MathStat474 - HW 8

Posted on Mon Oct 6. Due: WED Oct 22.

Problem 1.

Length (in days) of a randomly chosen human pregnancy is a normal random varialbe with \(\mu=266\) and \(\sigma=16\).

  1. Find Q1, the median, and Q3.
  2. What is the probability that a randomly chosen pregnancy will last less than 246 days?
  3. What is the probability that a randomly chosen pregnancy will last longer than 240 days?
  4. What is the probability that a randomly chosen pregnancy will last longer than 500 days?
  5. Suppose a pregnant person’s spose has scheduled their business trip so that they will be in town between 235th and 295th days. What is the probability that the birth will take place during that time?

Make sure you use standardized values to solve this problem. You can also solve integrals, for which you MUST show your work; or you can use software, in which case you MUST print out the code you used. Otherwise, look up the Z-tables and indicate how you arrived at the answers.

The standard normal Z-table is posted on the site (copy from the back of our textbook). If you are using it please highlight the cell in the table where you looked up the desired value!

Problem 2.

Solve exercise 6.1.

Given a continuous uniform distribution, show that:

  1. \(\mu=\frac{A+B}{2}\) and
  2. \(\sigma^2=\frac{(B-A)^2}{12}\).

Problem 3.

Solve exercise 6.3.

The daily amount of coffee, in liters, dispensed by a machine located in an airport lobby is a random variable X having a continuous uniform distribution with A = 7 and B = 10. Find the probability that on a given day the amount of coffee dispensed by this machine will be

  1. at most 8.8 liters;
  2. more than 7.4 liters but less than 9.5 liters;
  3. at least 8.5 liters

Problem 4.

Solve exercise 6.11.

A soft-drink machine is regulated so that it dis- charges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a stan- dard deviation equal to 15 milliliters,

  1. what fraction of the cups will contain more than 224 milliliters?
  2. what is the probability that a cup contains between 191 and 209 milliliters?
  3. how many cups will probably overflow if 230- milliliter cups are used for the next 1000 drinks?
  4. below what value do we get the smallest 25% of the drinks?

Problem 5.

Solve exercise 6.33.

Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk drivers will be

  1. less than 32?
  2. more than 49?
  3. at least 35 but less than 47?

Problem 6.

Solve exercise 4.12. In this problem we are practicing to compute the mean of a random variable.

If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function:

\(f(x)= 2(1−x)\) for \(0<x<1\) and \(f(x)=0\) elsewhere, find the average profit per automobile.

Problem 7.

Solve exercise 4.37. In this problem we are practicing to compute the variance of a random variable.

A dealer’s profit, in units of $5000, on a new automobile is a random variable X having the density function given in Exercise 4.12 on page 117. Find the variance of X.

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