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

Chapter 15

Confidence Intervals

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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?



Use confidence intervals to answer such questions
They convey information about the precision of the
estimates
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15.1 Ranges for Parameters
Two Parameters of Interest


p, the proportion who will return the application for the
credit card



µ, the average monthly balance that those who accept the
credit card will carry

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15.1 Ranges for Parameters
Summary Statistics (n = 1000)

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15.1 Ranges for Parameters
Confidence Interval for the Proportion


A confidence interval is a range of plausible values for a
parameter based on a sample.



Constructing confidence intervals relies on the sampling
distribution of the statistic.

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15.1 Ranges for Parameters
Confidence Interval for the Proportion


The Central Limit Theorem implies a normal model for the
sampling distribution of .
ˆ
p



E( ) = p and SE(

ˆ
p

)=

ˆ
p

p (1 − p ) / n

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15.1 Ranges for Parameters

95% Confidence Interval for p

The sample statistic in 95% of samples lies within 1.96
standard errors of the population parameter.

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15.1 Ranges for Parameters
95% Confidence Interval for p


For any of these samples, the interval formed by reaching
1.96 standard errors to the left and right of will contain
p. p
ˆ



The estimated standard error (se) is used in constructing
the confidence interval (i.e., is substituted for p).

ˆ
p

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15.1 Ranges for Parameters
95% Confidence Interval for p
The 100(1 – α)% confidence interval for p is

pˆ − za /2 pˆ ( 1 − pˆ ) / n to pˆ + za /2 pˆ ( 1 − pˆ ) / n
For a 95% confidence interval zα/2 = 1.96.

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15.1 Ranges for Parameters
Checklist for Confidence Interval for p


SRS condition. The sample is a simple random sample
from the relevant population.



Sample size condition (for proportion). Both n
and n
are larger than 10.

(1 − pˆ )

ˆ
p

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15.1 Ranges for Parameters
Credit Card Example


The estimated standard error is
se( ˆ) =
p

•

= 0.01097
0.14(1 − 0.14
)
1000

The 95% confidence interval is

0.14 ± 1.96(0.01097) = [0.1185 to 0.1615]

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15.1 Ranges for Parameters
Credit Card Example


With 95% confidence, the population proportion that will
accept the offer is between about 12% and 16%.



A larger sample size will reduce the estimated standard
error resulting in a narrower interval
(a more precise estimate of p).

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15.2 Confidence Interval for the Mean
Confidence Interval for µ


A similar procedure as that used for p is used to construct
a confidence interval for µ.



The estimated standard error for
se(

)=s/

X

is used.

X

n

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15.2 Confidence Interval for the Mean
Student’s t-Distribution


Used because we substitute s for σ in the estimated
standard error.



A parameter called degrees of freedom (df = n-1) controls
the shape of the distribution.



As n increases, the t-distribution more closely resembles
the standard normal distribution.
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15.2 Confidence Interval for the Mean

Student’s t-Distribution



Standard normal (black) and Student’s t-distributions (red)
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15.2 Confidence Interval for the Mean
Confidence Interval for µ
The 100(1 – α)% confidence interval for µ is

x

- tα/2, n-1 s /

to

n

+ t α/2, n-1 s /

x

.

n

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

degrees of freedom.

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15.2 Confidence Interval for the Mean
Checklist for Confidence Interval for µ


SRS condition. The sample is a simple random sample
from the relevant population.



Sample size condition. The sample size is larger than 10
times the squared skewness and 10 times the absolute
value of the kurtosis.

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15.2 Confidence Interval for the Mean
Percentiles of the t-Distribution

The t value for 95% confidence and 139 df = 1.98.
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15.2 Confidence Interval for the Mean
Credit Card Example


The estimated standard error is
se (



) = $2,833.33 /

X

= $239.46

140

The 95% confidence interval is
$1,990.50 ± 1.98($239.46)
[$1,516.37 to $2,464.63]

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15.2 Confidence Interval for the Mean
Credit Card Example
•

We are 95% confident that µ lies between $1,516.37 and
$2,464.63.

•

Might µ be $1,250? It could be, but based on the sample
results it’s not likely.

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15.3 Interpreting Confidence Intervals
Common Confusions: Wrong Interpretations


95% of all customers keep a balance of $1,520 to
$2,460.



The mean balance of 95% of samples of 140 accounts will
fall between $1,520 and $2,460.



The mean balance is between $1,520 and $2,460.

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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.

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15.4 Manipulating Confidence Intervals
Changing the Problem


Creating a new variable is preferable to combining
confidence intervals.



Consider the following:

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
10% on the revolving credit card balance.
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15.4 Manipulating Confidence Intervals
Creating a New Variable


Therefore, profit ($) earned from a customer is
yi =

-8 if offer is not accepted
0.10 (Balance) – 58 if offer is accepted.

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