Statistics& Financial Modelling Flashcards
Quantitative Finance
1
Q
Random Experiment
A
Random Experiment – a process leading to an uncertain outcome
2
Q
Basic Outcome
A
Basic Outcome – a possible outcome of a random experiment
3
Q
Sample Space (S)
A
Sample Space (S) – the collection of all possible outcomes of a random experiment
4
Q
Event (E)
A
Event (E) – any subset of basic outcomes from the sample space
5
Q
Intersection of Events
A
Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B
6
Q
Mutually Exclusive Events互斥
A
A and B are Mutually Exclusive Events if they have no basic outcomes in common
i.e., the set A ∩ B is empty
7
Q
Union of Events
A
Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either A or B
8
Q
Collectively Exhaustive
A
Events E1, E2, …,Ek are Collectively Exhaustive events if E1 U E2 U . . . U Ek = S
- i.e., the events completely cover the sample space
9
Q
Complement
A
The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted A
10
Q
Let the Sample Space be the collection of all possible outcomes of rolling one die: S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then
A = [2, 4, 6] and B = [4, 5, 6]
Q: Complements、Intersections、Unions、Mutually exclusive、Collectively exhaustive？
A
Mutually exclusive:
- A and B are not mutually exclusive
- The outcomes 4 and 6 are common to both
Collectively exhaustive:
- A and B are not collectively exhaustive
- AUB doesnotcontain1or3
11
Q
Probability
A
Probability – the chance that an uncertain event will occur (always between 0 and 1)
0 ≤ P(A) ≤ 1 For any event A
12
Q
Assessing Probability Methods
A
There are three approaches to assessing the probability of an uncertain event:
- classical probability
- relative frequency probability
- subjective probability
13
Q
Classical Probability Method
A
Assumes all outcomes in the sample space are equally likely to occur
Classical probability of event A:
14
Q
Permutations
A
Permutations: the number of possible arrangements when x objects are to be selected from a total of n objects and arranged in order [with (n – x) objects left over]
15
Q
Conditional probability
A
A conditional probability is the probability of one event, given that another event has occurred:
16
Q
Statistical Independence
A
- Two events are statistically independent if and only if: P(A∩ B)= P(A)P(B)
Events A and B are independent when the probability of one event is not affected by the other event
- If A and B are independent, then
P(A|B)= P(A)，if P(B)>0
P(B|A)= P(B)，if P(A)>0
17
Q
Joint and Marginal Probabilities
A
18
Q
Odds
A
- The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement
- The odds in favor of A are: below
19
Q
Bayes’ Theorem
A
对于贝叶斯公式，记住AB AB AB，然后再做分组：”AB = A×BA/B”。
贝叶斯定理虽然只是一个概率计算公式，但其最著名的一个用途便是“假阳性”和“假阴性”检测。
20
Q
Overinvolvement Ratio
A
21
Q
Using a Tree Diagram
A
22
Q
Random Variable
A
Represents a possible numerical value from a random experiment
23
Q
Discrete Random Variable
A
Takes on no more than a countable number of values
24
Q
Continuous Random Variable
A
- Can take on any value in an interval
- Possible values are measured on a continuum
25
Probability Distributions for Discrete Random Variables
Let X be a discrete random variable and x be one of its possible values.
* The probability that random variable X takes specific value x is denoted P(X = x)
* The **probability distribution function** of a random variable is a representation of the probabilities for all the possible outcomes.
* Can be shown algebraically, graphically, or with a table.
26
Probability Distribution Required Properties
* 0 ≤ P(x) ≤ 1 for any value of x
* The individual probabilities sum to 1;
27
Cumulative Probability Function
The **cumulative probability function**, denotedF(x0), shows the probability that X does not exceed the value x0.
**F(x0)=P(X≦x0)**
Where the function is evaluated at all values of x0.
28
Derived Relationship
The derived relationship between the probability distribution and the cumulative probability distribution.
Let X be a random variable with probability distribution P(x) and cumulative probability distribution F(x0). Then
29
Derived Properties
Derived properties of cumulative probability distributions for discrete random variables.离散随机变量累积概率分布的导出性质。
Let X be a discrete random variable with cumulative probability distribution F(x0). Then
1. 0 ≤ F(x0) ≤ 1 for every number x0
2. for x0 \< x1, then F(x0) ≤ F(x1)
30
Properties of Discrete Random Variables
Expected Value (or mean) of a discrete random variable X:
31
Variance and Standard Deviation( Formula)
32
Functions of Random Variables
If P(x) is the probability function of a discrete random variable X , and g(X) is some function of X , then the expected value of function g is
33
Linear Functions of Random Variables
* Let random variable X have mean μx and variance σ2x
* Let a and b be any constants.
