Chapter 3, Part A

1
ECO 3411

• Review Material
• Descriptive Statistics
• The Basics of Probability
• The Normal Probability Distribution

2
Descriptive Statistics - review

• Measures of Location
• Measures of Variability
• Measures of Relative Location
• Measures of Association Between Two Variables

3
Measures of Location

• Mean
• Median
• Mode

4
Example Apartment Rents

• Given below is a sample of monthly rent values
  ()
• for one-bedroom apartments. The data is a sample
  of 70
• apartments in a particular city. The data are
  presented
• in ascending order.

5
Mean

• The mean of a data set is the average of all the
  data values.
• If the data are from a sample, the mean is
  denoted by
• .
• If the data are from a population, the mean is
  denoted by µ (mu).

6
Example Apartment Rents

• Mean

7
Median

• 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.

8
Example Apartment Rents

• Median
• Median 50th percentile
• i (p/100)n (50/100)70 35.5
  Averaging the 35th and 36th data values
• Median (475 475)/2 475

9
Mode

• The mode of a data set is the value that occurs
  with greatest frequency.

10
Example Apartment Rents

• Mode
• 450 occurred most frequently (7 times)
• Mode 450

11
Using Excel to Computethe Mean, Median, and Mode

• Formula Worksheet

Note Rows 7-71 are not shown.
12
Using Excel to Computethe Mean, Median, and Mode

• Value Worksheet

Note Rows 7-71 are not shown.
13
Measures of Variability

• Range
• Variance
• Standard Deviation

14
Range

• 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.

15
Example Apartment Rents

• Range
• Range largest value - smallest value
• Range 615 - 425 190

16
Variance

• 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.
• If the data set is a population, the variance is
  denoted by ? 2.

17
Example Apartment Rents
18
Standard Deviation

• 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.
• If the data set is a population, the standard
  deviation is denoted ? (sigma).

19
Example Apartment Rents

• Variance
• Standard Deviation

20
Using Excel to Compute theSample Variance and
Standard Deviation

• Formula Worksheet

Note Rows 8-71 are not shown.
21
Using Excel to Compute theSample Variance and
Standard Deviation

• Value Worksheet

Note Rows 8-71 are not shown.
22
Using ExcelsDescriptive Statistics Tool

• Step 1 Select the Tools pull-down menu
• Step 2 Choose the Data Analysis option
• Step 3 Choose Descriptive Statistics from the
  list of
• Analysis Tools
• continued

23
Using ExcelsDescriptive Statistics Tool

• Step 4 When the Descriptive Statistics dialog
  box appears
• Enter B1B71 in the Input Range box Select
  Grouped By Columns
• Select Labels in First Row
• Select Output Range
• Enter D1 in the Output Range box
• Select Summary Statistics
• Select OK

24
Using ExcelsDescriptive Statistics Tool

• Value Worksheet (Partial)

25
Using ExcelsDescriptive Statistics Tool

• Value Worksheet (Partial)

26
Measures of Relative Locationand Detecting
Outliers

• z-Scores
• The Empirical Rule
• Detecting Outliers

27
z-Scores

• The z-score is often called the standardized
  value.
• It denotes the number of standard deviations a
  data value xi is from the mean.
• A data value less than the sample mean will have
  a z-score less than zero.
• A data value greater than the sample mean will
  have a z-score greater than zero.
• A data value equal to the sample mean will have a
  z-score of zero.

28
Example Apartment Rents

• z-Score of Smallest Value (425)

29
Example Apartment Rents

• z-Score of Smallest Value (425)
• Standardized Values for Apartment Rents

30
The Empirical Rule

• For data having a bell-shaped distribution
• 68.26 of the data values will be within one
  standard deviation of the mean.
• 95.44 of the data values will be within two
  standard deviations of the mean.
• 99.72 will be within three standard deviations
  of the mean.

31
Example Apartment Rents

• The Empirical Rule
• Interval in Interval
• Within /- 1s 436.06 to 545.54 48/70 69
• Within /- 2s 381.32 to 600.28 68/70 97
• Within /- 3s 326.58 to 655.02 70/70 100

32
Detecting Outliers

• An outlier is an unusually small or unusually
  large value in a data set.
• A data value with a z-score less than -3 or
  greater than 3 might be considered an outlier.
• It might be an incorrectly recorded data value.
• It might be a data value that was incorrectly
  included in the data set.
• It might be a correctly recorded data value that
  belongs in the data set !

