Unformatted text preview: Topic #2 Basic Statistics – Part 1
s s
s
s
s REVIEW ON OWN
• Summarizing Qualitative Data
• Summarizing Quantitative Data
• Measures of Location and Variability/Dispersion
Measures of Association Between Two Variables
Random Variables
Discrete Probability Distributions
Expected Value and Variance Slide 1 Summarizing Qualitative Data s
s
s
s
s Frequency Distribution
Relative Frequency Percent Frequency Distribution
Bar Graph
Pie Chart What does “Qualitative” mean? Slide 2 Frequency Distribution s s A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes.
The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data. Slide 3 Example: Marada Inn
Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The
ratings provided by a sample of 20 guests are shown
below.
Below Average Average Above Average
Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor
Above Average Excellent
Above Average Average Above Average
Average
Above Average Average How many rated worse than average?
How many rated better than average? Slide 4 Example: Marada Inn
s Frequency Distribution
Rating Frequency
Poor 2
Below Average 3
Average 5
Above Average 9
Excellent 1
Total 20
How many rated worse than average?
How many rated better than average? Slide 5 Example: Marada Inn
s Frequency Distribution
Rating Frequency
Poor 2
Below Average 3
Average 5
Above Average 9
Excellent 1
Total 20 The GM of Marada Inn has a goal that no more than 10% of all guests will rate their stay as worse than average. How is the inn doing?
Slide 6 Relative Frequency and
Percent Frequency Distributions
s s
s
s The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.
A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.
The percent frequency of a class is the relative frequency multiplied by 100.
A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. Slide 7 Example: Marada Inn
s Relative Frequency and Percent Frequency Distributions Relative Percent
Rating
Frequency Frequency
Poor .10
10
Below Average .15
15
Average .25
25
Above Average .45
45
Excellent .05 5
Total 1.00 100 Slide 8 Bar Graph
s s
s
s
s A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution.
On the horizontal axis we specify the labels that are used for each of the classes.
A frequency, relative frequency, or percent frequency scale can be used for the vertical axis.
Using a bar of fixed width drawn above each class label, we extend the height appropriately.
The bars are separated to emphasize the fact that each class is a separate category. Slide 9 Two Definitions
s HISTOGRAM
• Quantitative data s BAR GRAPH
• Qualitative (nonnumerical) data
• See examples that follow Slide 10 Example: Marada Inn Bar Graph
9
8
Frequency 7
6
5
4
3
2
1
Poor Below
Average Average Above Excellent
Average Rating Slide 11 ECO 6416 Grade Distribution
50% 60%
40%
20%
0% 25% 25%
0%
A 0%
B C D F Slide 12 ECO 6416 Grade Distribution
100%
100%
50%
0%
0% A B 0%
C 0%
D 0%
F Slide 13 Pie Chart
s s s The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. Slide 14 Pie Chart Slide 15 Example: Marada Inn
s Pie Chart
Exc.
Poor 5% 10% Above
Average 45% Below
Average 15%
Average 25% Quality Ratings Slide 16 Summarizing Quantitative Data s
s
s Frequency Distribution
Relative Frequency and Percent Frequency Distributions
Histogram Slide 17 Example: Hudson Auto Repair
The manager of Hudson Auto would like to get a
better picture of the distribution of costs for engine
tuneup parts. A sample of 50 customer invoices has
been taken and the costs of parts, rounded to the
nearest dollar, are listed below. 91
71
104
85
62 78
69
74
97
82 93
72
62
88
98 57
89
68
68
101 75
66
97
83
79 52
75
105
68
105 99
79
77
71
79 80
75
65
69
69 97
72
80
67
62 62
76
109
74
73 Slide 18 Example: Frequency Distribution Table
This is what a frequency distribution table looks like
Cumulative Relative Cumulative Percent Cost ($) Frequency Frequency Frequency Frequency
5059 2 .04 2
6069 13 .26 15
7079 16 .32 31
8089 7 .14 38
9099 7 .14 45
100109 5 .10 50
Totals 50 1.00 4 30 62 76 90 100 Slide 19 Frequency Distribution
s s Guidelines for Selecting Number of Classes
• Use between 5 and 20 classes.
