Measure of Variability (Dispersion, Spread) 1.Range
2.Inter-Quartile Range 3.Variance, standard deviation
4.Pseudo-standard deviation
Slide 8
1.Range R = Range = max - min 2.Inter-Quartile Range (IQR)
Inter-Quartile Range = IQR = Q 3 - Q 1
Slide 9
The Sample Variance Is defined as the quantity: and is denoted
by the symbol
Slide 10
The Sample Standard Deviation s Definition: The Sample Standard
Deviation is defined by: Hence the Sample Standard Deviation, s, is
the square root of the sample variance.
Slide 11
Interpretations of s In Normal distributions Approximately 2/3
of the observations will lie within one standard deviation of the
mean Approximately 95% of the observations lie within two standard
deviations of the mean In a histogram of the Normal distribution,
the standard deviation is approximately the distance from the mode
to the inflection point
Slide 12
s Inflection point Mode
Slide 13
s 2/3 s
Slide 14
2s
Slide 15
Computing formulae for s and s 2 The sum of squares of
deviations from the the mean can also be computed using the
following identity:
Slide 16
Then:
Slide 17
Slide 18
A quick (rough) calculation of s The reason for this is that
approximately all (95%) of the observations are between and
Thus
Slide 19
The Pseudo Standard Deviation (PSD) Definition: The Pseudo
Standard Deviation (PSD) is defined by:
Slide 20
Properties For Normal distributions the magnitude of the pseudo
standard deviation (PSD) and the standard deviation (s) will be
approximately the same value For leptokurtic distributions the
standard deviation (s) will be larger than the pseudo standard
deviation (PSD) For platykurtic distributions the standard
deviation (s) will be smaller than the pseudo standard deviation
(PSD)
Slide 21
Measures of Shape
Slide 22
Skewness Kurtosis
Slide 23
Skewness based on the sum of cubes Kurtosis based on the sum of
4 th powers
Slide 24
The Measure of Skewness
Slide 25
The Measure of Kurtosis
Slide 26
Interpretations of Measures of Shape Skewness Kurtosis g 1 >
0g 1 = 0 g 1 < 0 g 2 < 0 g 2 = 0 g 2 > 0
Slide 27
Inferential Statistics Making decisions regarding the
population base on a sample
Slide 28
Estimation by Confidence Intervals Definition An (100) P%
confidence interval of an unknown parameter is a pair of sample
statistics (t 1 and t 2 ) having the following properties: 1. P[t 1
< t 2 ] = 1. That is t 1 is always smaller than t 2. 2. P[the
unknown parameter lies between t 1 and t 2 ] = P. the statistics t
1 and t 2 are random variables Property 2. states that the
probability that the unknown parameter is bounded by the two
statistics t 1 and t 2 is P.
Slide 29
Confidence Interval for a Proportion
Slide 30
The sample size that will estimate p with an Error Bound B and
level of confidence P = 1 is: where: B is the desired Error Bound z
is the /2 critical value for the standard normal distribution p* is
some preliminary estimate of p. Determination of Sample Size
Slide 31
Confidence Intervals for the mean of a Normal Population,
Slide 32
The sample size that will estimate with an Error Bound B and
level of confidence P = 1 is: where: B is the desired Error Bound z
is the /2 critical value for the standard normal distribution s* is
some preliminary estimate of s. Determination of Sample Size
Slide 33
Hypothesis Testing An important area of statistical
inference
Slide 34
Definition Hypothesis (H) Statement about the parameters of the
population In hypothesis testing there are two hypotheses of
interest. The null hypothesis (H 0 ) The alternative hypothesis (H
A )
Slide 35
Type I, Type II Errors 1.Rejecting the null hypothesis when it
is true. (type I error) 2.accepting the null hypothesis when it is
false (type II error)
Slide 36
Decision Table showing types of Error H 0 is TrueH 0 is False
Correct Decision Type I Error Type II Error Accept H 0 Reject H
0
Slide 37
To define a statistical Test we 1.Choose a statistic (called
the test statistic) 2.Divide the range of possible values for the
test statistic into two parts The Acceptance Region The Critical
Region
Slide 38
To perform a statistical Test we 1.Collect the data. 2.Compute
the value of the test statistic. 3.Make the Decision: If the value
of the test statistic is in the Acceptance Region we decide to
accept H 0. If the value of the test statistic is in the Critical
Region we decide to reject H 0.
