handout seven

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Introduction to Probability and Statistics

Handout #7

Instructor: Lingzhou Xue

TA: Daniel Eck

The pdf file for this class is available on the class web page.

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Chapter 8

Fundamental Sampling Distributions and

Data Descriptions

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Sample Mean X and Sample Variance S2.

Histogram and Box Plot.

Central Limit Theorem (CLT).

2, t, and F Distributions.

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Example 1: Sample Distribution

The sample distribution is the distribution resulting from the

collection of actual data. A major characteristic of a sample is

that it contains a finite (countable) number of scores, the num-

ber of scores represented by the letter n. For example, supposethat the following data were collected:

15 14 15 18 15 20 15 16 17 14 17 13 11 14 18 12 17 12 21 8

14 17 14 12 13 15 15 16 17 14 16 13 14 15 18 16 16 17 14 15

16 15 17 12 14 14 13 13 13 14

These numbers constitute a sample distribution.

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Histogram

x

De

nsity

8 10 12 14 16 18 20

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

Sample Distribution.


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In addition to the frequency distribution, the sample distribu-

tion can be described with numbers, called statistics. Examples

of statistics are the mean, median, mode, standard deviation,

range, and correlation coefficient, among others.

If a different sample was taken, different scores would result.

However, there would also be some consistency in that while the

statistics would not be exactly the same, they would be simi-

lar. To achieve order in this chaos, statisticians have developed

probability models.


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Histogram

x

Density

8 10 12 14 16 18 20

0.0

0

0.0

5

0.1

0

0.1

5

0.2

0

Histogram

x

Density

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0.0

0

0.0

5

0.1

0

0

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Histogram

x

Density

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0.0

0

0.0

5

0.1

0

0.1

5

Histogram

x

Density

10 12 14 16 18 20

0.0

0

0.0

5

0.1

0

0.1

5

0.2

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Random Sampling

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Population

A population consists of the totality of the observations with

which we are concerned.

It is the entire group we are interested in, which we wish to

describe or draw conclusions about.

Sample

A sample is subset of a population.

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2008 Presidential Race from CNN.

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Example 2

If you wanted to find out the percentage of students at UMNwho enjoy reading Time. If we randomly select 20% of the pop-

ulation, this selection would be the sample in this experiment.

Therefore, the population would be all of the students who

attend UMN.

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A simple random sample of size n consists ofn individuals fromthe population chosen in such a way that every set ofn individuals

has an equal chance to be the sample actually selected.

Random Sample

Let X1, X2, . . . , X n be n independent random variables, each

having thesameprobability distributionf(x). DefineX1, X2, . . . , X n

to be a random sample of size n from the population f(x) and

write its joint probability distribution as

g(x1, x2, . . . , xn) =f(x1)f(x2) f(xn).

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Some Important Statistics

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Statistic

A statistic is a function of random variables that does not de-

pend upon any unknown parameter.

Sample Mean & Sample Variance

If X1, X2, . . . , X n represent a random sample of size n, then the

sample mean is defined by the statistic

X=1

n

ni=1

Xi

and the sample variance is defined by the statistic

S2 = 1

n 1n

i=1

(Xi X)2.

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Example 3

A comparison of coffee prices at 4 randomly selected grocery

stores in San Diego showed increases from the previous month

of 12, 15, 17, and 20 cents for 1-pound bag. Find the variance

of this random sample of price increases.Solution:

x=1

n

4i=1

xi = 16 cents.

s2 = 14 1

4

i=1

(xi x)2 =343

.

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Theorem

If S2 is the variance of a random sample of size n, we may write

S2 = 1

n(n

1)

n

n

i=1

X2i

n

i=1

Xi

2

.

Proof:

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Example 4

Find the sample mean and variance of the data 3, 4, 5, 6, 6, and

7, representing the number of trout caught by a random sample

of 6 fishermen on June 19, 1996, at Lake Muskoka.

Solution:

6i=1

x2i = 171,6

i=1

xi= 31, n= 6.

Hence,

s2 = 15 6[6 171 312] =136 .

Thus the sample standard deviation s=

13/6 = 1.47.

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Example 5

The numbers of incorrect answers on a true-false competency

test for a random sample of 15 students were recorded as follows:

2, 1, 3, 0, 1, 3, 6, 0, 3, 4. Find

the sample mean;

the sample variance.

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Mode

The mode in a list of numbers refers to the list of numbers thatoccur most frequently. A trick to remember this one is to

remember that mode starts with the same first two letters that

most does. Most frequently - Mode. Youll never forget that

one!

Median

The median is the middle value in your list. When the totals of

the list are odd, the median is the middle entry in the list after

sorting the list into increasing order. When the totals of the

list are even, the median is equal to the sum of the two middle

(after sorting the list into increasing order) numbers divided by

two. Thus, remember to line up your values, the middle number

is the median! Be sure to remember the odd and even rule.

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Data Displays and Graphical Methods

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Box Plot

A box plot (also known as a box-and-whisker diagram or plot orcandlestick chart) is a convenient way of graphically depicting the

five-number summary, which consists of 25% percentile (lower

quartile or first quartile (Q1)), median, 75% percentile (upper

quartile or third quartile (Q3)) and adjust values; in addition,

the boxplot indicates which observations, if any, are considered

unusual, or outliers.

Outlier

Outliers are observations that are considered to be unusually far

from the bulk of the data. Technically, one may view an outlieras being an observation that represents a rare event. If the

distance from the box exceeds 1.5 times the interquartile range,

Q3 Q1 (in either direction), the observation may be labeled anoutlier.