• Data sets with a larger number of elements usually require a larger number of classes.
• Smaller data sets usually require fewer classes.
Guidelines for Selecting Width of Classes
• USE CLASSES OF EQUAL WIDTH
• Approximate Class Width = Largest Data Value − Smallest Data Value
Number of Classes Slide 20 Example: Hudson Auto Repair
s Frequency Distribution
If we choose six classes: Approximate Class Width = (109 52)/6 = 9.5 ≅ 10
10 Frequency 2 13 16 7 7 5 Total 50
Would it be wrong if 5059,6079,8089,9099,100109?
Cost ($) 5059 6069 7079 8089 9099
100109 Slide 21 Example: Hudson Auto Repair
s Relative Frequency and Percent Frequency Distributions Relative Percent Cost ($) Frequency Frequency 5059
.04 4 6069 .26
26 7079
.32
32 8089 .14
14 9099
.14
14 100109 .10
10 Total 1.00 100 Slide 22 Histogram
s
s s s Another common graphical presentation of quantitative data is a histogram.
The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis.
A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. Slide 23 Example: Hudson Auto Repair
Histogram
18
16
14
Frequency s 12
10
8
6
4
2
50 60 70 80 90 100 110 Cost ($) Slide 24 Numer of Players NBA Salaries, few yrs ago
40
35
30
25
20
15
10
5
0 2 4 6 8 1012141618 2022 2426 28303234 36384042 Salaries ($100,000) Slide 25 Measures of Location s
s
s
s
s Mean
Median
Mode
Percentiles
Quartiles σ
µ%
x Slide 26 Example: Apartment Rents
Given below is a sample of monthly rent values ($)
for onebedroom apartments. The data is a sample of 70
apartments in a particular city. The data are presented
in ascending order. 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 27 Mean
The mean of a data set is the average of all the data values.
s If the data are from a sample, the mean is denoted by .
s x s ∑ xi
x=
n If the data are from a population, the mean is denoted by (mu). µ ∑ xi
µ=
N Slide 28 Example: Apartment Rents
s Mean ∑ xi 34 , 356
x=
=
= 490.80
n
70 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 29 Median s
s
s The median of a data set is the value in the middle when the data items are arranged in ascending order.
If there is an odd number of items, the median is the value of the middle item.
If there is an even number of items, the median is the average of the values for the middle two items. Slide 30 Example: Apartment Rents
Median Since 70 is even and ½ of 70 = 35, average 35th and 36th data values:
Median = (475 + 475)/2 = 475
s 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 31 Example: Apartment Rents
ALTERNATIVE METHOD Median Median = 50% percentile
i = (p/100)n = (50/100)70 = 35, average 35th and 36th data values: (see later slide)
Median = (475 + 475)/2 = 475
s
425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 32 Mode s The mode of a data set is the value that occurs with greatest frequency. Slide 33 Example: Apartment Rents
s Mode 450 occurred most frequently (7 times) Mode = 450 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 Mean = $491 Median = $475 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Mode = $450 Slide 34 Percentiles
s The pth percentile of a data set is a value such that at least p percent of the items take on this value or less
• (and at least (100 p) percent of the items take on this value or more.)
• Arrange the data in ascending order.
• Compute index i, the position of the pth percentile. i = (p/100)n
• If i is not an integer, round up. The pth percentile is the value in the ith position.
• If i is an integer, the pth percentile is the average of the values in positions i and i+1. Slide 35 Example: Apartment Rents
s 90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585
425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 36 Example: Apartment Rents
s 90th Percentile = 585
• So, 90% of sample values should be less than or equal to 585
63 values < 585, so above is true!