* Let Y = a + bX
* Then the mean and variance of Y are
μY= E(a+bX)=a+bμx
σ2Y= Var(a+bX)=b2σ2X
* so that the standard deviation of Y is
σY= |b|σx
34
Properties of Linear Functions of Random Variables
* Let a and b be any constants.
* a) E(a)=a and Var(a)=0
i. e., if a random variable always takes the value a, it will have mean a and variance 0
* b) E(bX)= bμx and Var(bX)= b2σ2x
i. e., the expected value of b·X is b·E(x)
35
Probability Distributions
36
Bernoulli Distribution伯努利分布=二项分布The Binomial Distribution
# * Consider only two outcomes: “**success**” or “**failure**”
* Let **p** denote the probability of success
* Let **1 – p** be the probability of failure
* Define random variable X:
x = 1 if success, x = 0 if failure
* Then t**he Bernoulli probability distribution** is
P(0)=(1--p) and P(1)=p
37
Mean and Variance of a Bernoulli Random Variable
The mean is μx = p
The variance is σ2x = p(1 – p)
38
Developing the Binomial Distribution
39
Binomial Probability Distribution
* A fixed number of observations, n
* e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse
* Two mutually exclusive and collectively exhaustive categories
* e.g., head or tail in each toss of a coin; defective or not defective light bulb
* Generally called “success” and “failure”
* Probability of success is P , probability of failure is 1 – P
* Constant probability for each observation
* e.g., Probability of getting a tail is the same each time we toss the coin
* Observations are independent
* The outcome of one observation does not affect the outcome of the other
40
The Binomial Distribution
P(x) = probability of x successes in n trials,
with probability of success p on each trial
x = number of ‘successes’ in sample,(x = 0, 1, 2, ..., n)
n = sample size (number of independent trials or observations)
p = probability of “success”
41
Shape of Binomial Distribution
The shape of the binomial distribution depends on the values of p and n
42
\*Mean and Variance of a Binomial Distribution
43
Using Binomial Tables
44
The Hypergeometric Distribution
* “n” trials in a sample taken from a finite population of size N
* Sample taken without replacement
* Outcomes of trials are dependent
* Concerned with finding the probability of “X” successes in the sample where there are “S”successes in the population
45
\* Hypergeometric Probability Distribution
46
Example ：Using the Hypergeometric Distribution
* 3 different computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded?
47
Jointly Distributed Discrete Random Variables
联合分布式离散随机变量
48
Properties of Joint Probability Distributions
Properties of Joint Probability Distributions of Discrete Random Variables
Let X and Y be discrete random variables with joint probability distribution P(x, y)
1. 0 ≤ P(x, y) ≤ 1 for any pair of values x and y
2. the sum of the joint probabilities P(x, y) over all possible pairs of values must be 1
49
\*Conditional Probability Distribution
50
Independence
51
Conditional Mean and Variance
52
Covariance
53
\*Correlation
54
Covariance and Independence
* The covariance measures the strength of the linear relationship between two variables
* If two random variables are statistically independent, the covariance between them is 0
* The converse is not necessarily true
55
\*Portfolio Analysis, mean, variance.
* Let random variable X be the price for stock A
* Let random variable Y be the price for stock B
* The market value, W, for the portfolio is given by the linear function
W= aX+ bY
(a is the number of shares of stock A, b is the number of shares of stock B)
56
\*Example: Investment Returns
57
\*Example：Portfolio
58
\*Interpreting the Results for Investment Returns
59
Continuous Random Variables
* A continuous random variable is a variable that can assume any value in an interval
* thickness of an item
* time required to complete a tasktemperature of a solution
* height, in inches
* These can potentially take on any value, depending only on the ability to measure accurately.
