Download 7.2 ap stats 08-09 filled.notebook

7.2 ap stats 08­09 filled.notebook
August 31, 2009
7.2 Means and Variances of Random Variables
examples:
What is the average number of moves from one spin?
0
0
12
2
0
1
1
3
2
1
7.2 ap stats 08­09 filled.notebook
August 31, 2009
The probability distribution looks like this:
Number of spaces to move X:
Probability:
0
1
2
3
1/4 1/2 1/8 1/8
The mean (of all possible turns) =
expected value E(x) for the number of spaces in a turn is
However, the average of a few observed spins would be x.
2
7.2 ap stats 08­09 filled.notebook
August 31, 2009
Mean or expected value:
E(x)=µx = Σ x • P(x)
Any time you see the greek capital letter sigma, Σ, it means to add the things that follow.
3
7.2 ap stats 08­09 filled.notebook
August 31, 2009
A psychologist studied the number of puzzles subjects were able to solve in a 3 minute period while subjected to the sound of a jackhammer outside. Let X be the number of puzzles completed successfully by a subject. X has the following distribution:
X
P(X)
0 1 2 3 4
.3 .3 __ .1 .1
What is the mean µx of X?
4
7.2 ap stats 08­09 filled.notebook
August 31, 2009
Variance • describes spread or variability in outcomes of X. • average of the squared deviations of the variable X from its mean µx. • σ^2 = Σ [(x ­ µ)^2 • P(x)]
(1/4)(0­9/8)^2+(1/2)(1­9/8)^2+(1/8)(2­9/8)^2+(1/8)(3­9/8)^2 = 5
7.2 ap stats 08­09 filled.notebook
August 31, 2009
The psychologist problem again:
X
P(X)
0 1 2 3 4
.3 .3 __ .1 .1
What is the mean µ of X?
What is the variance σ^2 of X?
What is the standard deviation σ of X?
You can now do through 7.29 in the book.
6
7.2 ap stats 08­09 filled.notebook
August 31, 2009
x ≈ µ each SRS gives a different x near µ.
As n ⇒the population size, x ⇒ µ. This is The Law of Large Numbers
ex.
mean car odometer readings
mean # of TVs per home
7
7.2 ap stats 08­09 filled.notebook
August 31, 2009
example:
Time to finish a test is normally distributed with mean 40 minutes and standard deviation 3 minutes. What is the probability that a randomly selected student will finish the test in less than 38 minutes?
P(38<X<42) = The probability is approximately .15 that the time it takes a randomly selected AP Stats student to finish the test will be more than...
8
7.2 ap stats 08­09 filled.notebook
August 31, 2009
9
7.2 ap stats 08­09 filled.notebook
August 31, 2009
10
7.2 ap stats 08­09 filled.notebook
August 31, 2009
Transforming data
example:
The avg. height for the coasters I rode in the last year is 119.78 feet & the standard deviation is 82.16 feet. Describe the mean and standard deviation in inches.
Conversion: feet ⇒ inches
(feet)(12)
11
7.2 ap stats 08­09 filled.notebook
August 31, 2009
Transforming data
example:
The avg. June high for some city is 78F and the standard deviation is 2F. Describe the mean and standard deviation in Celsius.
Conversion: F ⇒ C
(F­32)(5/9)
12
7.2 ap stats 08­09 filled.notebook
August 31, 2009
Means Rule 1: For linear transformations, transform the mean.
Rule 2: Add, subtract, multiply or divide means as indicated.
Variances Rule 1: For linear transformations, multiply by the square of the factor.
Rule 2: For independent random variables, "The Pythagorean Theorem of Statistics" add variances.
13
7.2 ap stats 08­09 filled.notebook
August 31, 2009
example:
A psychologist studied the number of puzzles subjects were able to solve in a 3 minute period while subjected to the sound of a jackhammer outside. The mean for X is 1.4
puzzles and the standard deviation is 1.28. What are the mean and standard deviation for the number of puzzles solved in 6 minutes?
14
7.2 ap stats 08­09 filled.notebook
August 31, 2009
example:
The caffeine in a serving of drink is a normally distributed
random variable. For a certain cola, the mean is 52 mg and
variance 3 mg2. For a certain brand of coffee, the mean is
120 mg and variance 8 mg2. Susan House (Maxwell's wife)
drinks one cup of this coffee and on her way to work, a can
of this cola. Find the mean & standard deviation for the mg of caffeine she consumes each morning.
What's the probability that she has over 175 mg of caffeine?
15
7.2 ap stats 08­09 filled.notebook
August 31, 2009
enter your probability 2 distribution in the lists: outcomes in L1 and probabilities in L2
1
3
mean
4
5
standard deviation
square that to get the variance.
16