ECON 245: Descriptive Statistics and Probability

Objective: practising using continuous probability distributions with Excel.

Continuous Distributions

• x

• µ, mean

• σ, standard deviation

• and cumulative.

– TRUE is used if we want the cumulative probability.

E.g.: Sup. a Tire Company developed a new tire to be sold in their stores. Since it is a new product, the managers believe that offering a mileage guarantee will be important for customers to accept the product. Thus, to finalize the tire mileage guarantee, the managers want probability information about x = the number of miles the tires will last.

• µ = 36, 500.

• σ = 5, 000.

• It is reasonable to assume a normal distribution.

The function NORM.D IST can be used to compute cumulative probabilities for a normal distribution.

To compute a cumulative probability that the tire mileage will be less than or equal to 40,00 miles we need to enter the following formula into any cell:

= NORM.D IST(40000, 36500, 5000, TRUE).

- The number that appears in the cell (0.758) will be the probability of tire mileage being less than or equal to 40,000.

• Thus, the probability that tire mileage will exceed 40,00 miles is:

1 − 0.758 = 0.2420

Note: TRUE can also be imputed as the number 1. E.g.: = NORM.D IST(40000, 36500, 5000, 1).

Excel also has the NORM.INV function which uses an inverse computation to find the x value corresponding to a given cumulative probability.

For instance, sup. we are interested instead in finding the guaranteed mileage that should be offered so that no more than 10% of the tires are eligible for the guarantee, then:

= NORM.INV(.1, 36500, 5000).

- The number that appears in the cell (30,092) will be the probability of tire lasting 30,092 or less is 10%

Excel’s NORM.S.D IST can be used to compute probabilities for r.v./ that follow a standard normal distribution. The general form of the function is: NORM.S > D IST(z, cumulative) where, z refers to the z value corresponding to the random variable and TRUE is specified for the second argument if a cumulative probability is what we are after. Similarly, NORM.S.INV function uses an inverse computation to find the z value corresponding to a given cumulative probability. The general form of the function is: NORM.S.INV(probability) where the only argument needed is the cumulative probability to be used. The function returns the corresponding z value.

Lastly, the function for computing exponential probabilities is EXPON.D IST, which requires three inputs:

E.g.: an exponential probability distribution with mean, µ = 15. The probability that the exponential variable is less

= EXPON.D IST(6, 1/15, TRUE).

- The number that appears in the cell (0.3297) indicates that the probability the exponential variable will be less than or equal to 6 is 0.3297.

a. What is the probability that a randomly selected exam will have a score of at least 71?

b. What percentage of exams will have scores between 89 and 92?

c. If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award?

a. What is the probability that a randomly selected LU graduate will have a starting salary of at least $38,400?

b. Individuals with starting salaries of less than $19,600 receive a low income tax break. What percentage of the graduates will receive the tax break?

c. What are the minimum and the maximum starting salaries of the middle 98% of the LU graduates?