CHAPTER 7



CHAPTER 7

Confidence Intervals and Sample Size

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.

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.

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.

7.1 Confidence Intervals for the Mean When [pic]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.

Point and Interval Estimates

A point estimate is a specific numerical value of a parameter. The best point estimate of the population mean [pic] is the sample mean [pic].

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.

I. Confidence Intervals

Even the best point estimate of the population mean is the sample mean, for the most part, the sample mean [pic] will be different from the population mean [pic] due to sampling error. For this reason, statisticians prefer an interval estimate. The confidence level of an interval estimate of a population mean [pic] is the probability that the interval estimate will contain[pic].

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.

Formula

Formula for the Confidence Interval of the Mean for a Specific [pic]

For a 90% confidence interval,

for a 95% confidence interval,

and for a 99% confidence interval,

Example 1: Find the critical values for each.

(a) [pic] for the 99% confidence interval

(b) [pic] for the 95% confidence interval

(c) [pic] for the 90% confidence interval

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 [pic] by [pic] from a large sample, the maximum error of the estimate, with level of confidence [pic] , is [pic]

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

[pic]

95% Confidence Interval

[pic]

For =0.05, 95% of the sample means will fall within the error value on either side of the population mean.

Example 1: Find the maximum error for ( based on [pic] =128.3, n = 64, [pic] = 32.4, and

confidence level of 98%.

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.

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

A physician wanted to estimate the mean length of time [pic] 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 of 23.4 minutes and a standard

deviation of 7.1 minutes. Find a 95% confidence interval for [pic].

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

A union official wanted to estimate the mean hourly wage [pic] of its members. A random sample of 100 members gave [pic] = $18.30 and [pic] = $3.25 per hour.

a) Find an 80% confidence interval for [pic].

(b) Find a 95% confidence interval for [pic].

(c) If you were to construct a 90% confidence interval for [pic] (do not construct it), would the interval be longer or shorter than the 80% confidence interval? Longer or shorter than the 95% confidence interval?

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

A restaurant owner believed that customer spending was below normal

at tables manned by one of waiters. The owner sampled 36 checks from

the waiter’s tables and got the following amounts (rounded to the nearest

dollar):

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.

IV. Determining the Sample Size for

Maximum Error of Estimate for [pic]

[pic]

Solve for n [pic]

[pic] Round the answer up to obtain a whole number.

Example 1: To estimate[pic], what sample size is required so that the maximum error of the estimate is only 8 square feet? Assume [pic] is 42 square feet.

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

Consider a population with unknown mean [pic] and population standard deviation ( = 15.

a) How large a sample size is needed to estimate [pic] to within five units

with 95% confidence?

(b) Suppose you wanted to estimate [pic] to within five units with 90% confidence. Without calculating, would the sample size required be larger or smaller than the found in part (a)?

(c) Suppose you wanted to estimate [pic] to within six units with 95%

confidence. Without calculating, would the sample size required

be larger or smaller than the found in part (a)?

7.2 Confidence Intervals for the Mean When[pic] Unknown

When the population sample size is less than 30, and the standard deviation is unknown, the t distribution must be used.

Characteristics of the t Distribution

The t distribution is similar to the standard normal distribution in the following ways:

It is bell shaped.

It is symmetrical about the mean.

The mean, median, and mode are equal to 0 and are located at the center of the distribution.

The curve never touches the x axis.

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

The variance is greater than 1.

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

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

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,

Formula

Formula for the Confidence Interval of the Mean When [pic] is Unknown and [pic]

Maximum Error for :[pic]

Degree of Freedom :[pic]

Example 1: Find [pic] with the following information.

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

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

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.

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.

When to Use the Z or t Distribution

[pic]

7.3 Inference Interverals and Sample Size for Proportion

I. Confidence Intervals for Proportions

Symbols Used in Proportion Notation

p = symbol for the population proportion

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

For a sample proportion,

[pic]

where X = number of sample units that possess the characteristics of interest and n = sample size.

Formula

Formula for a specific confidence interval for a proportion

when np and nq are each greater than or equal to 5.

II. Determining the Sample Size for p

[pic]

Round the answer up to obtain a whole number.

Since the sample has not yet been obtained, we do not know the value of [pic]and [pic].

However, it can be shown that regardless of the values of [pic]and [pic], the value of

[pic]([pic] will never be more than ¼. Therefore, to be on the safe side, we should take

the sample size to be at least

[pic] = [pic]

Round the answer up to obtain a whole number.

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

A city council commissioned a statistician to estimate to proportion

[pic]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 [pic]?

(b) What is the maximum error of estimate for [pic]?

(c) Find a 90% confidence interval for [pic].

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.

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?

Example 4: How large a sample is necessary to estimate the true proportion of adults

who are overweight to within 2 % with 95% confidence?

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.

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.

The following information is needed to determine the minimum sample size necessary to make an estimate of the mean:

The degree of confidence must be stated.

The population standard deviation must be known or be able to be estimated.

The maximum error of the estimate must be stated.

Conclusions

Estimation is an important aspect of inferential statistics.

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