AP Statistics

Linear Transformations

El DoradoCommunity College considers a student to be full-time if he or she is taking between 12 and 18 units. The number of units X that a randomly selected El DoradoCommunity College full-time student is taking in the fall semester has the following distribution.

Number of Units (X) / 12 / 13 / 14 / 15 / 16 / 17 / 18
Probability / 0.25 / 0.10 / 0.05 / 0.30 / 0.10 / 0.05 / 0.15
  1. At El Dorado Community College, the tuition for full-time students is $50 per unit. Let T = cost of tuition for the students. What is the expected cost and standard deviation for full-time students?

Cost of tuition (T)
Probability / 0.25 / 0.10 / 0.05 / 0.30 / 0.10 / 0.05 / 0.15

Notice:

  • ______
  • ______
  1. In addition to tuition charges, each full-time student at El Dorado Community College is assessed student fees of $100 per semester. If C = overall cost for a randomly selected full-time student, C = 100 + T.

Cost of tuition + fee (C)
Probability / 0.25 / 0.10 / 0.05 / 0.30 / 0.10 / 0.05 / 0.15

Notice:

  • ______
  • ______
  1. If a is being added to each value of X, what happens to µx ?
  1. If b is being multiplied with each value of X, what happens to µx ?

General statement about mean: ______

  1. If a is being added to each value of X, what happens to σx ?
  1. If b is being multiplied with each value of X, what happens to σx ?

General statement about standard deviation: ______

Example.

  1. Mrs. Prill’s AP Stat students have asked her to “curve” the grades on the last test. The tests had µx = 75 and σx =6.22.Jake requested that each grade be multiplied by 1.1 in order to get a 10% increase on each test grade. What would the new expected score and standard deviation be if Mrs. Prill were to agree to his proposal?
  1. In addition tothe 10% increase, Alex proposed that Mrs. Prill also add 5 points to each test. What would the new expected score and standard deviation be if Mrs. Prill were to agree to both proposals?

Tuition Charge (T) / 600 / 650 / 700 / 750 / 800 / 850 / 900
Probability / 0.25 / 0.10 / 0.05 / 0.30 / 0.10 / 0.05 / 0.15

The mean and standard deviation of this distribution are:

Notice that

  • the shapes of both distributions are the same
  • the mean of the distribution of T is 50 times bigger than the mean of X: 732.5 = 50(14.65)
  • the standard deviation of T is 50 times bigger than the standard deviation of X: 103 = 50(2.06)
  1. Here is the probability distribution for C:

Overall Cost (C) / 700 / 750 / 800 / 850 / 900 / 950 / 1000
Probability / 0.25 / 0.10 / 0.05 / 0.30 / 0.10 / 0.05 / 0.15

The mean and standard deviation of this distribution are:

Notice that

  • the shapes of both distributions are the same
  • the mean of the distribution of C is $100 larger than the mean of T: 832.5 = 100 + 732.5
  • the standard deviation of C is the same as the standard deviation of T
  1. We can add/subtract/multiply/divide to means.
  2. So then σy = |b|σx (you can only multiply/divide to standard deviations)