Category Archives: Statistics

What is the probability that the parcel was shipped express and arrived the next day?

1.  In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective. For which lot size n will A and B be more nearly independent: n = 10 or n = 10,000? Explain.

2.  A certain delivery service offers both express and standard delivery. Seventy-five percent of parcels are sent by standard delivery, and 25% are sent by express. Of those sent standard, 80% arrive the next day, and of those sent express, 95% arrive the next day. A record of a parcel delivery is chosen at random from the company’s files.

 

a.  What is the….

Given that the test is positive, what is the probability that the person has the disease?

1.  Refer to Exercise 28. Assume that both inspectors inspect every item and that if an item has no flaw, then neither inspector will detect a flaw.

a.  Assume that the probability that an item has a flaw is 0.10. If an item is passed by the first inspector, what is the probability that it actually has aflaw?

b.   Assume that the probability that an item has a flaw is 0.10. If an item is passed by both inspectors, what is the probability that it actually has a flaw?

 

2.   Refer to Example 2.26. Assume that the proportion of people in the community who have the disease is 0.05.

 

a.   Given that the test is positive, what is the probability that the person has the disease?

….

If a man tests negative, what is the probability that he actually has the disease?

1.  Refer to Example 2.26.

a.  If a man tests negative, what is the probability that he actually has the disease?

b.  For many medical tests, it is standard procedure to repeat the test when a positive signal is given. If repeated tests are independent, what is the probability that a man will test positive on two successive tests if he has the disease?

c.   Assuming repeated tests are independent, what is the probability that a man tests positive on two successive tests if he does not have the disease?

d.  If a man tests positive on two successive tests, what is the probability that he has the disease?

2.  If A and B are independent events, prove that the following pairs of events are independent: A c and B,….

Find the probability mass function of X

1.  Determine whether each of the following random variables is discrete or continuous.

a.  The number of heads in 100 tosses of a coin.

b.  The length of a rod randomly chosen from a day's production.

c.   The final exam score of a randomly chosen student from last semester's engineering statistics class.

d.  The age of a randomly chosen Colorado School of Mines student.

e.  The age that a randomly chosen Colorado School of Mines student will be on his or her next birthday.

 

 

2.   A survey of cars on a certain stretch of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a….

What is the smallest possible value for Y?

1.   Candidates for a job are interviewed one by one until a qualified candidate is found. Thirty percent of the candidates are qualified.

a.  What is the probability that the first candidate is qualified?

b.  What is the probability that the first candidate is unqualified and the second candidate is qualified?

c.   Let X represent the number of candidates interviewed up to and including the first qualified candidate.

d.  Find P(X = 4). Find the probability mass function of X.

 

2.   Refer to Exercise 10. The company has two positions open. They will interview candidates until two qualified candidates are found. Let Y be the number of candidates interviewed up to and including the second qualified candidate.

 

a.  What is the smallest possible value for Y?

b.  What is….

Find the mean and standard deviation of the length of time that four bulbs will last.

1.  If X and Y are independent random variables with means μX = 9.5 and μY = 6.8, and standard deviations σX = 0.4and σY = 0.1, find the means and standard deviations of the following:

a.  3X

b.  Y − X

c.   X + 4Y

 

2.   The bottom of a cylindrical container has an area of 10 cm2 . The container is filled to a height whose mean is 5cm, and whose standard deviation is 0.1 cm. Let V denote the volume of fluid in the container.

 

a.  Find μV .

b.  Find σV .

 

3.   The lifetime of a certain transistor in a certain application has mean 900 hours and standard deviation 30 hours. Find the mean and standard deviation of the length….

Find the standard deviation of the thickness of an item.

1.  Two batteries, with voltages V1 and V2 , are connected in series. The total voltage V is given by V = V1 + V2 . Assume that V1 has mean 12 V and standard deviation 1 V, and that V2 has mean 6 V and standard deviation 0.5 V.

a.  Find μV . Assuming V1 and V2 to be independent,

b.  find σV .

 

2.   A laminated item is composed of five layers. The layers are a simple random sample from a population whose thickness has mean 1.2 mm and standard deviation 0.04 mm.

a.  Find the mean thickness of an item.

b.  Find the standard deviation of the thickness of an item.

 

 

3.  Two independent measurements are made of the lifetime of a….

Find the standard deviation of the total volume of the beverage in the case.

1.  The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution (1 mole = 6.02 × 10 23 molecules). If X is the molarity of a solution of magnesium chloride (MgCl2 ), and Y is the molarity of a solution of ferric chloride (FeCl3 ), the molarity of chloride ion (Cl− ) in a solution made of equal parts of the solutions of MgCl2 and FeCl3 is given by M = X + 1.5Y. Assume that X has mean 0.125 and standard deviation 0.05, and that Y has mean 0.350 and standard deviation 0.10.

 

a.   Find μM.

b.   Assuming X and Y to be independent,find σM.

 

2.  A machine that fills bottles with a beverage….

Find the mean perimeter.

1.  The four sides of a picture frame consist of two pieces selected from a population whose mean length is 30 cm with standard deviation 0.1 cm, and two pieces selected from a population whose mean length is 45 cm with standard deviation 0.3 cm

a.  Find the mean perimeter.

b.  Assuming the four pieces are chosen independently, find the standard deviation of the perimeter.

 

2.   A gas station earns $2.60 in revenue for each gallon of regular gas it sells, $2.75 for each gallon of midgrade gas, and $2.90 for each gallon of premium gas. Let X1 , X2 , and X3 denote the numbers of gallons of regular, midgrade, and premium gasoline sold in a day. Assume that X1 , X2 , and X3 have….

What is the standard deviation of the total length?

1.  A certain commercial jet plane uses a mean of 0.15 gallons of fuel per passenger-mile, with a standard deviation of 0.01 gallons.

a.  Find the mean number of gallons the plane uses to fly 8000 miles if it carries 210 passengers.

b.  Assume the amounts of fuel used are independent for each passenger-mile traveled.

c.   Find the standard deviation of the number of gallons of fuel the plane uses to fly 8000 miles while carrying 210 passengers.

d.  The plane used X gallons of fuel to carry 210 passengers 8000 miles.

e.  The fuel efficiency is estimated as X/(210 × 8000). Find the mean of this estimate.

f.     Assuming the amounts of fuel used are independent for each passenger-mile, find the standard deviation of the estimate in….