine in South Dakota and a 162 meter wind turbine in Wyoming.

  1. Which wind turbine will be the tallest? Wyoming
  2. Explain why the wind turbine in South Dakota will be relatively taller than the wind turbine in Wyoming when the distribution within each state is considered.

It is further on the curve, higher z-score, fewer of its state that are shorter, etc.

  1. How tall would the wind turbine in Wyoming need to be for it to match the height of the wind turbine in South Dakota relative to other turbines in the state? 166.933

4)  The stop light at the corner of Grand and 19th street stays red for a random amount of time distributed N(180, 361) seconds.

  1. What is the 75th percentile for the time the stop light is red? 192.73
  2. I swear last Thursday that light hated me. It stayed red 229 seconds. How often does that happen? 0.004940
  3. What two times mark the most likely (middle) 95% of the times the light is red?

(as a hint a picture of what this is asking is shown below) 142.76, 217.24

5)  For each scenario listed below an inspector at the TechKnow company has a question that can be answered with a probability distribution. Say whether the distribution is most likely to be

Normal

Lognormal

Weibull

Gamma

Beta

Exponential

(as an added hint each distribution is used exactly once)

a)  An inspector knows if he can find more than 10 defective microchips at the TechKnow company then he can shut them down. He wondering how long it will take for him to find 10 defective chips gamma

b)  When a computer is loaded for the first time it has to load several components at the same time (thanks to multiple processors). The inspector waits until every component has finished loading. He wonders how long it will take the next computer to load weibull

c)  The employee roster at TechKnow has been experiencing the same slow increase as the rest of the company. The inspector wonders how many other companies have as many employees as TechKnow lognormal

d)  The electrical voltage through a microchip is supposed to be exactly 0.11 volts. If it is over 0.114 the chip will fry. The inspector wonders how likely the nextchip is to get fried by over voltage normal

e)  The inspector grabs a box of 250 microchips. He wonders if this box will have 10 defective chips beta

f)  He will plan his lunch break for when he finds his first defective chip. He’s wondering how long it will be before he can eat exponential

6)  When a 4-cylinder car is brought in for a tune up the number of cylinders that have low compression is between 0 and 4. The table below shows the probabilities for each

X / 0 / 1 / 2 / 3 / 4
f(x) / 0.80 / 0.10 / 0.05 / 0.03 / 0.02

a)  For a random car, what is the probability it will have any cylinders with low compression? .2

b)  For a random car, how many cylinders do you expect to have a problem? 0.37

c)  What is the standard deviation for cylinders with a problem? 0.8678

7)  The Wyoming Lottery (wyominglottery.com) lists the following probability for winning with a $1 MatchPrizeOdds ticket. What is the expected payout from a lottery ticket? 0.1248

Winnings / Probability
200,000 / 1 in 195,249,054
10,000 / 1 in 5,138,133
100 / 1 in 7,947
7 / 1 in 247
4 / 1 in 123
3 / 1 in 62

8)  Based on a sample of computer temperatures we can assume that the temperatures are normally distributed with a mean of 60.20oF and a standard deviation of 0.62oF. A computer technician wants to select a minimum temperature for requiring maintenance. What should that temperature be if he wants only 5% of computers to exceed it? 61.22

9)  The distribution for the number of people who use an elevator per work day is shown below.

Which of the following notations could be used to describe this distribution?

people ~ N(10, 352)

people ~ N(10, 35)

people ~ N(35, 102)

people ~ N(35, 10)

people ~ N(10, 60)

people ~ N(10, 602)

10)  How many pounds a statistician can bench press is normally distributed with a mean of 130 and standard deviation of 50. If Scott can bench 135 pounds, approximately what percentage of statisticians can bench more than Scott? 0.4602

11)  The cost of an phone depends on which type of phone you get. For a smart phone the prices are N($80, $252). Pierce only has $50 in his wallet. Which picture below shows the percentage of phones that Pierce can get? A

12)  My dad is a used car salesman that also rents cars. The rental cost is the same no matter how many miles you drive, until it hits a limit (where extra charges begin to apply). He believes that the number of miles driven by rentals is normally distributed with mean 4,500 and standard deviation 1,800. He wanted to set the mileage limit at a point such that 80% of the tourists drive fewer miles. Calculate the mileage limit to the nearest mile. 6014.88

13)  The price of a band saw in the US is N(500, 6400) in dollars ($). In England the price is N(800, 16384) in pounds (£). If I have a $124 band saw that I want to sell to a guy in England how much should I ask for in £? 198.4

14)  The IG-88 rocket is designed to hit a target 60 kilometers away. Unfortunately it doesn’t always hit right on target, in fact the distribution for where it hits is normal with a mean of 60 kilometers and a standard deviation of 0.12 kilometers. If the rocket misses by 0.4 kilometers (either too close or too far) civilians will be injured. What is the probability of civilian injuries? 0.0008

15)  Adam works at Computer Fixit where people can bring broken computers. When a customer enters the store there is a chance he will be paid to work on their computer. The probability of each amount from a random customer is given in the following table:

Job payment / $10 / $100 / $400 / $1,500
Probability / 0.40 / 0.08 / 0.06 / 0.01

a)  When the next person walks into Adam’s Place, how much money can Adam expect to get from them? 51

b)  What is the probability he will get less than $400 from a random customer? 0.93

c)  What is the standard deviation for the payments from the customers? 174.18

d) What is the median job payment amount? 10