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Slide 1

Chapter 8Descriptive Statistics Distributions Inferential Statistics

• Estimating a population mean

• Confidence Intervals

• Margin of error• Student’s t-distribution

Slide 2

Chips Ahoy! 1000 chips challenge

At least 1000 chocolate chips per 18 ounce bag?

Chip – “distinct piece of chocolate baked into

cookie dough”.

Stats class at US Air Force Academy

275 bags from all over country, 42 randomly

selected.

Cookies dissolved in water, chips counted.

Results…………

Slide 3

Results of 42 bags. . . . .

1200 1219 1103 1213 1258 1325 1295

1247 1098 1185 1087 1377 1363 1121

1279 1269 1199 1244 1294 1356 1137

1545 1135 1143 1215 1402 1419 1166

1132 1514 1270 1345 1214 1154 1307

1293 1546 1228 1239 1440 1219 1191

Estimate the mean number of chips per bag for all Chips Ahoy! Cookies?

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Slide 4

Point Estimate

Question: 1. Mean age of people in labour force2. Mean starting salary of Arts graduates.

Answer: Small population, take census, large population then we need to sample. Can achieve pretty accurate, cost effective results.

Mean

Slide 5

Confidence Interval

Sample mean will (probably) not exactly equal the population mean, some sampling error is to be expected.Need information on the accuracy of the point estimate……confidence level.

Confidence interval – abbreviate to CI

Slide 6

Confidence Interval – new mobile home prices

Random sample of 36 prices

= 42.2 σ = 7.2

Remember

is normally distributed

95.44% of sample lies within 2 s.d.

-2.4 = 39.8 + 2.4 = 44.6

2.136

2.7

nx

x

x x

39.8 44.6

95.44% confident that μ lies in here

x

x

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Slide 7

Simulation for new mobile home prices

Slide 8

Simulation for new mobile home prices

Green dot signifies sample mean

In 19/20 (95%) of the samples the population

mean lies inside the sample CI

If we simulate more than 40, say 1000 times

expect 95.44 to be in CI!

Assume normally distributed?

is approx. normally distributed by CLT,

then CI of approx norm distributed also.xx

Slide 9

Obtaining CI’s for a population mean when σ is known.

95.44% of all samples have means within 2 s.d. of μ

Generally

100(1-α)% of all

samples have

means within Zα/2

s.d’s of μ.

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Slide 10

More Formally

Slide 11

When to use the Z-interval procedure

For small samples (<15) the z-interval procedure should be used only when the variable under consideration is normally distributed (or close).

For samples of moderate size (15<30) the z-interval procedure can be used unless the data contains outliers or the variable under consideration is far frombeing normally distributed.

For large samples (>30) the z-interval procedure can be used essentially without restriction. However if outliers are present and their removal is not justified, the effect of the outliers on the CI should be examined, if significant – resample.

Slide 12

Fundamental Principle of Data Analysis

Before performing a statistical inference

procedure, examine the sample data. If any of

the conditions required for using the procedure

appear to be violated, do not apply the

procedure. Instead use a different, more

appropriate procedure, or, of you are unaware

of one consult a statistician.

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Slide 13

Age of the labour force

Sample of 50 people’s ages, σ = 12.1 years, x bar = 36.4

Find a 95% CI?

Table II - 95% CI so (1- α) = 0.05, or 0.025 in each tail – we get a value of 1.96.

36.4 ± 1.96 x 12.1/√50

33.0 to 39.9

We can be 95% confident that the mean age, μ, of all the people in the labour force is somewhere between 33.0 and 39.8 years.

nZx

.2/

Slide 14

Confidence and precision

For a fixed size, decreasing the confidence level increases the precision, and vice-versa.

Slide 15

Sample Size for estimating μ

The sample size required for a (1- α ) level

confidence interval for μ with a specified margin of

error, E, is given by the formula

Rounded up to the nearest whole number

e.g. 95% CI for μ within 0.5 year of x. (σ = 12.1 years)

2

2/ .

E

Zn

79.22495.0

1.1296.12

n

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Slide 16

CI for population mean when σ is unknown

n

xz

ns

xt

Students t-distribution

Standardised version of x

Different dist. for each sample size, dist. identified by name ‘degrees of freedom’, df = n-1

Slide 17

t-dist & t-curves

Thicker tails – converge on normal dist. in the limit

Slide 18

Properties

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Slide 19

Using the t-table

Slide 20

σ unknown

Pick pocket crimes

447 207 627 430 883

313 844 253 397 214

217 768 1064 26 587

833 277 805 653 549

649 554 570 223 443

Slide 21

Value ($) lost for sample of 25 pick pocket crimes

N= 25df = n-1 = 25-2=24X bar = 513.32S = 262.23

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Slide 22

Value lost for 25 pick pocket crimes

Normality?Outliers?

n

stx .2/

25

23.262.064.232.513

95% CI for mean value lost

405.07 to 621.57

Slide 23

Treatment of Outliers

Slide 24

Chips Ahoy! 1000 chips challenge

Variable Obs Mean Std. Err. [95% Conf. Interval]

Chips 42 1261.571 18.14284 1224.931 1298.212

n= 42; d.f.=41; tα/2 = t0.025 = 2.201; x = 1261.6; s = 117

n

stx .2/

42

6.117.021.26.1261

1224.9 to 1298.2

0.00

0.25

0.50

0.75

1.00

Norm

al F

[(va

r1-m

)/s]

0.00 0.25 0.50 0.75 1.00Chips

Chips Ahoy!

0

.001

.002

.003

.004

.005

Density

1100 1200 1300 1400 1500chips

Chips Ahoy!

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Slide 25

Department of FinanceMonthly Economic Bulletin October 2014

Monthly bulletin best source of Dept Finance general

economic information.

It contains many of the variables that we have looked

at in other publications.

This publication also contains some detailed

budgetary and economic statistics.

Slide 26

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