60
Cumulative Distribution Function
61
\*Probability Density Function
The probability density function, f(x), of random variable X has the following properties:
62
Probability as an Area
63
The Uniform Distribution统一分部
The uniform distribution is a probability distribution that has equal probabilities for all equal-width intervals within the range of the random variable
64
\*Expectations for Continuous Random Variables
* The mean of X, denoted μx , is defined as the expected value of X
μx= E[X]
* The variance of X, denoted σx2 , is defined as the expectation of the squared deviation, (X - μx)2, of a random variable from its mean
σx2 =E[(X-μx )2]
65
\*Mean and Variance of the Uniform Distribution
66
\*Linear Functions of Random Variables
67
The Normal Distribution
1. Bell Shaped
2. Symmetrical
3. Mean, Median and Mode are Equal
* Location is determined by the mean, μ
* Spread is determined by the standard deviation, σ
* The random variable has an infinite theoretical range:+∞ to -∞
* The normal distribution closely approximates the probability distributions of a wide range of random variables
* Distributions of sample means approach a normaldistribution given a “large” sample size
* Computations of probabilities are direct and elegant
* The normal probability distribution has led to good business decisions for a number of applications
68
Many Normal Distributions
By varying the parameters μ and σ, we obtain different normal distributions
69
The Normal Distribution Shape
70
\*The Normal Probability Density Function
The formula for the normal probability density function is
71
\*Cumulative Normal Distribution
72
\* Finding Normal Probabilities
73
\*TheStandard Normal Distribution
74
Comparing X and Z units
Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)
75
Appendix Table 1
* The Standard Normal Distribution table in the textbook (Appendix Table 1) shows values of the cumulative normal distribution function.
* For a given Z-value a , the table shows F(a). (the area under the curve from negative infinity to a )
76
\*The Standard Normal Table
77
General Procedure for Finding Probabilities查找概率的一般程序
To find **P(a \< X \< b)** when X is distributed normally:
* Draw the normal curve for the problem in terms of X；
* Translate X-values to Z-values；
* Use the Cumulative Normal Table.
78
\*Finding Normal Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0;
Find P(X \< 8.6)
79
\*Upper Tail Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0.
Now Find P(X \> 8.6)
80
\*Finding the X value for a Known Probability
Steps to find the X value for a known probability:
1. Find the Z value for the known probability
2. Convert to X units using the formula:
xa=μ+zaσ
**Example:**
* Suppose X is normal with mean 8.0 and standard deviation 5.0.
* Now find the X value so that only 20% of all values are below this X
81
Assessing Normality
Not all continuous random variables are normally distributed. 并非所有连续随机变量都是正态分布的。
It is important to evaluate how well the data is approximated by a normal distribution.评估数据与正态分布的近似程度非常重要
82
The Normal Probability Plot
* Arrange data from low to high values;
* Find cumulative normal probabilities for all values;
* Examine a plot of the observed values vs. cumulative probabilities (with the cumulative normal probability on the vertical axis and the observed data values on the horizontal axis);
* Evaluate the plot for evidence of linearity.
83
Normal Distribution Approximation for Binomial Distribution
* Recall the binomial distribution:
* n independent trials
* probability of success on any given trial = p
* Random variable X:
* Xi =1 if the ith trial is “success”
* Xi =0 if the ith trial is “failure”
**E[X]= μ= np**
**Var(X)=σ2= np(1-p)**
84
Example:Binomial Approximation
40% of all voters support ballot proposition A. What is the probability that between 76 and 80 voters indicate support in a sample of n = 200 ?
85
The Exponential Distribution指数分布
Used to model the length of time between two occurrences of an event (the time between arrivals)用于模拟两次事件发生之间的时间长度（到达之间的时间）
86
Example： Exponential Distribution
Customers arrive at the service counter at the rate of 15 per hour. What is the probability that the arrival time between consecutive customers is less than three minutes?
87
Jointly Distributed Continuous Random Variables
88
CH5-Covariance
89
Ch5- Correlation
90
\*Ch5-Sums of Random Variables
91
\*Differences Between a Pair of Random Variables
92
\*Linear Combinations of Random Variables
93
Example： Linear Combinations of Random Variables
Two tasks must be performed by the same worker.
X = minutes to complete task 1; μx = 20, σx = 5
Y = minutes to complete task 2; μy = 30, σy = 8
X and Y are normally distributed and independent
* What is the mean and standard deviation of the time to complete both tasks?