Slide 39
Probability ofhe two types of error Definitions: For any
statistical testing procedure define = P[Rejecting the null
hypothesis when it is true] = P[ type I error] = P[accepting the
null hypothesis when it is false] = P[ type II error]
Slide 40
Determining the Critical Region 1.The Critical Region should
consist of values of the test statistic that indicate that H A is
true. (hence H 0 should be rejected). 2.The size of the Critical
Region is determined so that the probability of making a type I
error, , is at some pre-determined level. (usually 0.05 or 0.01).
This value is called the significance level of the test.
Significance level = P[test makes type I error]
Slide 41
To find the Critical Region 1.Find the sampling distribution of
the test statistic when is H 0 true. 2.Locate the Critical Region
in the tails (either left or right or both) of the sampling
distribution of the test statistic when is H 0 true. Whether you
locate the critical region in the left tail or right tail or both
tails depends on which values indicate H A is true. The tails
chosen = values indicating H A.
Slide 42
3.the size of the Critical Region is chosen so that the area
over the critical region and under the sampling distribution of the
test statistic when is H 0 true is the desired level of =P[type I
error] Sampling distribution of test statistic when H 0 is true
Critical Region - Area =
Slide 43
The z-tests Testing the probability of success Testing the mean
of a Normal Population
Slide 44
The Alternative Hypothesis H A The Critical Region Critical
Regions for testing the probability of success, p
Slide 45
The Alternative Hypothesis H A The Critical Region Critical
Regions for testing mean, of a normal population
Slide 46
You can compare a statistical test to a meter Value of test
statistic Acceptance Region Critical Region Critical Region
Critical Region is the red zone of the meter
Slide 47
Value of test statistic Acceptance Region Critical Region
Critical Region Accept H 0
Slide 48
Value of test statistic Acceptance Region Critical Region
Critical Region Reject H 0
Slide 49
Acceptance Region Critical Region Sometimes the critical region
is located on one side. These tests are called one tailed
tests.
Slide 50
Whether you use a one tailed test or a two tailed test depends
on: 1.The hypotheses being tested (H 0 and H A ). 2.The test
statistic.
Slide 51
If only large positive values of the test statistic indicate H
A then the critical region should be located in the positive tail.
(1 tailed test) If only large negative values of the test statistic
indicate H A then the critical region should be located in the
negative tail. (1 tailed test) If both large positive and large
negative values of the test statistic indicate H A then the
critical region should be located both the positive and negative
tail. (2 tailed test)
Slide 52
Usually 1 tailed tests are appropriate if H A is one-sided. Two
tailed tests are appropriate if H A is two - sided. But not
always
Slide 53
The p-value approach to Hypothesis Testing
Slide 54
Definition Once the test statistic has been computed form the
data the p-value is defined to be: p-value = P[the test statistic
is as or more extreme than the observed value of the test statistic
when H 0 is true] more extreme means giving stronger evidence to
rejecting H 0
Slide 55
Properties of the p -value 1.If the p-value is small (
The approximate test for a comparing two means of Normal
Populations (unequal variances) Null HypothesisAlt.
