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Confidence Intervals

Chapter 6

6.1

- Confidence Intervals for the Mean (Large

Samples)

Point Estimate for Population µ

A point estimate is a single value estimate for a

population parameter. The most unbiased point

estimate of the population mean, ?, is the sample

mean, ?.

Example A random sample of 32 textbook prices

(rounded to the nearest dollar) is taken from a

local college bookstore. Find a point estimate

for the population mean, ?.

34 34 38 45 45 45 45 54

56 65 65 66 67 67 68 74

79 86 87 87 87 88 90 90

94 95 96 98 98 101 110 121

The point estimate for the population mean of

textbooks in the bookstore is 74.22.

Interval Estimate

An interval estimate is an interval, or range of

values, used to estimate a population parameter.

How confident do we want to be that the interval

estimate contains the population mean, µ?

Level of Confidence

The level of confidence c is the probability that

the interval estimate contains the population

parameter.

c is the area beneath the normal curve between

the critical values.

Use the Standard Normal Table to find the

corresponding z-scores.

The remaining area in the tails is 1 c .

Common Levels of Confidence

If the level of confidence is 90, this means

that we are 90 confident that the interval

contains the population mean, µ.

zc 1.645

?zc ? 1.645

The corresponding z-scores are 1.645.

Common Levels of Confidence

If the level of confidence is 95, this means

that we are 95 confident that the interval

contains the population mean, µ.

zc 1.96

?zc ? 1.96

The corresponding z-scores are 1.96.

Common Levels of Confidence

If the level of confidence is 99, this means

that we are 99 confident that the interval

contains the population mean, µ.

zc 2.575

?zc ? 2.575

The corresponding z-scores are 2.575.

Margin of Error

The difference between the point estimate and the

actual population parameter value is called the

sampling error.

When µ is estimated, the sampling error is the

difference µ ?. Since µ is usually unknown,

the maximum value for the error can be calculated

using the level of confidence.

Given a level of confidence, the margin of error

(sometimes called the maximum error of estimate

or error tolerance) E is the greatest possible

distance between the point estimate and the value

of the parameter it is estimating.

Margin of Error

Example A random sample of 32 textbook prices is

taken from a local college bookstore. The mean

of the sample is ? 74.22, and the sample

standard deviation is s 23.44. Use a 95

confidence level and find the margin of error for

the mean price of all textbooks in the bookstore.

We are 95 confident that the margin of error for

the population mean (all the textbooks in the

bookstore) is about 8.12.

Confidence Intervals for µ

A c-confidence interval for the population mean µ

is ? ? E lt µ lt ? E. The probability that

the confidence interval contains µ is c.

Example A random sample of 32 textbook prices is

taken from a local college bookstore. The mean

of the sample is ? 74.22, the sample standard

deviation is s 23.44, and the margin of error

is E 8.12. Construct a 95 confidence interval

for the mean price of all textbooks in the

bookstore.

Continued.

Confidence Intervals for µ

Example continued Construct a 95 confidence

interval for the mean price of all textbooks in

the bookstore.

? 74.22

s 23.44

E 8.12

? ? E 74.22 8.12

? E 74.22 8.12

66.1

82.34

With 95 confidence we can say that the cost for

all textbooks in the bookstore is between 66.10

and 82.34.

Finding Confidence Intervals for µ

Finding a Confidence Interval for a Population

Mean (n ? 30 or s known with a normally

distributed population)

In Words In Symbols

- Find the sample statistics n and ?.
- Specify ?, if known. Otherwise, if n ? 30, find

the sample standard deviation s and use it as an

estimate for ?. - Find the critical value zc that corresponds to

the given level of confidence. - Find the margin of error E.
- Find the left and right endpoints and form the

confidence interval.

Use the Standard Normal Table.

Left endpoint ??E Right endpoint ? E Interval

??E lt µ lt ? E

Confidence Intervals for µ (? Known)

Example A random sample of 25 students had a

grade point average with a mean of 2.86. Past

studies have shown that the standard deviation is

0.15 and the population is normally distributed.

Construct a 90 confidence interval for the

population mean grade point average.

n 25

? 2.86

? 0.15

zc 1.645

2.81 lt µ lt 2.91

? E 2.86 0.05

With 90 confidence we can say that the mean

grade point average for all students in the

population is between 2.81 and 2.91.

Sample Size

Given a c-confidence level and a maximum error of

estimate, E, the minimum sample size n, needed to

estimate ?, the population mean, is If ? is

unknown, you can estimate it using s provided you

have a preliminary sample with at least 30

members.

Example You want to estimate the mean price of

all the textbooks in the college bookstore. How

many books must be included in your sample if you

want to be 99 confident that the sample mean is

within 5 of the population mean?

Continued.

Sample Size

Example continued You want to estimate the mean

price of all the textbooks in the college

bookstore. How many books must be included in

your sample if you want to be 99 confident that

the sample mean is within 5 of the population

mean?

? 74.22

? ? s 23.44

zc 2.575

(Always round up.)

You should include at least 146 books in your

sample.