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Box Plot

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Example 6

The following set of numbers are the amount of marbles fifteen

different boys own (they are arranged from least to greatest).

18 27 34 52 54 59 61 68 78 82 85 87 91 93 100.

Find the median.

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

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Box-and-Whisker Plot for Example 6.

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Sampling Distribution of Means

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Sampling Distribution

The probability distribution of a statistic is called a sampling

distribution.

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If we are sampling from a population with unknown distribu-

tion, either finite or infinite, the sampling distribution of X will

be approximately normal with mean and variance 2/n pro-

vided that the sample size is large (n >30).

Central Limit Theorem

If X is the mean of a random sample of size n taken from a

population with mean and finite variance 2, then the limiting

form of the distribution of

Z=

X /

n

as n , is the standard normal distribution N(0, 1).

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Example 7

Let Xbe the sample mean of a random sample of size 100 drawn

from an exponential distribution with its graph given by

f(x) =1

4ex/4, x >0

Exponential p.d.f with = 4.

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Decide which of the graphs labeled (a)-(d) would most closely

resemble the sampling distribution of the sample mean X. Ex-

plain briefly your reasoning.


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Example 8

An electrical firm manufactures light bulbs that have a length of

life that is approximately normally distributed, with mean equal

to 800 hours and a standard deviation of 40 hours. Find the

probability that a random sample of 16 bulbs will have an average

life of less than 775 hours?Solution:

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Solution:

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Sampling Distribution: Difference Between Two Averages

If independent samples of size n1 and n2 are drawn at random

from two populations, discrete or continuous, with means 1and 2, and variances

21 and

22, respectively, then the sampling

distribution of the differences of means, X1X2, is approximatelynormally distributed with mean and variance given by

X1X2 =1 2, and 2X1X2 =

21n1

+22n2

.

Hence

Z=

(X1

X2)

X1

X2

21n1

+22n2

is approximately a standard normal variable.

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Example 10

The television picture tubes of manufacture A have a mean life-time of 6.5 years and a standard deviation 0.9 year, while those

of manufacturer B have a mean lifetime of 6.0 years and a stan-

dard deviation of 0.8 year. What is the probability that a random

sample of 36 tubes from manufacturer A will have a mean life-

time that is at least 1 year more than the mean lifetime of asample of 49 tubes from manufacturer B?

Solution:


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Example 11

The mean score for freshmen on an aptitude test at a certain

college is 540, with a standard deviation of 50. What is the

probability that two groups of students selected at random, con-

sisting of 64 and 100 students, respectively, will differ in theirmean scores by

1. more than 10 points?

2. an amount between 5 and 10 points?

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Solution:

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Sampling Distribution of S2

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Sampling Distribution of S2

If S2 is the variance of a random sample of size n taken from a

normal population having the variance 2, then the statistic

2 =(n 1)S2

2 =

ni=1

(Xi X)22

has a chi-squared distribution with =n

1 degrees of free-

dom.

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Example 12

Find the probability that a random sample of 21 observations,from a normal population with variance 2 = 5, will have a

variance s2

1. greater than 2.065;

2. between 2.065 and 3.6445.

Solution:

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tDistribution

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tDistribution

Let Z be a standard normal random variable and V a chi-

squared random variable with degrees of freedom. If Z and V

are independent, then the distribution of the random variable

T, where

T = ZV /

is given by the density function

h(t) =[(+ 1)/2]

(/2)

(1 +t2

)(+1)/2,

< t 2(v)). That is, 2

represent the 2-value above which we find an area equal to .

Note for t(v)

In the textbook, we use =P(T > t(v)). That is, t represent

the t-value above which we find an area equal to .

Note for f(v1,v2)

In the textbook, we have = P(F > f(v1, v2)). That is, f

represent the f-value above which we find an area equal to .

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Example 13b

Consider T = Xs/n for a random sample of size n= 8.

Calculate P(T < 2.517) and P(2.998< T


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F-Distribution

Let U and V be two independent random variables having chi-

squared distribution with 1 and 2 degrees of freedom, respec-

tively. Then the distribution of the random variable F = U/1

V /2is

given by the density

f(x) =

[(1+2)/2](1/2)1/2

(1/2)(2/2)x(1/2)1

(1+1x/2)(1+2)/2

, x >0;

0, x 0.

This is known as the Fdistribution with 1 and 2 degrees offreedom.

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0 1 2 3 4 5

0.

0

0.

5

1.

0

1.

5

2.

0

F Distributions

x

f(x)

v1=100, v2=100

v1=6, v2=10

v1=10, v2=30

The F-Distribution curves.

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Theorem

IfS21 and S22 are the variances of independent random samples

of size n1 and n2 taken from normal populations with variances

21 and 22, respectively, then

F =S21/

21

S22/22

=22S

21

21S22

has an F-distribution with 1 =n1 1 and 2 = n2 1 degreesof freedom.

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What Is the F-Distribution Used for?

The F-distribution is used in two-sample situations to draw in-

ferences about the population variances. However, the F-

distribution is applied to many other types of problems in which

the sample variances are involved. In fact, the F-distribution is

called the variance ratio distribution.

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Example 15

IfS

2

1 andS

2

2 represent the variances of independent random sam-ples of size n1 = 31 and n2 = 25, taken from normal populationswith variances 21 = 20 and

22 = 10, respectively, find:

1. P

S22 3.88

.

Solution:

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Solution:

1.

P(S22 36.4152

= 1 0.05= 0.95

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2.

P

S21S22

>3.88

=P

S21S22

22

21>3.88

22

21

=P

S21S22

2221

>3.88 12

=P

F(30,24) >1.94

= 0.05

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