Download Confidence Intervals

Chapter 15
Confidence Intervals
Copyright © 2011 Pearson Education, Inc.
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 p̂.

E( p̂) = p and SE( p̂) =
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 p̂will contain p.

The estimated standard error (se) is used in
constructing the confidence interval (i.e., p̂ is
substituted for 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 np̂
and n(1  pˆ ) are larger than 10.
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15.1 Ranges for Parameters
Credit Card Example

The estimated standard error is
se( p̂ ) =
•
0.14(1  0.14)
= 0.01097
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 X is used.
se( X ) = s / 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 /
n
to
x
+ tα/2, n-1 s / 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 ( X ) = $2,833.33 / 140 = $239.46

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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15.4 Manipulating Confidence Intervals
Summary Statistics for Profit Earned
For 100,000 offers, the 95% confidence interval for
total profit is from $557,000 to $2,017,000.
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15.5 Margin of Error

A precise confidence interval has a small margin
of error.

Margin of error is affected by (1) level of
confidence, (2) variation in the data and (3)
number of observations.

It is used in determining sample size.
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15.5 Margin of Error
Determining Sample Size

For a study about µ with 95% coverage, find the
sample size using
n = 4σ2 / (Margin of Error)2

Obtain an estimate for σ2 using a pilot sample
(since we have to choose n before collecting
data)
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15.5 Margin of Error
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(4302) / 502 = 295.8 or 300 customers
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15.5 Margin of Error
Determining Sample Size

For a study about p, no need for a pilot sample.

Use p = 0.5 which results in largest possible
value for σ of ½.
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15.5 Margin of Error
Sample Sizes for Various Margins of Error
(95% coverage)
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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?
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4M Example 15.1: PROPERTY TAXES
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.
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4M Example 15.1: PROPERTY TAXES
Mechanics: Statistics on Lease Costs
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4M Example 15.1: PROPERTY TAXES
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.
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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?
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4M Example 15.2: A POLITICAL POLL
Method
Construct a 95% confidence interval for the
population proportion, p. Check SRS and
sample size conditions.
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4M Example 15.2: A POLITICAL POLL
Mechanics
Find the estimated standard error, se ( p̂ )
se ( p̂ ) =
(0.4)(0.6)
400
= 0.0245
The 95% confidence interval is
0.40 ± 1.96 (0.0245) = [0.352 to 0.448]
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4M Example 15.2: A POLITICAL POLL
Message
The mayor can be 95% certain that 35% to
45% of registered voters think he is doing a
good job.
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Best Practices

Be sure that the data are an SRS from the
population.

Stick to 95% confidence intervals.

Round the endpoints of intervals when presenting
the results.

Use full precision for intermediate calculations.
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Pitfalls

Do not claim that a 95% confidence interval holds
µ.

Do not use a confidence interval to describe other
samples.

Do not manipulate the sampling to obtain a
particular confidence interval.
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