Section 7 - 1 Introduction

Chapter 7

Section 7 - 1 Introduction

Inferential statistics is the process of making judgments about a population based on properties of a sample from the population.

One aspect of inferential statistics is estimation.

The best point estimate of the population mean is the sample mean .

The best point estimate of the population proportion is the sample proportion .

An important question in estimating the population mean and proportion is that of sample size.

How large should the sample be in order to make an accurate estimate?

The procedures for estimating the population mean, estimating the population proportion, and estimating a sample size will be explained. The other type of inference is called hypothesis testing, which is a decision-making process for evaluating claims about a population. The procedures for hypothesis testing for the population mean and proportion will be explained in Chapter 8.

Section 7 - 2Estimating a population mean (Large-Sample Case)

I.Level of Confidence 1 - 

Even 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 due to sampling error. For this reason, statisticians prefer an interval estimate. The confidence level of an interval estimate of a population mean is the probability that the interval estimate will contain .

Example 1: Find the critical values for each.

(a) for the 99% confidence interval

(b) for the 95% 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 by from a large sample, the maximum error of the estimate, with

level of confidence 1- , is

When is unknown, we can estimate it by ; as long as n  30  .

Example 1: Find the maximum error for  based on =128.3, n = 64, = 32.4, and

confidence level of 98%.

III.Confidence Interval for 

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 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 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 .

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

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.

(a)Find an 80% confidence interval for .

(b)Find a 95% confidence interval for .

(c)If you were to construct a 90% confidence interval for

(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 the 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

Solve for n

Round the answer up to obtain a whole number.

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.

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

Consider a population with unknown mean and population standard

deviation  = 15.

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

with 95% confidence?

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

confidence. Without calculating, would the sample size required

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

confidence. Without calculating, would the sample size required

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

Section 7-3

I.Confidence interval for  when  is unknown and n is small

When estimating by from a small sample, the maximum error of the estimate, with

level of confidence 1- , is

Maximum Error for 

Confidence Interval for

t - table

-- bell shape with thick tails

-- t value depends on the degree of freedom ( df = n -1)

n = 5

n=10

Example 1: Find 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.

Section 7 - 4Inference Concerning a Population Proportion

I.Confidence Interval for

When estimating by , the maximum error of the estimate with confidence 1- is

where

n will be sufficiently large if both x and n–x are at least 5.

Confidence Interval for

II.Determining the Sample Size for

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.

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

A city council commissioned a statistician to estimate to 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 ?

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?

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