94
Financial Investment Portfolios
A financial portfolio can be viewed as a linear combination of separate financial instruments
95
Example: Portfolio Analysis
* Consider two stocks, A and B
* The price of Stock A is normally distributed with mean 12 and variance 4
* The price of Stock B is normally distributed with mean 20 and variance 16
* The stock prices have a positive correlation, ρAB = .50
* Suppose you own
* 10 shares of Stock A
* 30 shares of Stock B
Z值在表中查F（Z）的值
96
\*Example： cumulative distribution probabilities
易错
97
\*Example：Z表和Za表
98
Confidence Intervals 置信区间
Contents of this chapter:
* Confidence Intervals for the **Population Mean, μ**
* when Population Variance σ2 is Known
* when Population Variance σ2 is Unknown
* Confidence Intervals for the **Population Proportion, P** (large samples)
* Confidence interval estimates for the **variance** of a normal population
* Finite population corrections
* Sample-size determination
99
Properties of Point Estimators
* An **estimator** of a population parameter is
* a random variable that depends on sample
information . . .
* whose value provides an approximation to this unknown parameter
* A specific value of that random variable is called an **estimate**
100
Point and Interval Estimates
* A point estimate is a single number,
* a confidence interval provides additional information about variability
101
Point Estimates
102
Unbiasedness
103
Ch-7： Bias
104
Most Efficient Estimator
* Suppose there are several unbiased estimators of θ
* The **most efficient estimator** or the **minimum variance unbiased estimator** of θ is the unbiased estimator with the **smallest variance**
105
Confidence Interval Estimation
How much uncertainty is associated with a point estimate of a population parameter?
* An interval estimate provides more information about a population characteristic than does a point estimate
* Such interval estimates are called **confidence interval estimates.**
* An interval gives a range of values:
* Takes into consideration variation in sample statistics from sample to sample
* Based on observation from 1 sample
* Gives information about closeness to unknown population parameters
* Stated in terms of level of confidence
* Can never be 100% confident
106
Confidence Interval and Confidence Level
* If P(a \<θ \< b) = 1 -α, then the interval from a to b is called a 100(1 - α)% confidence interval of θ.
* The quantity 100(1 - α)% is called theconfidence level of the interval
* α is between 0 and 1
* In repeated samples of the population, the true value of the parameter θ would be contained in 100(1 -α )% of intervals calculated this way.
* The confidence interval calculated in this manner is written as a \< θ\< b with 100(1 - α)% confidence
107
\* General Formula置信区间
The general form for all confidence intervals is:
108
Confidence Intervals 分布图σ2 Known
109
Confidence Interval Estimation for the Mean (σ2 Known)
* Assumptions
* Population variance σ2 is known
* Population is normally distributed
* If population is not normal, use large sample
110
Confidence Limits置信区间限制
111
Margin of Error置信区间误差幅度
112
Reducing the Margin of Error 减小误差
113
\* Finding Z1-α/2
114
Common Levels of Confidence
Commonly used confidence levels are 90%, 95% and 99%
115
Intervals and Level of Confidence间隔和置信度
116
Example：置信区间 confidence interval
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 95% confidence interval for the true mean resistance of the population.
Solution:
117
Confidence Interval Estimation for the Mean (σ2 Unknown)
If the population standard deviation σ is unknown, we can **substitute the sample standard deviation, s**
This introduces extra uncertainty, since s is variable from sample to sample
So we **use the t distribution** instead of the normal distribution
118
Student’s **t Distribution**
119
Student’s t Table
120
t distribution values，
With comparison to the Z value
121
σ2 Unknown --Margin of Error
122
Example：T-Table
A random sample of n = 25 has x- = 50 and s = 8. Form a 95% confidence interval for μ
123
Confidence Interval Estimation for **Population Proportion**
124
Confidence Interval Endpoints置信区间端点
125
Example：置信区间 Population proportion
A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers
126
Confidence Intervals for the **Population Variance人口差异**
* Goal: Form a confidence interval for the population variance, σ2
* The confidence interval is based on the sample variance, s2
* Assumed: the population is normally distributed
127
Example：Population Variance
You are testing the speed of a batch of computer processors. You collect the following data (in Mhz):
Sample size 17，Sample mean 3004，Sample std dev 74.
Assume the population is normal.
Determine the 95% confidence interval for σx2
128
Confidence Interval Estimation: Finite Populations
Finite Population Correction Factor
If the sample size is more than 5% of the population size (and sampling is without replacement) then a finite population correction factor must be used when calculating the standard error.
Suppose sampling is without replacement and the sample size is large relative to the population size.
Assume the population size is large enough to apply the central limit theorem.
Apply the finite population correction factorwhen estimating the population variance
129
Estimating the Population Mean
130
Finite Populations: Mean 有限人口的平均值
131
Estimating the Population Total
Consider a simple random sample of size n from a population of size N
The quantity to be estimated is the population total Nμ
An unbiased estimation procedure for the population total Nμ yields the point estimate Nx-
132
Example：Confidence Interval for Population Total
A firm has a population of 1000 accounts and wishes to estimate the value of the total population balance.