HypothesisCritical Region H 0 : 1 = 2 H 0 : 1 2 t t H 0 : 1 > 2
t > t H 0 : 1 < 2 t < -t Test statistic
Slide 93
Confidence intervals for the difference in two means of normal
populations (small samples, unequal variances) (1 )100% confidence
limits for 1 2 with
Slide 94
The paired t-test An example of improved experimental
design
Slide 95
The matched pair experimental design (The paired sample
experiment) Prior to assigning the treatments the subjects are
grouped into pairs of similar subjects. Suppose that there are n
such pairs (Total of 2n = n + n subjects or cases), The two
treatments are then randomly assigned to each pair. One member of a
pair will receive treatment 1, while the other receives treatment
2. The data collected is as follows: (x 1, y 1 ), (x 2,y 2 ), (x
3,y 3 ),, , (x n, y n ). x i = the response for the case in pair i
that receives treatment 1. y i = the response for the case in pair
i that receives treatment 2.
Slide 96
Let x i = the measurement of the response for the subject in
pair i that received treatment 1. Let y i = the measurement of the
response for the subject in pair i that received treatment 2.
x1y1x1y1 The data x2y2x2y2 x3y3x3y3 xnynxnyn
Slide 97
To test H 0 : 1 = 2 is equivalent to testing H 0 : d = 0. (we
have converted the two sample problem into a single sample
problem). The test statistic is the single sample t-test on the
differences d 1, d 2, d 3, , d n namely df = n - 1
Slide 98
Testing for the equality of variances The F test
Slide 99
The test statistic (F) The sampling distribution of the test
statistic If the Null Hypothesis (H 0 ) is true then the sampling
distribution of F is called the F-distribution with 1 = n - 1
degrees in the numerator and 2 = m - 1 degrees in the
denominator
Slide 100
The F distribution 1 = n - 1 degrees in the numerator 2 = m - 1
degrees in the denominator F ( 1, 2 )
Slide 101
(Two sided alternative) Reject H 0 if or Critical region for
the test:
Slide 102
Reject H 0 if Critical region for the test (one tailed): (one
sided alternative)
Slide 103
Summary of Tests
Slide 104
One Sample Tests p = p 0 p > p 0 p p 0 p < p 0
Slide 105
Two Sample Tests
Slide 106
Two Sample Tests - continued SituationTest statisticH0H0 HAHA
Critical Region Two independent Normal samples with unknown means
and variances (unequal) t t df = * >> t > t df = * F (m-1,
n -1) >> F > F (n-1, m -1)
The paired t test SituationTest statisticH0H0 HAHA Critical
Region n matched pair of subjects are treated with two treatments.
d i = x i y i has mean = t t df = n - 1 >> t > t df = n -
1
The test for independence (zero correlation) The test
statistic: Reject H 0 if |t| > t a/2 (df = n 2) H 0 : X and Y
are independent H A : X and Y are correlated The Critical region
This is a two-tailed critical region, the critical region could
also be one-tailed
Slide 166
Spearmans rank correlation coefficient (rho)
Slide 167
Spearmans rank correlation coefficient (rho) Spearmans rank
correlation coefficient is computed as follows: Arrange the
observations on X in increasing order and assign them the ranks 1,
2, 3, , n Arrange the observations on Y in increasing order and
assign them the ranks 1, 2, 3, , n. For any case (i) let ( x i, y i
) denote the observations on X and Y and let ( r i, s i ) denote
the ranks on X and Y.
Slide 168
Spearmans rank correlation coefficient is defined as follows:
For each case let d i = r i s i = difference in the two ranks. Then
Spearmans rank correlation coefficient ( ) is defined as
follows:
Slide 169
Properties of Spearmans rank correlation coefficient 1.The
value of is always between 1 and +1. 2.If the relationship between
X and Y is positive, then will be positive. 3.If the relationship
between X and Y is negative, then will be negative. 4.If there is
no relationship between X and Y, then will be zero. 5.The value of
will be +1 if the ranks of X completely agree with the ranks of Y.
6.The value of will be -1 if the ranks of X are in reverse order to
the ranks of Y.
