Download AP Stat Ch. 7 Day 1 Lesson Worksheet 08

AP STATISTICS
Chapter 7 – Random Variables
Section 7.1: Discrete and Continuous Random Variables
Name _______________________
Date __________
Period _____
Day 1 Assignment: Exercises 7.2, 7.6, 7.8, 7.10, 7.12, 7.16
A. Key vocabulary: random variable, probability distribution, discrete random variable,
continuous random variable, density curve, normal density curve, probability histogram
B. Review Chapter 6 Test
C. A random variable is a variable whose value is a numerical outcome of a random phenomenon.
Example 1:
Toss 3 coins and count the number of heads. Let X equal the number of heads counted.
X will vary randomly from trial to trial. X will take on the values 0, 1, 2, or 3. So, X is a discrete random
variable because its’ value is a numerical outcome of a random phenomenon and there are a countable
number of possible values it can take on.
Experimental versus theoretical probability:
Experiment: Each student tosses 3 coins 10 times and records number of heads each trial.
Heads 0
Count
1
2
3
Theory: Make a tree and determine theoretical probability of each outcome.
Tree:
Probability Histogram:
Heads
0
Theoretical
Probability
1
2
3
A discrete random variable X has a countable number (finite or infinite) of possible values. The
probability distribution of X lists the values and their probabilities:
Value of X:
x1
x2
x3
…
xk
Probability:
p1
p2
p3
…
pk
Of course, the probabilities must all be between 0 and 1 (inclusive) and have a sum of 1.
Example 2: Generate a random number on your calculator using RAND. Let the value generated be X.
X will be a value between 0 and 1 inclusive, and it will vary randomly from trial to trial. So, X is a
continuous random variable because its’ value is a numerical outcome of a random phenomenon and the
values it takes on are values drawn from a part of a continuous “number line”.
Probability of a number between 0.3 and 0.7
P(0.3 < X < 0.7) = 0.4

If we used a spinner scaled as in the graphic above, we could accomplish the same thing as using RAND.
The essence of this situation is that every number from 0 to 1 is possible and equally likely to occur.
A continuous random variable X takes on all values in an interval of numbers. The probability
distribution of X is described by a density curve. The probability of any event is the area under the density
curve and above the values of X that make up that event.
D. Normal Distributions as Probability Distributions
Recall:
N(,)
and
Z
X 

Normal curve with mean and standard deviation , and that Z standardizes by quantifying how many
standard deviations a value is above or below the mean.
Example 7.4 (page 400) – Drugs in Schools
SRS of n = 1500 American adults finds 30% believing drug usage is our schools most important problem.
Sample proportions ( p̂ ) are random variables that, under the right conditions, is and unbiased estimator that
distributes normally around the true population proportion.
For the sake of our discussion, assume that p̂ has the distribution N(0.3, 0.0118).
Find P( p̂ < 0.28)
Find P(0.28 < p̂ < 0.32)
Find P( p̂ > 0.31)
Example: Suppose adult American females have heights N(54,2). What is the probability that a randomly
selected gal has X>55?