Download Stats 4.1

Statistics Lesson 4.1.notebook
April 05, 2016
Probability Distributions
The outcome of a probability experiment is often a count or a measure.
A random variable x represents a numerical value associated with each outcome of a probability experiment.
The word random indicates that x is determined by chance. There are two types of random variables: discrete and continuous.
A random variable is discrete when it has a finite or countable number of possible outcomes that can be listed.
A random variable is continuous when it has an uncountable number ot possible outcomes, represented by an interval on a number line
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Statistics Lesson 4.1.notebook
April 05, 2016
You conduct a study of the number of telephone calls a telemarketer makes in one day. The possible values of the random variable x are 0,1,2,3,4, and so on. Because the set of possible outcomes can be listed, x is a discrete random variable. We could represent its values on a number line.
Another study measured the time (in hours) a telemarketer spends making calls in one day. Because the time spent making calls in one day can be any number from 0 to 24
(including fractions and decimals), x is a continuous random variable. Its values can be represented by an interval on the number line.
(a) Let x represent the number of Fortune 500 companies that lost money in the previous year.
(b) Let x represent the volume of gasoline in a 21­gallon truck.
(c) Let x represent the speed of a rocket.
(d) Let x represent the number of calves born on a farm in one year.
You need to be able to distinguish between discrete and continuous random variables because different techniques are use to analyze each.
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Statistics Lesson 4.1.notebook
April 05, 2016
Discrete Probability Distributions
Each value of a discrete random variable can be assigned a probability. By listing each value of the random variable with its corresponding probability, you are forming a discrete probability distribution.
A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A discrete probability distribution must satisfy these conditions.
IN WORDS
IN SYMBOLS
1. The probability of each value of the discrete
random variable is between 0 and 1, inclusive.
2. The sum of all the probabilities is 1.
Constructing a Discrete Probability Distribution
Let x be a discrete random variable with possible outcomes x1,x2....xn.
1. Make a frequency distribution for the possible outcomes.
2. Find the sum of the frequencies.
3. Find the probability of each possible outcome by dividing its frequency by the
sum of the frequencies.
4. Check that each probability is between 0 and 1, inclusive, and that the sum
of all the probabilities is 1.
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Statistics Lesson 4.1.notebook
April 05, 2016
A industrial psychologist administered a personality inventory test for passive­aggressive traits to 150 employees. Each individual was given a score from 1 to 5, where 1 is extremely passive and 5 is extremely aggressive. A score of 3 indicated neither trait. The results are shown in the given table. Construct a probability distribution for the random variable x. Then graph the distribution using a histogram.
Frequency Distribution
Score, x Frequency, f
1
2
3
4
5
24
33
42
30
21
x
1
P(x)
0.16 0.22 0.28 0.20 0.14
2
3 4 5
All probabilities are between 0 and 1 and the sum equals 1
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Statistics Lesson 4.1.notebook
April 05, 2016
TRY IT YOURSELF 2 P. 192
Days of rain, x
0
1
2
3
Probability, P(x)
0.216
0.432
0.288
0.064
Verify that the table on the left is a probability distribution.
1. Is each probability between 0 and 1?
2. Is the sum of the probabilities equal to 1?
Are each of the following distributions a probability distribution?
x 1 2 3 4
x 5 6 7 8
P(x) 0.28 0.21 0.43 0.15
P(x) 5
Statistics Lesson 4.1.notebook
April 05, 2016
Mean, Variance, and Standard Deviation
Mean of a Discrete Random Variable
The mean of a discrete random variable is given by
Each value of x is multiplied by its corresponding probability and the products are added.
Remember the personality inventory test from an earlier example. We had a discrete probability distribution table of 1
2
3
4
5
0.16
0.22
0.28
0.20
0.14
1
2
3
4
5
0.16
0.22
0.28
0.20
0.14
1(0.16) = 0.16
2(0.22) = 0.44
3(0.28) = 0.84
4(0.20) = 0.80
5(0.14) = 0.70
The mean score of this personality test is 2.9.
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Statistics Lesson 4.1.notebook
April 05, 2016
Variance and Standard Deviation of a Discrete Random Variable
The variance of a discrete random variable is The standard deviation is
Again using the personality inventory test where before rounding μ = 2.94
x
1
0.16
­1.94
3.7636
0.602176
2
0.22
­0.94
0.8836
0.194392
3
0.28
0.06
0.0036
0.001008
4
0.20 1.06
1.1236
0.224720
5
0.14
2.06
4.2436
0.594104
.
The variance is The standard deviation is Most of the data values differ from the mean by no more than 1.3.
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Statistics Lesson 4.1.notebook
April 05, 2016
EXPECTED VALUE
The mean of a random variable represents what you would expect to happen ove thousands of trials. It is also called the expected value.
The expected value of a discrete random variable is equal to the mean of the random variable.
Probabilities can never be negative, but the expected value of the random variable can be negative.
Example: At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value of your gain?
If you buy one $2 ticket and win a prize, you will win $498, $248, $148 or $73. Make a probability distribution for the possible gains.
This represents buying a $2 ticket and winning nothing.
$498 $248 $148
$73
­$2
Gain
Probability,
P(x)
The Expected value E(x) = ΣxP(x)
(this means you can expect to lose $1.35 for each ticket you buy)
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Statistics Lesson 4.1.notebook
April 05, 2016
At a raffle, 2000 tickets are sold a $5 each for five prizes of $2000, $1000, $500, $250, and $100. You buy one ticket. What is the expected value of your gain?
P. 197 5 ­ 37 odds
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