Slide 170
Relationship between Regression and Correlation
Slide 171
Recall Also since Thus the slope of the least squares line is
simply the ratio of the standard deviations the correlation
coefficient
Slide 172
The coefficient of Determination
Slide 173
Sums of Squares associated with Linear Regresssion = SS
unexplained
Slide 174
It can be shown: (Total variability in Y) = (variability in Y
explained by X) + (variability in Y unexplained by X)
Slide 175
It can also be shown: = proportion variability in Y explained
by X. = the coefficient of determination
Slide 176
Further: = proportion variability in Y that is unexplained by
X.
Slide 177
Regression (in general)
Slide 178
In many experiments we would have collected data on a single
variable Y (the dependent variable ) and on p (say) other variables
X 1, X 2, X 3,..., X p (the independent variables). One is
interested in determining a model that describes the relationship
between Y (the response (dependent) variable) and X 1, X 2, , X p
(the predictor (independent) variables. This model can be used for
Prediction Controlling Y by manipulating X 1, X 2, , X p
Slide 179
The Model: is an equation of the form Y = f(X 1, X 2,...,X p |
1, 2,..., q ) + where 1, 2,..., q are unknown parameters of the
function f and is a random disturbance (usually assumed to have a
normal distribution with mean 0 and standard deviation .
Slide 180
The Multiple Linear Regression Model
Slide 181
In Multiple Linear Regression we assume the following model Y =
0 + 1 X 1 + 2 X 2 +... + p X p + This model is called the Multiple
Linear Regression Model. Again are unknown parameters of the model
and where 0, 1, 2,..., p are unknown parameters and is a random
disturbance assumed to have a normal distribution with mean 0 and
standard deviation .
Slide 182
Summary of the Statistics used in Multiple Regression
Slide 183
The Least Squares Estimates: - the values that minimize
Slide 184
The Analysis of Variance Table Entries a) Adjusted Total Sum of
Squares (SS Total ) b) Residual Sum of Squares (SS Error ) c)
Regression Sum of Squares (SS Reg ) Note: i.e. SS Total = SS Reg
+SS Error
Slide 185
The Analysis of Variance Table SourceSum of Squaresd.f.Mean
SquareF RegressionSS Reg pSS Reg /p = MS Reg MS Reg /s 2 ErrorSS
Error n-p-1SS Error /(n-p-1) =MS Error = s 2 TotalSS Total n-1
Slide 186
Uses: 1.To estimate 2 (the error variance). - Use s 2 = MS
Error to estimate 2. 2.To test the Hypothesis H 0 : 1 = 2 =... = p
= 0. Use the test statistic - Reject H 0 if F > F
(p,n-p-1).
Slide 187
3.To compute other statistics that are useful in describing the
relationship between Y (the dependent variable) and X 1, X 2,...,X
p (the independent variables). a)R 2 = the coefficient of
determination = SS Reg /SS Total = = the proportion of variance in
Y explained by X 1, X 2,...,X p 1 - R 2 = the proportion of
variance in Y that is left unexplained by X 1, X2,..., X p = SS
Error /SS Total.
Slide 188
b)R a 2 = "R 2 adjusted" for degrees of freedom. = 1 -[the
proportion of variance in Y that is left unexplained by X 1, X
2,..., X p adjusted for d.f.]
Slide 189
c) R= R 2 = the Multiple correlation coefficient of Y with X 1,
X 2,...,X p = = the maximum correlation between Y and a linear
combination of X 1, X 2,...,X p Comment: The statistics F, R 2, R a
2 and R are equivalent statistics.
Slide 190
Logistic regression
Slide 191
The dependent variable y is binary. It takes on two values
Success (1) or Failure (0) This is the situation in which Logistic
Regression is used We are interested in predicting a y from a
continuous dependent variable x.
Slide 192
The logisitic Regression Model Let p denote P[y = 1] =
P[Success]. This quantity will increase with the value of x. The
ratio: is called the odds ratio This quantity will also increase
with the value of x, ranging from zero to infinity. The quantity:
is called the log odds ratio
Slide 193
The logisitic Regression Model i. e. : In terms of the odds
ratio Assumes the log odds ratio is linearly related to x.
