Stats Aabr(Ch17)

8/6/2019 Stats Aabr(Ch17)

1/44

What does Statistics Mean? Descriptive statistics

Number of people

Trends in employment

Data

Inferential statistics

Make an inference about a population from a

sample


2/44

Population Parameter VersusSample Statistics


3/44

Population Parameter Variables in a population

Measured characteristics of a population

Greek lower-case letters as notation


4/44

Sample Statistics Variables in a sample

Measures computed from data

English letters for notation


5/44

Making Data Usable Frequency distributions

Proportions

Central tendency

Mean

Median

Mode

Measures of dispersion


6/44

Frequency (number of

people making depositsAmount in each range)

less than $3,000 499$3,000 - $4,999 530$5,000 - $9,999 562$10,000 - $14,999 718$15,000 or more 811

3,120

Frequency Distribution of

Deposits


7/44

Amount Percent

less than $3,000 16$3,000 - $4,999 17$5,000 - $9,999 18$10,000 - $14,999 23

$15,000 or more 26100

Percentage Distribution of

Amounts of Deposits


8/44

Amount Probability

less than $3,000 .16$3,000 - $4,999 .17$5,000 - $9,999 .18$10,000 - $14,999 .23

$15,000 or more .261.00

Probability Distribution of

Amounts of Deposits


9/44

Measures of Central Tendency Mean - arithmetic average

, Population; , sample

Median - midpoint of the distribution

Mode - the value that occurs most often

X


10/44

Population Mean

NXi7!Q


11/44

n

XXi

7!

Sample Mean


12/44

Product A Product B

196 150198 160

199 176199 181200 192200 200200 201201 202201 213201 224202 240

202 261

Sales forProducts A and B,

Both Average 200


13/44


14/44

The Range

as a Measure of Spread

The range is the distancebetween the smallest

and the largest value in the set.

Range = largest value smallest value


15/44

Mean Squared Deviation

n

XXi 2

)(


16/44

The Variance

2

2

S

Sample

Population

W


17/44

Variance

1

)22

7!

n

XXS


18/44

Variance The variance is given in squared units

The standard deviation is the square root of

variance:


19/44

Sample Standard Deviation

1

2

7!

n

XXiS


20/44

Class Data Histogramless than $3,000 499$3,000 - $4,999 530$5,000 - $9,999 562$10,000 - $14,999 718$15,000 or more 811

3,120

0

100

200

300

400

500

600

700

800

900

less than$3,000

$3,000 -$4,999

$5,000 -$9,999

10,000 -$14,999

$15,000 ormore


21/44

Mean

n

X Xm.f7!


22/44

Standard Deviation

1

/..2

2

7 !

n

nXfXfSmm


23/44

Other Measures

Harmonic Mean

Geometric Mean

Quadratic Mean

!

x

nHM

/1

n nGM xxxx ...........3.2.1!

n

xQM

!2


24/44

The Normal Distribution Normal curve

Bell shaped

Almost all of its values are within plus or

minus 3 standard deviations

I.Q. is an example

Skewness: Pearsons index of skewness (-1to+1)

s

medianxPI

!

3


25/44

Normal Distribution

2

2

2W

Q

x


26/44

Standardized Normal Distribution

Symetrical about its mean

Mean identifies highest point

Infinite number of cases - a continuous

distribution

Area under curve has a probability density = 1.0

Mean of zero, standard deviation of 1


27/44

The Standardized Normal is the

Distribution of Z

z +z


28/44

Linear Transformation of Any Normal

Variable Into a Standardized Normal Variable

-2 -1 0 1 2

Sometimes the

scale is stretchedSometimes thescale is shrunk

QQ

WW

X

W

Q!x

z

Also called Z scores


29/44

Parameter Estimates Point estimates

Confidence interval estimates


30/44

Population distribution

Sample distribution

Sampling distribution


31/44

Distribution n tandardDeviationPopulation Q W

ample X

amplingX

Q

XS


32/44

Central Limit Theorem

nS

x

W!

n

S

S x!

Estimating the Standard Error

of the Mean

Standard Error of the Mean

QQ !X


33/44

n

SZX cls!Q


34/44

Random Sampling Error and

Sample Size are Related


35/44

Sample Size Variance (standard

deviation)

Magnitude of error Confidence level


36/44


37/44

Sample Size Formula - Example

Suppose a survey researcher, studying

expenditures on juices, wishes to have a95 percent confident level (Z) and a

range of error (E) of less than $2.00. The

estimate of the standard deviation is$29.00.


38/44

2

Ezsn

!

2

00.200.2996.1

!

2

00.284.56

!

242.28! 808!

Sample Size Formula - Example


39/44

Finite Population Correction Factor

1

N

nN

Multiply the sample size calculated by

this factor


40/44

Proportions

npq

np

!

!

W

Q

Standard Error of the proportion. Sp

Normal Approximation to Binomial Distribution

If np5, nq5 where q=1-p


41/44

n

pqzpp

n

npqz

n

np

n

x

npqznpx

zx

s!

s!

s!

s!

2

WQ

Confidence Interval for a

Proportion


42/44

2

2

E

pqZn !

Sample Size for a Proportion

Where:n = Number of items in samples

Z2 = The square of the confidence interval

in standard error units.

p = Estimated proportion of success

q = (1-p) or estimated the proportion of failures

E2 = The square of the maximum allowance for error

between the true proportion and sample proportion


43/44

Calculating Sample Size

at the 95% Confidence Level

753!001225.

922.!

001225

)24)(.8416.3(!

)035( .

)4)(.6(.)961.(

n4.q

6.p

2

2

!

!

!


44/44

Stats Aabr(Ch17)

Documents

Transcript of Stats Aabr(Ch17)