Math 227 Elementary Statistics

Bluman 5th edition

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CHAPTER 7

Confidence Intervals

and Sample Size

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Objectives

• Find the confidence interval for the mean

when is known or n > 30.

• Determine the minimum sample size for

finding a confidence interval for the mean.

• Find the confidence interval for the mean

when is unknown and n < 30.

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Objectives (cont.)

• Find the confidence interval for a proportion.

• Determine the minimum sample size for finding a confidence interval for a proportion.

• Find a confidence interval for a variance and a standard deviation.

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Introduction

• Estimation is the process of estimating the

value of a parameter from information

obtained from a sample.

• The procedures for estimating the

population mean, estimating the population

proportion, and estimating a sample size

will be explained.

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7.1 Confidence Intervals for the Mean

When Known and Sample Size

Three Properties of a Good Estimator • The estimator should be an unbiased estimator. That is,

the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.

• The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.

• The estimator should be a relatively efficient estimator; that is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

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Point and Interval Estimates

• A point estimate is a specific numerical value

of a parameter. The best point estimate of the

population mean m is the sample mean .

• An interval estimate of a parameter is an

interval or a range of values used to estimate

the parameter. This estimate may or may not

contain the value of the parameter being

estimated.

X

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I. Confidence Intervals

• Even though the best point estimate of the

population mean is the sample mean, for the

most part, the sample mean will be different

from the population mean m due to sampling

error. For this reason, statisticians prefer an

interval estimate. The confidence level of an

interval estimate of a population mean m is the

probability that the interval estimate will contain m.

X

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Confidence Level and

Confidence Interval

• The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.

• A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.

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Formula

• Formula for the Confidence Interval of the Mean for a Specific a

For a 90% confidence interval,

for a 95% confidence interval,

and for a 99% confidence interval,

α 2z =1.96

α 2z = 2.58

nzX

nzX

m

aa 2/2/

2 1.65za

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• Greek letter (alpha) represents the area in both tails of the standard normal distribution curve, and represents the area of each one of the tails.

• The relationship between and the confidence level is that the stated confidence level is the percentage equivalent to the decimal value of , and vice versa.

Ex) For 95% confidence interval,

For 90% confidence interval,

a

2a

a

1 a0.05a 0.10a

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Example 1 : Find the critical values for each.

(a) for the 99% confidence interval

22 58a / .z

%99

2/az

0 995.0 005.

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(b) for the 95% confidence interval

21 96a / .z

95%

2/az

0 975.0 025.

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(c) for the 90% confidence interval

21 65a / .z

90%

2/az

0 95.0 05.

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II. Maximum Error of the Estimate

The maximum error of estimate is the maximum

difference between the point estimate of a parameter and

the actual value of the parameter.

Definition :

When estimating by from a large sample, the maximum error of the

estimate, with level of confidence , is

When is unknown, we can estimate it by , as long as →

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95% Confidence Interval

For =0.05, 95% of the sample means will fall within the

error value on either side of the population mean.

a

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Example 1 :

Find the maximum error for based on = 128.3, n = 64 , s = 32.4 and

confidence level of 98%.

First , figure out what formula to use :

is unknown, but →

%98

22 33a / .z

0 99.0 01.

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III. Rounding Rule for a Confidence

Interval for a Mean

• When you are computing a confidence interval

for a population mean by using raw data, round

off to one more decimal place than the number

of decimal places in the original data.

• When you are computing a confidence interval

for a population mean by using a sample mean

and a standard deviation, round off to the same

number of a decimal places as given for the

mean.

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Example 1 : (Ref General Statistics by Chase/Bown, 4th Ed.)

A physician wanted to estimate the mean length of time that a patient had to

wait to see him after arriving at the office. A random sample of 50 patients

showed a mean waiting time 23.4 minutes and a standard deviation of 7.1

minutes. Find a 95% confidence interval for .

is unknown, but →

0.975

Now

Formula to use : , but first find .

0 025.

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A union official wanted to estimate the mean hourly wage of its members. A

random sample of 100 members gave = $18.30 and = $3.25 per hour.

Example 2 : (Ref General Statistics by Chase/Bown, 4th Ed.)

(a) Find an 80% confidence interval for .

is unknown, but → Now

Formula to use : , but first find .

0.900 10.

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(b) Find an 95% confidence interval for .

Now

0.9750 025.

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(c) If you were to construct a 90% confidence interval for (do not construct it),

would the interval be longer or shorter than 80% confidence interval? Longer

or shorter than the 95% confidence interval?

%80

%90

%90

%95

Interval of 90% is longer than interval

of 80%.

Interval of 90% is shorter than interval

of 95%.

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A restaurant owner believed that customer spending was below normal at tables

manned by one of the waiters. The owner sampled 36 checks from the waiter’s

tables and got the following amounts (rounded to the nearest dollar) :

Example 3 : (Ref General Statistics by Chase/Bown, 4th Ed.)

47 46 56 70 52 58 48 57 49 61 52 40 60 22 74 59 60 30

61 44 62 41 53 57 50 52 57 59 69 51 58 56 44 36 47 51

Find a 95% confidence interval for the true mean amount of money spent at the

waiter’s tables.

Using calculator,

0.9750 025.

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IV. Determining the Sample Size for

Maximum Error of the estimate for

Solve for n

Round the answer up to obtain a whole number

Note : If level of confidence is not given, then use 95%.

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Example 1 :

To estimate , what sample size is required so that the maximum error of the

estimate is only 8 square feet? Assume is 42 square feet. (Use 95%

confidence level.)

0.9750 025.