Slide 194
The logisitic Regression Model Solving for p in terms x.
Slide 195
Interpretation of the parameter 0 (determines the intercept) p
x
Slide 196
Interpretation of the parameter 1 (determines when p is 0.50
(along with 0 )) p x when
Slide 197
Interpretation of the parameter 1 (determines slope when p is
0.50 ) p x
Slide 198
The Multiple Logistic Regression model
Slide 199
Here we attempt to predict the outcome of a binary response
variable Y from several independent variables X 1, X 2, etc
Slide 200
Nonparametric Statistical Methods
Slide 201
Definition When the data is generated from process (model) that
is known except for finite number of unknown parameters the model
is called a parametric model. Otherwise, the model is called a non-
parametric model Statistical techniques that assume a non-
parametric model are called non-parametric.
Slide 202
Nonparametric Statistical Methods
Slide 203
The sign test A nonparametric test for the central location of
a distribution
Slide 204
To carry out the The Sign test: S = the number of observations
that exceed 0 = s observed p-value = P [S s observed ] ( = 2 P [S s
observed ] for 2-tailed test) where S is binomial, n = sample size,
p = 0.50 1.Compute the test statistic: 2.Compute the p-value of
test statistic, s observed : 3.Reject H 0 if p-value low (<
0.05)
Slide 205
Sign Test for Large Samples
Slide 206
If n is large we can use the Normal approximation to the
Binomial. Namely S has a Binomial distribution with p = and n =
sample size. Hence for large n, S has approximately a Normal
distribution with mean and standard deviation
Slide 207
Hence for large n,use as the test statistic (in place of S)
Choose the critical region for z from the Standard Normal
distribution. i.e. Reject H 0 if z z /2 two tailed ( a one tailed
test can also be set up.
Slide 208
Nonparametric Confidence Intervals
Slide 209
Now arrange the data x 1, x 2, x 3, x n in increasing order
Assume that the data, x 1, x 2, x 3, x n is a sample from an
unknown distribution. Hence x (1) < x (2) < x (3) < < x
(n) x (1) = the smallest observation x (2) = the 2 nd smallest
observation x (n) = the largest observation Consider the k th
smallest observation and the k th largest observation in the data x
1, x 2, x 3, x n x (k) and x (n k + 1)
Slide 210
Hence P[x (k) < median < x (n k + 1) ] = p(k) + p(k + 1)
+ + p(n-k) = P = P[k the no. of obs greater than the median n-k]
where p(i)s are binomial probabilities with n = the sample size and
p =1/2. This means that x (k) to x (n k + 1) is a P(100)%
confidence interval for the median Choose k so that P = p(k) + p(k
+ 1) + + p(n-k) is close to.95 (or 0.99)
Slide 211
Summarizing where P = p(k) + p(k + 1) + + p(n-k) and p(i)s are
binomial probabilities with n = the sample size and p =1/2. x (k)
to x (n k + 1) is a P(100)% confidence interval for the median
Slide 212
For large values of n one can use the normal approximation to
the Binomial to find the value of k so that x (k) to x (n k + 1) is
a 95% confidence interval for the median.
Slide 213
Slide 214
The Wilcoxon Signed Rank Test An Alternative to the sign
test
Slide 215
For Wicoxons signed-Rank test we would assign ranks to the
absolute values of (x 1 0, x 2 0, , x n 0 ). A rank of 1 to the
value of x i 0 which is smallest in absolute value. A rank of n to
the value of x i 0 which is largest in absolute value. W + = the
sum of the ranks associated with positive values of x i 0. W - =
the sum of the ranks associated with negative values of x i 0.
Slide 216
To carry out Wilcoxons signed rank test We 1.Compute T = W + or
W - (usually it would be the smaller of the two) 2.Let t observed =
the observed value of T. 3.Compute the p-value = P[T t observed ]
(2 P[T t observed ] for a two-tailed test). i.For n 12 use the
table. ii.For n > 12 use the Normal approximation. 4.Conclude H
A (Reject H 0 ) if p-value is less than 0.05 (or 0.01).