A sample of 80 accounts is selected with average balance of $87.60 and standard deviation of $22.30.
Find the 95% confidence interval estimate of the total balance.
133
Estimating the Population Proportion: Finite Population
134
Confidence Intervals for Population Proportion: Finite Population
135
Sample Size Determination: Population Proportion
136
Example：Required Sample Size: Population Proportion
How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?
137
Sample-Size Determination:Finite Populations
138
Example: Sample Size to Estimate Population Proportion
How large a sample would be necessary to estimate within ±5% the true proportion of college graduates in a population of 850 people with 95% confidence?
139
\*考试例题： 置信区间sigma unknown
140
Ch-9 Concepts of Hypothesis Testing
A hypothesis is a claim (assumption) about a population parameter:
* population mean
* Example: The mean monthly cell phone bill ofthiscityis μ=$52
* population proportion
* Example: The proportion of adults in this city with cell phones is p = .88
141
The Null Hypothesis, H0
142
The Alternative Hypothesis, H1
* Is the opposite of the null hypothesis
* e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 )
* Challenges the status quo
* Never contains the “=” , “≤” or “≥” sign
* May or may not be supported
* **Is generally the hypothesis that the researcher is trying to support**
143
Hypothesis Testing Process
144
**Level of Significance, α**
**and the Rejection Region**
* Defines the unlikely values of the sample statistic if the null hypothesis is true
* Defines **rejection region** of the sampling distribution
* Is designated by **α** , (level of significance)
* Typical values are 0.01, 0.05, or 0.10
* Is selected by the researcher at the beginning
* Provides the **critical value(s)** of the test
145
Ch-9 Errors in Making Decisions
**Type I Error**
* Reject a true null hypothesis
* Considered a serious type of error
_The probability of Type I Error is **α**_
* Called **level of significance** of the test
* Set by researcher in advance
**Type II Error**
* Fail to reject a false null hypothesis
* _The probability of Type II Error is β_
146
Ch-9 Outcomes and Probabilities
147
Consequences of Fixing the Significance Level of a Test
148
Type I & II Error Relationship
149
Power of the Test
150
Hypothesis Tests for the Mean
151
Tests of the Mean of a Normal Distribution (σ Known)
152
p-Value Approach to Testing
* p-value: Probability of obtaining a test statistic more extreme ( ≤ or ≥) than the observed sample value given H0 is true
* Also called observed level of significance
* Smallest value of for which H0 can be
rejected
153
\* 不懂：Example: Upper-Tail Z Test for Mean (Known)
A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume σ= 10 is known)
Form hypothesis test:
H0: μ ≤ 52 the average is not over $52 per month；
H1: μ \> 52 the average is greater than $52 per month(i.e., sufficient evidence exists to support the
manager’s claim)
154
One-Tail Tests
155
Upper-Tail Tests
There is only one critical value, since the rejection area is in only one tail
156
Lower-Tail Tests
157
Two-Tail Tests
158
Example:Hypothesis Testing
Test the claim that the true mean # of TV sets in US homes is equal to 3. (Assume σ = 0.8)
* State the appropriate null and alternative hypotheses.
* H0: μ = 3 , H1: μ ≠ 3 (This is a two tailed test)
* Specify the desired level of significance
* Suppose that α= .05 is chosen for this test
* Choose a sample size
* Suppose a sample of size n = 100 is selected
* Determine the appropriate technique
* σ is known so this is a z test
* Set up the critical values
* For α= .05 the critical z values are ±1.96
* Collect the data and compute the test statistic
* Suppose the sample results are n = 100, x- = 2.84 (σ = 0.8 is assumed known)
159
Example: p-Value
Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is μ = 3.0?
160
Tests of the Mean of a Normal Population (σ Unknown)
161
Example: Two-Tail Test（σ Unknown）
The average cost of a hotel room in Chicago is said to be $168 per night.
A random sample of 25 hotels resulted in
x- = $172.50 and s = $15.40. Test at the α= 0.05 level.
H0: μ = 168， H1: μ≠ 168
162
Tests of the Population Proportion
* Involves categorical variables
* Two possible outcomes
* “Success” (a certain characteristic is present)
* “Failure” (the characteristic is not present)
* Fraction or proportion of the population in the“success” category is denoted by P
* Assume sample size is large
163
Hypothesis Tests for Proportions
164
Example: Z Test for Proportion
A marketing company claims that it receives 8% responses from its mailing.To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the α= .05 significance level.