33
Example Apartment Rents

• Detecting Outliers
• The most extreme z-scores are -1.20 and 2.27.
• Using z gt 3 as the criterion for an outlier,
• there are no outliers in this data set.
• Standardized Values for Apartment Rents

34
Example

• Suppose annual salaries for sales associates
  from a particular store have a mean of 32,500
  and a standard deviation of 2,500.
• Calculate and interpret the z-score for a sales
  associate who makes 36,000.
• Suppose that the distribution of annual salaries
  for sales associates at this store is
  bell-shaped. Use the empirical rule to calculate
  the percentage of sales associates with salaries
  between 27,500 and 37,500.

35
Measures of Association Between Two Variables

• Covariance
• Correlation Coefficient

36
Covariance

• The covariance is a measure of the linear
  association between two variables.
• Positive values indicate a positive relationship.
• Negative values indicate a negative relationship.

37
Covariance

• If the data sets are samples, the covariance is
  denoted by sxy.
• If the data sets are populations, the covariance
  is denoted by ?xy.

38
Covariance
A high school guidance counselor collected the
following data about the grade point averages
(GPA) and the SAT mathematics test scores for six
seniors.

• Compute and interpret the sample covariance for
  the data

39
Correlation Coefficient

• 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.
• If the data sets are samples, the coefficient is
  rxy.
• If the data sets are populations, the coefficient
  is .

40
Correlation Coefficient
A high school guidance counselor collected the
following data about the grade point averages
(GPA) and the SAT mathematics test scores for six
seniors.

• Compute and interpret the sample covariance for
  the data
• Compute and interpret the correlation coefficient
  (sx 0.385, sy 85.323)

41
Using Excel to Compute theCovariance and
Correlation Coefficient

• Formula Worksheet

42
Using Excel to Compute theCovariance and
Correlation Coefficient

• Value Worksheet

43
Introduction to Probability

• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a scale
  from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is almost
  certain to occur.
• A probability of 0.5 indicates the occurrence of
  the event is just as likely as it is unlikely.

44
An Experiment and Its Sample Space

• An experiment is any process that generates
  well-defined outcomes.
• The sample space for an experiment is the set of
  all experimental outcomes.
• A sample point is an element of the sample space,
  any one particular experimental outcome.

45
Example Bradley Investments

• Bradley has invested in two stocks, Markley Oil
  and
• Collins Mining. Bradley has determined that the
• possible outcomes of these investments three
  months
• from now are as follows.
• Investment Gain or Loss
• in 3 Months (in 000)
• Markley Oil Collins Mining
• 10 8
• 5 -2
• 0
• -20

Sample Point
Sample Space
46
Assigning Probabilities

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

47
Classical Method

• If an experiment has n possible outcomes, this
  method
• would assign a probability of 1/n to each
  outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance
• of occurring.

48
Relative Frequency Method

• Example Lucas Tool Rental
• Lucas would like to assign probabilities to the
• number of floor polishers it rents per day.
  Office
• records show the following frequencies of daily
  rentals
• for the last 40 days.
• Number of Number
• Polishers Rented of Days
• 0 4
• 1 6
• 2 18
• 3 10
• 4 2

49
Relative Frequency Method

• Example Lucas Tool Rental
• The probability assignments are given by
  dividing
• the number-of-days frequencies by the total
  frequency
• (total number of days).
• Number of Number
• Polishers Rented of Days Probability
• 0 4 .10 4/40
• 1 6 .15 6/40
• 2 18 .45 etc.
• 3 10 .25
• 4 2 .05
• 40 1.00

50
Subjective Method

• When economic conditions and a companys
  circumstances change rapidly it might be
  inappropriate to assign probabilities based
  solely on historical data.
• We can use any data available as well as our
  experience and intuition, but ultimately a
  probability value should express our degree of
  belief that the experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

51
Example Bradley Investments

• Applying the subjective method an analyst
• made the following probability assignments.
• Exper. Outcome (Markley, Collins)
  Net Gain/Loss Probability
• ( 10, 8) 18,000 Gain
  .20
• ( 10, -2) 8,000 Gain
  .08
• ( 5, 8) 13,000 Gain
  .16
• ( 5, -2) 3,000 Gain
  .26
• ( 0, 8) 8,000 Gain
  .10
• ( 0, -2) 2,000 Loss
  .12
• (-20, 8) 12,000 Loss
  .02
• (-20, -2) 22,000 Loss
  .06

52
Events and Their Probability

• An event is a collection of sample points.
• The probability of any event is equal to the sum
  of the probabilities of the sample points in the
  event.