Slide 217
For sample sizes, n > 12 we can use the fact that T (W + or
W - ) has approximately a normal distribution with
Slide 218
1.The t test i.This test requires the assumption of normality.
ii.If the data is not normally distributed the test is invalid The
probability of a type I error may not be equal to its desired value
(0.05 or 0.01) iii.If the data is normally distributed, the t-test
commits type II errors with a smaller probability than any other
test (In particular Wilcoxons signed rank test or the sign test)
2.The sign test i.This test does not require the assumption of
normality (true also for Wilcoxons signed rank test). ii.This test
ignores the magnitude of the observations completely. Wilcoxons
test takes the magnitude into account by ranking them Comments
Slide 219
Two-sample Non-parametic tests
Slide 220
Mann-Whitney Test A non-parametric two sample test for
comparison of central location
Slide 221
The Mann-Whitney Test This is a non parametric alternative to
the two sample t test (or z test) for independent samples. These
tests (t and z) assume the data is normal The Mann- Whitney test
does not make this assumption. Sample of n from population 1 x 1, x
2, x 3, , x n Sample of m from population 2 y 1, y 2, y 3, , y
m
Slide 222
The Mann-Whitney test statistics U 1 and U 2 Arrange the
observations from the two samples combined in increasing order
(retaining sample membership) and assign ranks to the observations.
Let W 1 = the sum of the ranks for sample 1. Let W 2 = the sum of
the ranks for sample 2. Then and
Slide 223
The distribution function of U (U 1 or U 2 ) has been tabled
for various values of n and m (
The Mann-Whitney test for large samples For large samples (n
> 10 and m >10) the statistics U 1 and U 2 have approximately
a Normal distribution with mean and standard deviation
Slide 225
Thus we can convert U i to a standard normal statistic And
reject H 0 if z z /2 (for a two tailed test)
Slide 226
The Kruskal Wallis Test Comparing the central location for k
populations An nonparametric alternative to the one-way ANOVA
F-test
Slide 227
Situation: Data is collected from k populations. The sample
size from population i is n i. The data from population i is:
Slide 228
The computation of The Kruskal-Wallis statistic We group the N
= n 1 + n 2 + + n k observation from k populations together and
rank these observations from 1 to N. Let r ij be the rank
associated with with the observation x ij. Handling of tied
observations If a group of observations are equal the ranks that
would have been assigned to those observations are averaged
Slide 229
The Kruskal-Wallis statistic where = the sum of the ranks for
the i th sample
Slide 230
The Kruskal-Wallis test Reject H 0 : the k populations have
same central location
Slide 231
Probability Theory Probability Models for random phenomena
Slide 232
Definitions
Slide 233
The sample Space, S The sample space, S, for a random phenomena
is the set of all possible outcomes.
Slide 234
An Event, E The event, E, is any subset of the sample space, S.
i.e. any set of outcomes (not necessarily all outcomes) of the
random phenomena S E Venn diagram
Slide 235
The event, E, is said to have occurred if after the outcome has
been observed the outcome lies in E. S E
Slide 236
Set operations on Events Union Let A and B be two events, then
the union of A and B is the event (denoted by A B) defined by: A B
= {e| e belongs to A or e belongs to B} A B AB
Slide 239
AB The event A B occurs if the event A occurs and the event and
B occurs.
Slide 240
Complement Let A be any event, then the complement of A
(denoted by ) defined by: = {e| e does not belongs to A} A
Slide 241
The event occurs if the event A does not occur A
Slide 242
In problems you will recognize that you are working with:
1.Union if you see the word or, 2.Intersection if you see the word
and, 3.Complement if you see the word not.