• True? 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 37 Quartiles s
s
s
s Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile Slide 38 Measures of Variability s
s
s Range
Variance
Standard Deviation Slide 39 Range
s
s
s The range of a data set is the difference between the largest and smallest data values.
It is the simplest measure of variability.
It is very sensitive to the smallest and largest data values. Slide 40 Example: Apartment Rents
s Range Range = largest value smallest value Range = 615 425 = 190 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 41 Variance
s
s The variance is the average of the squared differences between each data value and the mean.
If the data set is a sample, the variance is denoted by s2. s 2 2 ∑ ( xi − x )
s=
n −1
If the data set is a population, the variance is denoted by σ 2. ∑ ( xi − µ )
σ=
N
2 2 Slide 42 Variance Example
Salaries (x)
x x ($1000) (x ) (x )2 200
200
40,000
300
100
10,000
400 0 0
500 100
10,000
600 200
40,000
x = 400 sum = 100,000
2 ( xi − x )2
∑ s= = 100,000/4 = $25,000
n− 1 Slide 43 Standard Deviation
s
s
s s The standard deviation of a data set is the positive square root of the variance.
It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean.
If the data set is a sample, the standard deviation is denoted s. s = s2
If the data set is a population, the standard deviation is denoted σ (sigma). 2
σ= σ Slide 44 Standard Deviation Example
s Recall s2 = $25,000 s s = s 2 = $158 x
Salaries (x ) 200 200 300 100 400 0 500 100 600 200 Average distance from mean = 600/4 = $150 Slide 45 More About Variance & Std. Deviation s Standard Deviation
• this measure is a number that shows how widely the values are dispersed about the mean of the distribution. • it shows (loosely) the average distance any variable value is from the variable's mean. • in the same units of measurement as the original distribution values. • For instance, if the variable is "revenue per month," measured in dollars, then the standard deviation will also be in dollars. Slide 46 Dispersion
s A small standard deviation shows that your values are grouped closely about the mean • s = 1 s a large standard deviation shows that your values are dispersed widely about the mean. • s = 5 Slide 47 Standard Deviation Example Season winning percentages for two teams
NOTE: same mean winning percentages. A: .450, .555, .345, .650 mean = .500 standard deviation = 0.13 B: .200, .350, .800 , .650 mean = .500 standard deviation = 0.27
Which team is more unpredictable?
How can you tell? Slide 48 Dispersion s Variance
• This is the square of the standard deviation. • This is obviously, then, a measure of how widely values are dispersed about the mean of a distribution. Variance is the “holy grail” of academic research. A primary goal of statistical testing is to explain most of the variance (or variation) in the variable that you are examining or explaining. Slide 49 Who Uses This s s Qualitycontrol specialists have traditionally regarded observations more than three standard deviations from the mean as candidates for further examination.
A common measure of risk in investments is the Beta, a calculated value based upon the standard deviation. Slide 50 Does blood pressure predict life expectancy?
Do SAT scores predict college performance?
Does understanding statistics make you a better person? Slide 51 Measures of Association between Two Variables
s
s Scatter Diagrams
Correlation Coefficient Slide 52 Scatter Diagrams
s Thus far we have focused on methods that are used to summarize the data for one variable at a time. s Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables.
• scatter diagram graphical Slide 53 Example: Pittsburgh Panthers Football Team
s Scatter Diagram The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored (in a game.)
x = Number of
y = Number of Interceptions Points Scored
1
14
3
24
2
18
1
17
3
27 (Note: these data are for 5 different games. For example: in game #1, they made 1 interception and scored a total of 14 points in that game.) Slide 54 Example: Panthers Football Team Number of Points Scored s What happens to points scored as interceptions rise?
Scatter Diagram
y
30
25
20
15
10
5
0 0 1
2
3
Number of Interceptions x Slide 55 Correlation Coefficient
s s The coefficient can take on values between 1 and +1.
• Values near 1 indicate a strong negative linear relationship.
• Values near +1 indicate a strong positive linear relationship.
• Zero value indicates no relationship.