Check:
Our approximation for p is pˆ = 25/500 = .05
np(1 - p) = (500)(.05)(.95) = 23.75 \> 5
165
Example: Z Test for Proportion--use p-Value Solution
A marketing company claims that it receives 8% responses from its mailing.To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the α= .05 significance level.
Check:
Our approximation for p is pˆ = 25/500 = .05
np(1 - p) = (500)(.05)(.95) = 23.75 \> 5
p-Value Solution
166
Assessing the Power of a Test
167
Type II Error
168
Example：Type II Error
Type II error is the probability of failing to reject a false H0
Suppose we fail to reject H0: μ ≥ 52
when in fact the true mean is μ\* = 50
Suppose we do not reject H0: μ 52 when in fact the true mean is μ\* = 50
Find β
169
Example：Calculate β, 接上一例题
Suppose n = 64 , σ = 6 , and α= .05
170
Example：Power of the Test
接上一例题
If the true mean is μ\* = 50,
The probability of Type II Error = β = 0.1539
The power of the test = 1 – β = 1 – 0.1539 = 0.8461
171
Tests of the Variance of a Normal Distribution
172
Decision Rules: Variance
173
Appendix: Guidelines for Decision Rule：Population Mean
174
Appendix: Guidelines for Decision Rule：Population Proportion
175
Two Population Hypothesis Tests
176
Ch-10：Dependent Samples
177
Example:Matched Pairs
two-tail test
Assume you send your salespeople to a “customer service” training workshop. Has the training made adifference in the number of complaints? You collect the following data:
Has the training **made a difference** in the number of complaints (at the α= 0.05 level)?
178
Tests of the Difference Between Two Normal **Population Means**: Independent Samples
**Goal:** Form a confidence interval for the difference between two population means, μx – μy
* Different populations
* Unrelated
* Independent
* Sample selected from one population has no effect on the sample selected from the other population
* Normally distributed
179
Tests of the Difference Between Two Normal Population Means: σx2 and σy2 Known
180
σx2 and σy2 UnKnown，Population means, independent samples，**Assumed Equal**
181
Example：Pooled Variance t Test
You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data:
Assuming both populations are approximately normal with equal variances, is there a difference in average yield (α= 0.05)?
182
Population means, independent samples: σx2 and σy2 Unknown, Assumed Unequal
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Tests of the Difference Between **Two Population Proportions** (Large Samples)
Goal: Test hypotheses for the difference between two population proportions, **Px – Py**
Assumptions: Both sample sizes are large, **nP(1 – P) \> 5**
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\*Example: Two Population Proportions
Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes
Test at the .05 level of significance
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\* Tests of Equality of Two Variances
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\* Example: F Test
You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:
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Some Comments on Hypothesis Testing
* A test with low power can result from:
* Small sample size
* Large variances in the underlying populations
* Poor measurement procedures
* If sample sizes are large it is possible to find significant differences that are not practically important
* Researchers should select the appropriate level of significance before computing p-values
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Ch-10 Chapter Summary
* Compared two dependent samples (paired samples)
* Performed paired sample t test for the mean difference
* Compared two independent samples
* Performed z test for the differences in two means
* Performed pooled variance t test for the differences in two means
* Compared two population proportions
* Performed z-test for two population proportions
* Performed F tests for the difference between two population variances
* Used the F table to find F critical values
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Ch-11 Overview of Linear Models
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Least Squares Regression
Introduction to Regression Analysis
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Linear Regression Model
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Linear Regression Assumptions
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Least Squares Coefficient Estimators
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Computer Computation of Regression Coefficients
* The coefficients b0 and b1 , and other regression results in this chapter, will be found using a computer
* Hand calculations are tedious
* Statistical routines are built into Excel
* Other statistical analysis software can be used
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Interpretation of the Slope and the Intercept
P577 用电脑检验Regression
* b0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values)
* b1 is the estimated change in the average value of y as a result of a one- unit change in x
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Example：single variable，σ Unknown，T-test
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Example： Correlation coefficient，simple linear Regression， R2，confidence interval
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Explanatory Power of a Linear Regression Equation
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Coefficient of Determination, R2
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Correlation and R2
The coefficient of determination, R2, for a simple regression is equal to the simple correlation squared.
R2= r2
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Estimation of Model Error Variance