Slide 243
Definition: mutually exclusive Two events A and B are called
mutually exclusive if: A B
Slide 244
If two events A and B are are mutually exclusive then: A B
1.They have no outcomes in common. They cant occur at the same
time. The outcome of the random experiment can not belong to both A
and B.
Slide 245
Rules of Probability
Slide 246
The additive rule P[A B] = P[A] + P[B] P[A B] and if A B = P[A
B] = P[A] + P[B]
Slide 247
The Rule for complements for any event E
Slide 248
Conditional probability
Slide 249
The multiplicative rule of probability and if A and B are
independent. This is the definition of independent
Slide 250
Counting techniques
Slide 251
Summary of counting rules Rule 1 n(A 1 A 2 A 3 . ) = n(A 1 ) +
n(A 2 ) + n(A 3 ) + if the sets A 1, A 2, A 3, are pairwise
mutually exclusive (i.e. A i A j = ) Rule 2 n 1 = the number of
ways the first operation can be performed n 2 = the number of ways
the second operation can be performed once the first operation has
been completed. N = n 1 n 2 = the number of ways that two
operations can be performed in sequence if
Slide 252
Rule 3 n 1 = the number of ways the first operation can be
performed n i = the number of ways the i th operation can be
performed once the first (i - 1) operations have been completed. i
= 2, 3, , k N = n 1 n 2 n k = the number of ways the k operations
can be performed in sequence if
Slide 253
Basic counting formulae 1.Orderings 2.Permutations The number
of ways that you can choose k objects from n in a specific order
3.Combinations The number of ways that you can choose k objects
from n (order of selection irrelevant)
Slide 254
Random Variables Numerical Quantities whose values are
determine by the outcome of a random experiment
Slide 255
Random variables are either Discrete Integer valued The set of
possible values for X are integers Continuous The set of possible
values for X are all real numbers Range over a continuum.
Slide 256
The Probability distribution of A random variable A
Mathematical description of the possible values of the random
variable together with the probabilities of those values
Slide 257
The probability distribution of a discrete random variable is
describe by its : probability function p(x). p(x) = the probability
that X takes on the value x. This can be given in either a tabular
form or in the form of an equation. It can also be displayed in a
graph.
Slide 258
Comments: Every probability function must satisfy: 1.The
probability assigned to each value of the random variable must be
between 0 and 1, inclusive: 2.The sum of the probabilities assigned
to all the values of the random variable must equal 1: 3.
Slide 259
Probability Distributions of Continuous Random Variables
Slide 260
Probability Density Function The probability distribution of a
continuous random variable is describe by probability density curve
f(x).
Slide 261
Notes: The Total Area under the probability density curve is 1.
The Area under the probability density curve is from a to b is P[a
< X < b].
Slide 262
Normal Probability Distributions (Bell shaped curve)
Slide 263
Mean, Variance and standard deviation of Random Variables
Numerical descriptors of the distribution of a Random Variable
Slide 264
Mean of a Discrete Random Variable The mean, , of a discrete
random variable x is found by multiplying each possible value of x
by its own probability and then adding all the products together:
Notes: The mean is a weighted average of the values of X. The mean
is the long-run average value of the random variable. The mean is
centre of gravity of the probability distribution of the random
variable
Slide 265
2 Variance of a Discrete Random Variable: Variance, 2, of a
discrete random variable x is found by multiplying each possible
value of the squared deviation from the mean, (x ) 2, by its own
probability and then adding all the products together: Standard
Deviation of a Discrete Random Variable: The positive square root
of the variance:
Slide 266
The Binomial distribution An important discrete
distribution
Slide 267
X is said to have the Binomial distribution with parameters n
and p. 1. X is the number of successes occurring in the n
repetitions of a Success-Failure Experiment. 2.The probability of
success is p. 3. The probability function
Slide 268
Mean,Variance & Standard Deviation of the Binomial
Ditribution The mean, variance and standard deviation of the
binomial distribution can be found by using the following three
formulas:
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Mean of a Continuous Random Variable (uses calculus) The mean,
, of a discrete random variable x Notes: The mean is a weighted
average of the values of X. The mean is the long-run average value
of the random variable. The mean is centre of gravity of the
probability distribution of the random variable
Slide 270
Variance of a Continuous Random Variable Standard Deviation of
a Continuous Random Variable: The positive square root of the
variance:
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The Normal Probability Distribution Points of Inflection
Slide 272
Main characteristics of the Normal Distribution Bell Shaped,
symmetric Points of inflection on the bell shaped curve are at and
+ That is one standard deviation from the mean Area under the bell
shaped curve between and + is approximately 2/3. Area under the
bell shaped curve between 2 and + 2 is approximately 95%.