If the data sets are samples, the coefficient is rxy. rxy =
s sxy sx s y (std. devs. of x & y) If the data sets are populations, the coefficient is . ρ xy
σ xy
ρ xy =
σ xσ y Slide 56 Correlation Coefficient (cont.) s Examples
• Hours of study vs. course grades
• correlation = 0.84
• What does this mean?
• turnovers vs. number of points scored
• correlation = 0.67
• What does this mean? Slide 57 Example: Panthers Football Team Number of Points Scored Describe relationship between points scored & interceptions
s Scatter Diagram & Correlation Coefficient
y
30
25
20
15
10
5
0 0 Correlation = .94 1
2
3
Number of Interceptions x Slide 58 Correlations
s See QB Ratings handout Slide 59 Exercise s College Athletic Departments • Questions #6 #9 Slide 60 Random Variables
s
s A random variable is a numerical description of the outcome of an experiment.
A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes.
• A discrete random variable may assume either a finite number of values or an infinite sequence of values.
• A continuous random variable may assume any numerical value in an interval or collection of intervals. Slide 61 Discrete vs. Continuous Random Variables (cont.)
During the 200304 NFL season, the Tampa Bay Bucs had 2 passes intercepted in three games. Game Opponent
No. of Interceptions
#1 Philadelphia 1
#2
Carolina 1
#3
Atlanta 0
s s They averaged giving up = .67 interceptions per game • 2/3 = .666666. . . Which is discrete and which is continuous? Slide 62 Example: JSL Appliances
s Discrete random variable with a finite number of values
Let x = number of TV sets sold at the store in one day where x can take on 5 values (0, 1, 2, 3, 4) s Discrete random variable with an infinite sequence of values
Let x = number of customers arriving in one day where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no finite upper limit on the number that might arrive. Slide 63 Discrete Probability Distributions
s s s s The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.
The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable.
The required conditions for a discrete probability function are: 0 ≤ f(x) ≤ 1 Σ f(x) = 1
We can describe a discrete probability distribution with a table, graph, or equation. Slide 64 Example: JSL Appliances
s Using past data on TV sales (below left), a tabular representation of the probability distribution for TV sales (below right) was developed. Number Units Sold of Days 0 80 1 50 2 40 3 10 4 20 200 x f(x) 0 .40 (=80/200) 1 .25 (=50/200) 2 .20 3 .05 4 .10 1.00 Slide 65 Example: JSL Appliances
Graphical Representation of the Probability Distribution .50
Probability s .40 .30
.20
.10
0 1 2 3 4 Values of Random Variable x (TV sales) Slide 66 Example: JSL Appliances
s Use probability function f(x) to represent the Probability Distribution
• f(0) = .40
0 ≤ f(x) ≤ 1 • f(1) = .25
Σ f(x) = 1
• f(2) = .20
• f(3) = .05
• f(4) = .10 On any one day, what’s the probability that JSL will sell 3 TV sets? Slide 67 Discrete Uniform Probability Distribution
s s The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula.
The discrete uniform probability function is f(x) = 1/n s where: n = the number of values the random variable may assume
Note that the values of the random variable are equally likely. Slide 68 Discrete Uniform Probability Distribution
Example
Suppose you have 3 coins in your wallet & you’re rummaging around in it to get some money .
s If the chance of grabbing 1 or 2 or 3 coins is the same, then • f(1 coin) = 1/3
• f(2 coins) = 1/3
• f(3 coins) = 1/3
Suppose you have 5 coins in your purse and the chance of grabbing 1 or 2 or 3 or 4 or 5 coins is the same. What is the probability of pulling out 2 coins?
s f(2 coins) = 1/5 Slide 69 Expected Value and Variance
s s s The expected value, or mean, of a random variable is a measure of its central location.
• Expected value of a discrete random variable:
E(x) = µ = Σ xf(x)
The variance summarizes the variability in the values of a random variable.
