Statistics for business decision making and analysis robert stine and foster chapter 15

15.1 Ranges for ParametersTwo Parameters of Interest  p, the proportion who will return the application for the credit card credit card will carry... 15.1 Ranges for ParametersConfide

Trang 2

Confidence Intervals

Chapter 15

Trang 3

15.1 Ranges for Parameters

Before deciding to offer an affinity credit card

to alumni of a university, the credit

company wants to know how many

customers will accept the offer and how

large a balance they will carry?

estimates

Trang 4

Two Parameters of Interest

 p, the proportion who will return the application for the

credit card

credit card will carry

Trang 5

Summary Statistics (n = 1000)

Trang 6

Confidence Interval for the Proportion

parameter based on a sample

distribution of the statistic.

Trang 7

Confidence Interval for the Proportion

sampling distribution of

 E( ) = p and SE( ) =

p ˆ

p ˆ p ˆ p ( 1 − p ) / n

Trang 8

95% Confidence Interval for p

The sample statistic in 95% of samples lies within 1.96

standard errors of the population parameter.

Trang 9

1.96 standard errors to the left and right of will contain p.

the confidence interval (i.e., is substituted for p)

p ˆ

Trang 10

The 100(1 – α)% confidence interval for p is

For a 95% confidence interval z α/2 = 1.96.

ˆ a ˆ 1 ˆ / to ˆ a ˆ 1 ˆ /

p z − p − p n p z + p − p n

Trang 11

Checklist for Confidence Interval for p

from the relevant population

) ˆ 1

( − p

Trang 12

Credit Card Example

0 1

( 14

Trang 13

accept the offer is between about 12% and 16%

error resulting in a narrower interval

(a more precise estimate of p).

Trang 14

15.2 Confidence Interval for the Mean

Confidence Interval for µ

a confidence interval for µ

se( ) = s /

X

Trang 15

Student’s t-Distribution

standard error.

the shape of the distribution.

the standard normal distribution.

Trang 16

Student’s t-Distribution

Trang 17

Confidence Interval for µ

The 100(1 – α)% confidence interval for µ is

- t α/2, n-1 s / to + t α/2, n-1 s / .

The value of t depends on the level of confidence and n – 1 degrees of freedom.

Trang 18

Checklist for Confidence Interval for µ

from the relevant population

times the squared skewness and 10 times the absolute

value of the kurtosis

Trang 19

Percentiles of the t-Distribution

The t value for 95% confidence and 139 df = 1.98.

Trang 20

se ( ) = $2,833.33 / = $239.46

$1,990.50 ± 1.98($239.46) [$1,516.37 to $2,464.63]

Trang 21

$2,464.63.

results it’s not likely.

Trang 22

15.3 Interpreting Confidence Intervals

Common Confusions: Wrong Interpretations

$2,460.

fall between $1,520 and $2,460.

Trang 23

15.4 Manipulating Confidence Intervals

Obtaining Ranges for Related Quantities

If [L to U] is a 100(1 – α)% confidence interval for µ,

then [c x L to c x U] is a 100 (1 – α)% confidence interval for c x µ and [c + L to c + U] is a

100(1 – α)% confidence interval for c + µ.

Trang 24

Changing the Problem

confidence intervals.

Let Y = profit earned from each customer A

customer who does not accept the card costs the bank $8 Each customer who accepts the

card costs the bank $58 but the bank earns

Trang 25

Creating a New Variable

y i = -8 if offer is not accepted

0.10 (Balance) – 58 if offer is accepted.

Trang 26

Summary Statistics for Profit Earned

For 100,000 offers, the 95% confidence interval for total

Trang 27

15.5 Margin of Error

 Margin of error is affected by (1) level of confidence, (2)

variation in the data and (3) number of observations.

Trang 28

Determining Sample Size

size using

n = 4σ 2 / (Margin of Error) 2

have to choose n before collecting data)

Trang 29

Example.

A nutritionist wants to know the average calorie intake for

female customers to within ± 50 calories with 95%

confidence A pilot study gives an estimate of 430

calories for σ Find n.

n = 4(430 2 ) / 50 2 = 295.8 or 300 customers

Trang 30

Determining Sample Size

½.

Trang 31

Sample Sizes for Various Margins of Error

(95% coverage)

Trang 32

4M Example 15.1: PROPERTY TAXES

Motivation

A mayor is considering a tax on business that is

proportional to the amount spent to lease

property in his city How much revenue would a 1% tax generate?

Trang 33

Method

Need a confidence interval for µ (average cost of

a lease) to obtain a confidence interval for the

amount raised by the tax Check conditions

(SRS and sample size) before proceeding.

Trang 34

Mechanics: Statistics on Lease Costs

Trang 35

Message

We are 95% confident that the average cost of a lease is between $410,000 and $550,000 The 95% confidence interval for tax raised per business is therefore [$4,100

to $5,500] Since the number of businesses leased in

the city is 4,500, we are 95% confident that the amount raised will be $18,450,000 to $24,750,000.

Trang 36

4M Example 15.2: A POLITICAL POLL

Motivation

The mayor is seeking reelection Only 40% of

registered voters think he is doing a good job (n

= 400) What does this indicate about the

attitudes of all voters in the city?

Trang 37

Method

Construct a 95% confidence interval for the

population proportion, p Check SRS and

sample size conditions.

Trang 38

4 0 (

Trang 39

Message

The mayor can be 95% certain that 35% to

45% of registered voters think he is doing a good job

Trang 40

Best Practices

results.

 Use full precision for intermediate calculations

Trang 41

samples.

confidence interval.

Định dạng
Số trang	41
Dung lượng	694 KB