53
Example Bradley Investments

• Events and Their Probabilities
• Event M Markley Oil Profitable
• M (10, 8), (10, -2), (5, 8), (5,
  -2)
• P(M) P(10, 8) P(10, -2) P(5, 8)
  P(5, -2)
• .2 .08 .16 .26
• .70
• Event C Collins Mining Profitable
• C (10, 8), (5, 8), (0, 8), (-20,
  8)
• P(C) .48 (found using the same logic)

54
Basic Concepts of Probability

• Complement of an Event
• Union of Two Events
• Intersection of Two Events
• Mutually Exclusive Events

55
Complement of an Event

• The complement of event A is defined to be the
  event consisting of all sample points that are
  not in A.
• The complement of A is denoted by Ac.
• The Venn diagram below illustrates the concept of
  a complement.

Sample Space S
Event A
Ac
56
Union of Two Events

• The union of events A and B is the event
  containing all sample points that are in A or B
  or both.
• The union is denoted by A ??B?
• The union of A and B is illustrated below.
• P(A ??B) The probability of the occurrence of
  Event A or Event B.

Sample Space S
57
Example Bradley Investments

• Union of Two Events
• Event M Markley Oil Profitable
• Event C Collins Mining Profitable
• M ??C Markley Oil Profitable
• or Collins Mining Profitable
• M ??C (10, 8), (10, -2), (5, 8), (5, -2),
  (0, 8), (-20, 8)
• P(M ??C) P(10, 8) P(10, -2) P(5, 8) P(5,
  -2)
• P(0, 8) P(-20, 8)
• .20 .08 .16 .26 .10 .02
• .82

58
Intersection of Two Events

• The intersection of events A and B is the set of
  all sample points that are in both A and B.
• The intersection is denoted by A ????
• The intersection of A and B is the area of
  overlap in the illustration below.
• P(A ???) The probability of the occurrence of
  Event A and Event B.

Sample Space S
Intersection
Event A
Event B
59
Example Bradley Investments

• Intersection of Two Events
• Event M Markley Oil Profitable
• Event C Collins Mining Profitable
• M ??C Markley Oil Profitable
• and Collins Mining Profitable
• M ??C (10, 8), (5, 8)
• P(M ??C) P(10, 8) P(5, 8)
• .20 .16
• .36

60
Mutually Exclusive Events

• Two events are said to be mutually exclusive if
  the events have no sample points in common. That
  is, two events are mutually exclusive if, when
  one event occurs, the other cannot occur.
• Addition Law for Mutually Exclusive Events
• P(A ??B) P(A) P(B)

Sample Space S
Event A
Event B
61
Roll the Dice

• If you roll 2 dice, whats the probability of
  rolling a 7 or 11?

62
Die 1
Die 2
63
Die 1
Die 2
P(7) 6/36 .167
P(11) 2/36 .056
64
Roll the Dice

• If you roll 2 dice, whats the probability of
  rolling a 7 or 11?

65
Continuous Probability Distributions

• Normal Probability Distribution

f(x)
x
?
66
Continuous Probability Distributions

• A continuous random variable can assume any value
  in an interval on the real line or in a
  collection of intervals.
• It is not possible to talk about the probability
  of the random variable assuming a particular
  value.
• Instead, we talk about the probability of the
  random variable assuming a value within a given
  interval.
• The probability of the random variable assuming a
  value within some given interval from x1 to x2 is
  defined to be the area under the graph of the
  probability density function between x1 and x2.

67
Normal Probability Distribution

• Graph of the Normal Probability Density Function

x1
x2
68
Normal Probability Distribution

• Graph of the Normal Probability Density Function

x1
x2
69
Normal Probability Distribution

• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• Two parameters, m (mean) and s (standard
  deviation), determine the location and shape of
  the distribution.
• The highest point on the normal curve is at the
  mean, which is also the median and mode.
• The mean can be any numerical value negative,
  zero, or positive.
• continued

Characteristics of the Normal Probability
Distribution
70
Normal Probability Distribution

• The normal curve is symmetric.
• The standard deviation determines the width of
  the curve larger values result in wider, flatter
  curves.
• The total area under the curve is 1 (.5 to the
  left of the mean and .5 to the right).
• Probabilities for the normal random variable are
  given by areas under the curve.

