Applied Statistics 270 - Chapter 4

Front 1

random variable
(df)

Back 1

a random variable is a variable that assumes NUMERICAL VALUES associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample

Front 2

example of random variable

Back 2

Discover a frequency distribution for the number of students that use the walkup ATM on campus each day over the course of a year

Front 3

What is the experimental unit

Back 3

the ATM

Front 4

What is the experiment

Back 4

counting the # of students using the ATM of a particular day

Front 5

What is the random variable

Back 5

# of students

Front 6

What are the sample points for this random variable?

Back 6

0,1,...# of students at UNR

Front 7

Discrete random variables
(df)

Back 7

Discrete random variables can assume a countable number of values

i.e. has to be a whole integer, doesn't use fractions.

ex. you can't cast half of a vote, can't have 3.5 visits to the ATM

Front 8

Continuous random variables
(df)

Back 8

Continuous random variables can have values corresponding to any of the points contained in one or more intervals

i.e. can be in between integers, can go to fractions

Front 9

Examples of discrete random variables

Back 9

1. whether a hunter gets an elk tag in 2007
2. the number of semesters a student takes to graduate in your major
3. whether a driver gets caught speeding

Front 10

Examples of continuous random variables

Back 10

1. a swimmer's time in the 50 meter butterfly
2. the time it takes a student to do the apst270 homework
3. the amount of acreage burned in wildfires within a 100 mile radius of Reno each summer

Front 11

When studying a discrete random variable, we want to know:
(2)

Back 11

1. the possible values it can have
2. the probability of each value

Front 12

Values can be expressed as
(2)

Back 12

fractions or percentages

ex. .25, 25%

Front 13

The probability distribution of a discrete random variable
(df)

Back 13

The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume

Front 14

Example of finding a probability distribution

Back 14

1.Experiment: tossing 2 coins simultaneously
2. random variable x: # of heads observed
3. x can assume values of 0, 1, and 2

HH: x=2, prob=.25
TH: x=1, prob=.25
HT: x=1, prob=.25
TT; x=0, prob=.25

Front 15

Requirements for the probability distribution of a discrete random variable x
(2)

Back 15

1. p(x) must be greater than or equal to 0 for all values of x

2. The sum of the probabilities must equal 1

(no probability is greater than 1, and probability can't be negative)

Front 16

signs of x and p(x)

Back 16

x can be neg,

p(x) can't be negative

Front 17

If you don't know N, calculate the mean using the formula:

Back 17

mean equals the sums of p(x) * x (for all values of x)


note: mean can be neg

Front 18

Formula for the mean, or expected value of a discrete random variable

Back 18

u (mu) = E(x) = sum of x * p(x)

Front 19

Formula for the variance of a discrete random variable

Back 19

o^2 = sum of (x-u)^2 * p(x)

Front 20

Formula for standard deviation

Back 20

o = square root of o^2

Front 21

When we know the mean (u) and standard deviation (o) of the probability distribution of x,

Back 21

1. we can use Chebyshev's rule
2. and the Empirical rule

to make statements about the likelihood of values of x falling within the intervals:

(u +/- o), (u +/- 2o), (u +/- 3o)

Front 22

Why do we use those intervals?

Back 22

We use those intervals because we often want to describe HOW FAR an observation is away from its mean in terms of the standard deviation

Front 23

Example of using the intervals

Back 23

Suppose your score on the midterm is 70%. You hear that the mean was 50% and the standard deviation was 10%. Your score is 2 standard deviations ABOVE the mean

Front 24

Chebyshev's Rule

Back 24

applies to any data set, regardless of the shape of the frequency distribution. The rule states that the minimum amount of data that will lie within K (k>1) standard deviation of the mean for any distribution of data.

There will be at least 75% of the data within 2 standard deviations of the mean and at least 88% of the data within 3 standard deviations of the mean.

Front 25

Empirical Rule

Back 25

only works for bell-shaped symmetric data

1. aboout 68% of the values lie within one standard deviation of the mean
2. about 95% of the values will lie within 2 standard deviations of the mean
3. about 99.7% of the values will lie within 3 standard deviations of the mean

Front 26

Binomial Random Variable

(characteristics)

Back 26

1. An experiment of n identical trials
2. 2 possible outcomes on each trial, denoted as S(success) and F(failure)
3. Probability of success (p) is constant from trial to trial. Probability of failure (q) is 1-p or P^c
4. Trials are independent
5. Binomial random variable - number of S's in n trials

Front 27

Multiplicative Rule using example numbers

Back 27

P(3 of the 4 adults pass the test) = 4(.1)^3(.9)

Front 28

Formula for the probability distribution p(x)

Back 28

p(x) = (n -over- x) * p^x * q ^(n-x)

where p = probability of success on single trial
q=1-p
n= number of trials
x= number of successes in n trials

Front 29

Example where n = 4, x = 0

Back 29

p(0) = (4 -over- 0) *p^0 * q^4

Front 30

Formula for the mean

Back 30

u = np

Front 31

formula for the Variance

Back 31

o^2 = npq

Front 32

Formula for standard deviation

Back 32

o = square root of (n*p*g)

Front 33

Instructions for making a binomial distribution on Minitab

Back 33

calc
-->probability
---> binomial
a. select probability, enter # of trials, and probability of success, input where to put calculations (i.e. c1)

then do bar chart to show data