• Variance of a discrete random variable: Var(x) = σ 2 = Σ (x µ )2f(x)
The standard deviation, σ , is defined as the positive square root of the variance. Slide 70 Example: JSL Appliances
s Expected Value of a Discrete Random Variable col. 1 col. 2
NOTE: (outcomes) (probabilities) x
0
1
2
3
4 f(x) xf(x) .40 .00 (=col. 1 x col. 2) .25 .25 (=col. 1 x col. 2) .20 .40 .05 .15 .10 .40 1.20 = E(x)
The expected number of TV sets sold in a day is 1.2
What’s average number of TV sets JSL sells each day? Slide 71 Example: JSL Appliances
s Expected Value of a Discrete Random Variable col. 1 col. 2
NOTE: (outcomes) (probabilities) x
0
1
2
3
4 f(x) xf(x) .40 .00 (=col. 1 x col. 2) .25 .25 (=col. 1 x col. 2) .20 .40 .05 .15 .10 .40 1.20 = E(x)
The expected number of TV sets sold in a day is 1.2
What’s min number of TV sets JSL should stock each day? Slide 72 Expected Value vs. Mean
s s s JSL Appliances sets sold per day
• 0, 1, 2, 3, 4
Expected value = 1.2
• Σ xf(x) Mean = 2
N ∑x 0 +1+ 2 + 3 + 4
=
=2
N
5 i =1 s i Why the difference in values? Which statistic gives a more realistic picture of average number of TV sets sold per day? WHY? Slide 73 Binomial Probability Distribution
s What’s the probability of seeing x successes in the next n experiments?
• Probability that 2 of next 3 customers will buy?
• Probability of 2 scores (TD or field goal) in next five possessions
• Probability that 5 of next 10 shots will go through basket
• Probability that 12 of next 15 loan applicants will be approved?
• Probability that 30 of next 150 Accounting grads will be employed in public accounting? Slide 74 Binomial Probability Distribution
s Properties of a Binomial Experiment
• The experiment consists of a sequence of n identical trials.
• Two outcomes, success and failure, are possible on each trial. • The probability of a success, denoted by p, does not change from trial to trial.
• The trials are independent. Slide 75 Example: Evans Electronics
s Binomial Probability Distribution Evans is concerned about a low retention rate for employees. On the basis of past experience, management has seen a turnover of 10% of the hourly employees annually. If 3 hourly employees are chosen for training, what is the probability that 1 of them will leave the company this year? Slide 76 Binomial Probability Distribution
s Binomial Probability Function n!
f ( x) =
p x (1 − p ) ( n − x )
x !( n − x ) !
where:
f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial Slide 77 Example: Evans Electronics
s Using the Binomial Probability Function Let: p =.10, n = 3, x = 1 n!
f ( x) =
p x (1 − p ) ( n − x )
x !( n − x ) !
3!
f (1) =
( 0.1)1 ( 0. 9 ) 2
1!( 3 − 1) ! = (3)(0.1)(0.81) = .243 Choosing 3 hourly employees at random, the probability that 1 of them will leave the company this year is 24.3% Is this probability high enough for the company to take action?