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Example 2 : (Ref General Statistics by Chase/Bown, 4th Ed.)

Consider a population with unknown mean and population standard deviation

(a) How large a sample size is needed to estimate to within five units with

95% confidence?

= 15.

(b) Suppose you wanted to estimate to within five units with 90% confidence.

Without calculating, would the sampling size required be larger or smaller than the

found in part (a) ?

90% confidence means smaller → n is smaller.

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(c) Suppose you wanted to estimate to within six units with 95% confidence.

Without calculating, would the sampling size required be larger or smaller the found

in part (a) ?

n is smaller because when you divide by a bigger E.

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7.2 Confidence Intervals for the Mean

( Unknown and )

• When the population sample size is less

than 30, and the standard deviation is

unknown, the t distribution must be used.

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Characteristics of the t

Distribution

• The t distribution is similar to the standard

normal distribution in the following ways:

1. It is bell shaped.

2. It is symmetrical about the mean.

3. The mean, median, and mode are equal to 0

and are located at the center of the

distribution.

4. The curve never touches the x axis.

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Characteristics of the t

Distribution (cont.)

• The t distribution differs from the standard normal distribution in the following ways.

1. The variance is greater than 1.

2. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size.

3. As the sample size increases, the t distribution approaches the standard normal distribution.

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t Distribution

• The degrees of freedom are the number of

values that are free to vary after a sample

statistic has been computed.

• The degrees of freedom for the confidence

interval for the mean are found by

subtracting 1 from the sample size. That

is,

1.. nfd

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Formula

• Formula for the Confidence Interval of the Mean When is Unknown and

/ 2 / 2

s sX t X t

n na am

Maximum Error for :

. . 1d f n Degree of Freedom :

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Example 1 : Find with the following information.

(a) Level of confidence is 98% with n = 19

%98

%12/ a

Look up on t – table

(b) Level of confidence is 90% with n = 25

%90

%52/ a

Look up on t – table

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Example 2 :

A sample of 25 two-year-old chickens shows that they lay an average of 21 eggs

per month. The standard deviation of the sample was 2 eggs. Assume the

population is approximately normal. Construct a 99% confidence interval for the

true mean.

%99

%5.02/ a

Look up on t – table

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Example 3 :

A random sample of 20 parking meters in a large municipality showed the following

incomes for a day.

$2.60 $1.05 $2.45 $2.90 $1.30 $3.10 $2.35

$2.00 $2.40 $2.35 $2.40 $1.95 $2.80 $2.50

$2.10 $1.75 $1.00 $2.75 $1.80 $1.95

Assume the population is approximately normal. Find the 95% confidence interval

of the true mean.

%95

%5.22/ a

Look up on t – table

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When to Use the Z or t Distribution

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7.3 Confidence Intervals and

Sample Size for Proportions

I. Confidence Intervals for Proportions

• Symbols Used in Proportion Notation

p = symbol for the population proportion

(read p “hat”) = symbol for the sample proportion

For a sample proportion,

where X = number of sample units that possess the

characteristics of interest and n = sample size.

and or 1ˆ ˆ ˆX n X

p q pn n

p̂

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Formula

• Formula for a specific confidence interval for

a proportion

when np and nq are each greater than or

equal to 5.

n

qpzpp

n

qpzp aa

ˆˆˆ

ˆˆˆ

2/2/

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II. Determining the Sample Size for p

Round the answer up to obtain a whole number.

Since the sample has not yet been obtained, we do not know the value of

and . However, it can be shown that regardless of the values of and ,

the value of will never be more than ¼ . Therefore, to be on the safe

side, we should take the sample size to be at least

Round the answer up to obtain a whole number.

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Example 1 : (Ref: General Statistics by Chase/Bown, 4th Ed.)

A city council commissioned a statistician to estimate the proportion of voters in

favor of a proposal to build a new library. The statistician obtained a random

sample of 200 voters, with 112 indicating approval of the proposal.

(a) What is a point estimate for ?

(b) What is the maximum error of estimate for with

a 90% confidence level?

(c) Find a 90% confidence interval for .

Example 2 :

A Roper poll of 2,000 American adults showed that 1,440 thought that chemical

dumps are among the most serious environmental problems. Estimate with a 98%

confidence interval the proportion of population who consider chemical dumps

among the most serious environmental problem.

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22 33a / .z

0 99.0 01.

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Example 3 :

A recent study indicated that 29% of the 100 women over age 55 in the study were

widows.

(a) How large a sample must one take to be 90% confident that the estimate is

within 0.05 of the true proportion of women over 55 who are widows?

(b) If no estimate of the sample proportion is available, how large should the

sample be?

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Example 4 :

How large a sample is necessary to estimate the true proportion of adults who are

overweight to within 2% with a 95% confidence?

0.9750 025.

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Summary

• A good estimator must be unbiased, consistent, and relatively efficient.

• There are two types of estimates of a parameter: point estimates and interval estimates.

• A point estimate is a single value. The problem with point estimates is that the accuracy of the estimate cannot be determined, so the interval estimate is preferred.

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Summary (cont.)

• By calculating a 95% or 99% confidence interval

about the sample value, statisticians can be 95%

or 99% confident that their estimate contains the

true parameter.

• Once the confidence interval of the mean is

calculated, the z or t values are used depending

on the sample size and whether the standard

deviation is known.

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Summary (cont.)

• The following information is needed to

determine the minimum sample size

necessary to make an estimate of the

mean:

1. The degree of confidence must be stated.

2. The population standard deviation must be

known or be able to be estimated.

3. The maximum error of the estimate must be

stated.

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Conclusions

Estimation is an important aspect of

inferential statistics.