Characteristics of the Normal Probability
Distribution
71
Normal Probability Distribution

• of Values in Some Commonly Used Intervals
• 68.26 of values of a normal random variable are
  within /- 1 standard deviation of its mean.
• 95.44 of values of a normal random variable are
  within /- 2 standard deviations of its mean.
• 99.72 of values of a normal random variable are
  within /- 3 standard deviations of its mean.

72
x4
x6
x1
x2
x3
x5
z
-1
-2
1
-3
2
3
.6826
73
x4
x6
x1
x2
x3
x5
z
-1
-2
1
-3
2
3
.9544
74
x4
x6
x1
x2
x3
x5
z
-1
-2
1
-3
2
3
.9972
75
Standard Normal Probability Distribution

• A random variable that has a normal distribution
  with a mean of zero and a standard deviation of
  one is said to have a standard normal probability
  distribution.
• The letter z is commonly used to designate this
  normal random variable.
• Converting to the Standard Normal Distribution

• We can think of z as a measure of the number of
  .....standard deviations x is
  from ?.

76
Standard Normal Probability Distribution
? 0 ? 1
P 1.0
77
Standard Normal Probability Distribution
? 0 ? 1
P .5
P .5
78
Standard Normal Probability Distribution
? 0 ? 1
f(z)
z
1
0
79

• Using the Standard Normal Probability Table
  (Table 1)

80

• Using the Standard Normal Probability Table
  (Table 1)

81

• Using the Standard Normal Probability Table
  (Table 1)

82
Standard Normal Probability Distribution
? 0 ? 1
f(z)
z
1
0
83
Standard Normal Probability Distribution
? 0 ? 1
f(z)
z
1
0
84
Standard Normal Probability Distribution
? 0 ? 1
f(z)
z
1
0
85
Standard Normal Probability Distribution
? 0 ? 1
f(z)
z
1
0
86
Using the Standard Normal Probability Table
87
Example Pep Zone

• Standard Normal Probability Distribution
• Pep Zone sells auto parts and supplies including
  a
• popular multi-grade motor oil. When the stock of
  this
• oil drops to 20 gallons, a replenishment order is
  placed.
• The store manager is concerned that sales are
  being
• lost due to stockouts while waiting for an order.
  It has
• been determined that leadtime demand is normally
• distributed with a mean of 15 gallons and a
  standard
• deviation of 6 gallons.
• The manager would like to know the probability
  of a
• stockout, P(x gt 20).

88
Example Pep Zone
? 15 ? 6
P(x gt 20)
20
89
Example Pep Zone
? 15 ? 6
20
z
0
90
Example Pep Zone
? 15 ? 6
20
z
.83
0
91
Example Pep Zone

• Standard Normal Probability Distribution
• The Standard Normal table shows an area of .7967
  for the region below z .83. The shaded tail
  area is 1.00 - .7967 .2033. The probability of
  a stock-out is .2033.

Area .7967
Area 1.00 - .7967 .2033
z
0
.83
92
Example Pep Zone

• Standard Normal Probability Distribution
• If the manager of Pep Zone wants the
  probability of a stockout to be no more
  than .05, what should the reorder point be?
• Let z.05 represent the z value cutting the .05
  tail area.

Area .05
Area .95
z.05
0
93
Example Pep Zone

• Using the Standard Normal Probability Table
• We now look-up the .9500 area in the Standard
  Normal Probability table to find the
  corresponding z.05 value.
• z.05 1.645 is a reasonable estimate.

94
Example Pep Zone

• Standard Normal Probability Distribution
• If the manager of Pep Zone wants the probability
  of a stockout to be no more than .05,
  what should the reorder point be?

Area .05
Area .95
z
1.645
0
95
Example Pep Zone

• Standard Normal Probability Distribution
• The corresponding value of x is given by
• A reorder point of 24.87 gallons will place the
  probability of a stockout during leadtime at .05.
  Perhaps Pep Zone should set the reorder point
  at 25 gallons to keep the probability under .05.

96
End of Review