See “Scooter’s Struggles” Slide 78 Binomial Probability: Scooter’s Struggles Slide 79 Example: Evans Electronics
s Using a Tree Diagram First
Worker Second
Worker Third
Worker Value
of x Probab. Leaves (.1) 3 .0010 S (.9) Leaves (.1) L (.1) 2 .0090 2 .0090 1 .0810 2 .0090 1 .0810 L (.1) 1 .0810 S (.9) 0 .7290 L (.1) Stays (.9)
L (.1)
Leaves (.1)
Stays (.9)
Stays (.9) S (.9) S (.9) Slide 80 Example: Evans Electronics
s Using a Tree Diagram First
Worker Second
Worker Third
Worker Value
of x Probab. Leaves (.1) 3 .0010 S (.9) Leaves (.1) L (.1) 2 .0090 2 .0090 1 .0810 2 .0090 1 .0810 L (.1) 1 .0810 S (.9) 0 .7290 L (.1) Stays (.9)
L (.1)
Leaves (.1)
Stays (.9)
Stays (.9) S (.9) S (.9) Slide 81 Example: Evans Electronics
s Using a Tree Diagram First
Worker Second
Worker Third
Worker Value
of x Probab. Leaves (.1) 3 .0010 S (.9) Leaves (.1) L (.1) 2 .0090 2 .0090 1 .0810 2 .0090 1 .0810 L (.1) 1 .0810 S (.9) 0 .7290 L (.1) Stays (.9)
L (.1)
Leaves (.1)
Stays (.9)
Stays (.9) S (.9) S (.9) Slide 82 Example: Evans Electronics
s Using a Tree Diagram First
Worker Second
Worker Third
Worker Value
of x Probab. Leaves (.1) 3 .0010 S (.9) Leaves (.1) L (.1) 2 .0090 2 .0090 1 .0810 2 .0090 1 .0810 L (.1) 1 .0810 S (.9) 0 .7290 L (.1) Stays (.9)
L (.1)
Leaves (.1)
Stays (.9)
Stays (.9) S (.9) S (.9) Slide 83 Slide 84 Binomial Probability Distribution
s s s Expected Value E(x) = µ = np Variance Var(x) = σ 2 = np(1 p)
Standard Deviation
SD ( x ) = σ = np (1 − p ) Slide 85 Example: Evans Electronics
s Binomial Probability Distribution
• Expected Value E(x) = µ = 3(.1) = 0.3 employees out of 3
• Variance Var(x) = σ 2 = 3(.1)(.9) = .27
• Standard Deviation SD( x) = σ = 3(.1)(.9) = .52 employees
Using the expected value result, on average, how many employees will leave within one year out of each 9 hired?
Is this a problem the company should address? Slide 86 Poisson Probability Distribution s What will be the number of occurrences over a specific time or distance interval?
• How many points will be scored in 30 minutes?
• How many fans will enter gate #3 in 5 minutes?
• How many patients will arrive at the ER in 30 minutes?
• How many repairs will be needed in 10 miles of highway? Slide 87 Poisson Probability Distribution
s Properties of a Poisson Experiment
• The probability of an occurrence is the same for any two intervals of equal length.
• The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval. Slide 88 Poisson Probability Distribution
s Poisson Probability Function µ x e −µ
f ( x) =
x!
where:
f(x) = probability of x occurrences in an interval µ = mean number of occurrences in an interval e = 2.71828 Slide 89 Example: Bright House Networks Stadium
s Using the Poisson Probability Function
Fans arrive at gate #3 of the Bright House Networks Stadium at the average rate of 6 per minute 15 minutes before UCF football games. What is the probability of 4 arrivals in 30 seconds 15 minutes before a game? (note) µ = 6 per minute = 3 per 30 seconds, x = 4
34 ( 2. 71828) −3
f ( 4) =
=.1680
4! Slide 90 After Class: Read Article About Shark Attacks Slide 91 Exercises
1. Binomial and Poisson Distributions
a) Inclass
b) Homework Slide 92 Tabular and Graphical Procedures
Data
Qualitative Data Quantitative Data Tabular
Methods Graphical Methods Tabular
Methods Graphical Methods •Frequency Distribution
•Rel. Freq. Dist.
•% Freq. Dist.
•Crosstabulation •Bar Graph
•Pie Chart •Frequency Distribution
•Rel. Freq. Dist.
•Cum. Freq. Dist.
•Cum. Rel. Freq. Distribution •StemandLeaf Display
•Crosstabulation •Dot Plot
•Histogram
•Ogive
•Scatter Diagram Slide 93 OneMinute Essay s PRINT NAME ON PAGE
s “What concept in today’s class was hardest to understand?” s ONE OR TWO SENTENCES MAX Slide 94 ...
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 Fall '11
 RichardHofler
 Econometrics, Self number, 496, 175

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