Slide 273
Normal approximation to the Binomial distribution Using the
Normal distribution to calculate Binomial probabilities
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Normal Approximation to the Binomial distribution X has a
Binomial distribution with parameters n and p Y has a Normal
distribution
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Sampling Theory Determining the distribution of Sample
statistics
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The distribution of the sample mean
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Thus if x 1, x 2, , x n denote n independent random variables
each coming from the same Normal distribution with mean and
standard deviation . Then has Normal distribution with
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The Central Limit Theorem The Central Limit Theorem (C.L.T.)
states that if n is sufficiently large, the sample means of random
samples from any population with mean and finite standard deviation
are approximately normally distributed with mean and standard
deviation. Technical Note: The mean and standard deviation given in
the CLT hold for any sample size; it is only the approximately
normal shape that requires n to be sufficiently large.
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Graphical Illustration of the Central Limit Theorem Original
Population 30 Distribution of x: n = 10 Distribution of x: n = 30
Distribution of x: n = 2 30
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Implications of the Central Limit Theorem The Conclusion that
the sampling distribution of the sample mean is Normal, will to
true if the sample size is large (>30). (even though the
population may be non- normal). When the population can be assumed
to be normal, the sampling distribution of the sample mean is
Normal, will to true for any sample size. Knowing the sampling
distribution of the sample mean allows to answer probability
questions related to the sample mean.
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Sampling Distribution of a Sample Proportion
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Sampling Distribution for Sample Proportions Let p = population
proportion of interest or binomial probability of success. Let is
approximately a normal distribution with = sample proportion or
proportion of successes.
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Sampling distribution of a differences
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If X, Yare independent normal random variables, then : X Y is
normal with Note
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Sampling distribution of a difference in two Sample means
Slide 286
Situation We have two normal populations (1 and 2) Let 1 and 1
denote the mean and standard deviation of population 1. Let 2 and 2
denote the mean and standard deviation of population 2. Let x 1, x
2, x 3, , x n denote a sample from a normal population 1. Let y 1,
y 2, y 3, , y m denote a sample from a normal population 2.
Objective is to compare the two population means
Slide 287
Then
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Sampling distribution of a difference in two Sample
proportions
Slide 289
Situation Suppose we have two Success-Failure experiments Let p
1 = the probability of success for experiment 1. Let p 2 = the
probability of success for experiment 2. Suppose that experiment 1
is repeated n 1 times and experiment 2 is repeated n 2 Let x 1 =
the no. of successes in the n 1 repititions of experiment 1, x 2 =
the no. of successes in the n 2 repititions of experiment 2.
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Then
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The Chi-square ( 2 ) distribution
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The Chi-squared distribution with degrees of freedom Comment:
If z 1, z 2,..., z are independent random variables each having a
standard normal distribution then U = has a chi-squared
distribution with degrees of freedom.
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The Chi-squared distribution with degrees of freedom - degrees
of freedom
Slide 294
2 d.f. 3 d.f. 4 d.f.
Slide 295
Statistics that have the Chi-squared distribution: This
statistic is used to detect independence between two categorical
variables d.f. = (r 1)(c 1)
Slide 296
Let x 1, x 2, , x n denote a sample from the normal
distribution with mean and standard deviation , then has a
chi-square distribution with d.f. = n 1.
