Statistics Test

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Lecture Slides

Elementary Statistics Eleventh Edition

and the Triola Statistics Series

by Mario F. Triola

Chapter 7

Estimates and Sample Sizes

7-1 Review and Preview

7-2 Estimating a Population Proportion

7-3 Estimating a Population Mean: σ Known

7-4 Estimating a Population Mean: σ Not Known

7-5 Estimating a Population Variance

Section 7-1

Review and Preview

Review

❖ Chapters 2 & 3 we used “descriptive statistics” when we summarized data using tools such as graphs, and statistics such as the mean and standard deviation.

❖ Chapter 6 we introduced critical values: z denotes the z score with an area of  to its right. If  = 0.025, the critical value is z0.025 = 1.96. That is, the critical value z0.025 = 1.96 has an area of 0.025 to its right.

Preview

❖ The two major activities of inferential statistics are (1) to use sample data to estimate values of a population parameters, and (2) to test hypotheses or claims made about population parameters.

❖ We introduce methods for estimating values of these important population parameters: proportions, means, and variances.

❖ We also present methods for determining sample sizes necessary to estimate those parameters.

This chapter presents the beginning of inferential statistics.

Section 7-2

Estimating a Population

Proportion

Key Concept In this section we present methods for using a

sample proportion to estimate the value of a

population proportion.

• The sample proportion is the best point

estimate of the population proportion.

• We can use a sample proportion to construct a

confidence interval to estimate the true value

of a population proportion, and we should

know how to interpret such confidence

intervals.

• We should know how to find the sample size

necessary to estimate a population proportion.

Definition

A point estimate is a single value (or

point) used to approximate a population

parameter.

The sample proportion p is the best

point estimate of the population

proportion p.

Definition

Example:

Because the sample proportion is the best point estimate of the population proportion, we conclude that the best point estimate of p is 0.70. When using the sample results to estimate the percentage of all adults in the United States who believe in global warming, the best estimate is 70%.

In the Chapter Problem we noted that in a Pew Research Center poll, 70% of 1501 randomly selected adults in the United States believe in global warming, so the sample proportion is

= 0.70. Find the best point estimate of the proportion of all adults in the United States who believe in global warming.

p̂

Definition

A confidence interval (or interval

estimate) is a range (or an interval)

of values used to estimate the true

value of a population parameter. A

confidence interval is sometimes

abbreviated as CI.

A confidence level is the probability 1 –  (often

expressed as the equivalent percentage value)

that the confidence interval actually does contain

the population parameter, assuming that the

estimation process is repeated a large number of

times. (The confidence level is also called degree

of confidence, or the confidence coefficient.)

Most common choices are 90%, 95%, or 99%.

( = 10%), ( = 5%), ( = 1%)

Definition

We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval 0.677 < p < 0.723.

“We are 95% confident that the interval from 0.677 to 0.723 actually does contain the true value of the population proportion p.”

This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p.

(Note that in this correct interpretation, the level of 95% refers to the success rate of the process being used to estimate the proportion.)

Interpreting a Confidence Interval

Know the correct interpretation of a confidence interval.

Caution

Confidence intervals can be used informally to compare different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of proportions.

Caution

Critical Values A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. Critical values are based on the following observations:

1. Under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution.

Critical Values

2. A z score associated with a sample proportion has a probability of /2 of falling in the right tail.

Critical Values

3. The z score separating the right-tail region is

commonly denoted by z/2 and is referred to

as a critical value because it is on the

borderline separating z scores from sample

proportions that are likely to occur from those

that are unlikely to occur.

Definition

A critical value is the number on the

borderline separating sample statistics

that are likely to occur from those that are

unlikely to occur. The number z/2 is a

critical value that is a z score with the

property that it separates an area of /2 in

the right tail of the standard normal

distribution.

The Critical Value z2

Notation for Critical Value

The critical value z/2 is the positive z value

that is at the vertical boundary separating an

area of /2 in the right tail of the standard

normal distribution. (The value of –z/2 is at

the vertical boundary for the area of /2 in the

left tail.) The subscript /2 is simply a

reminder that the z score separates an area of

/2 in the right tail of the standard normal

distribution.

Finding z2 for a 95%

Confidence Level

-z2 z2

Critical Values

 2 = 2.5% = .025

 = 5%

z2 = 1.96−+

Use Table A-2 to find a z score of 1.96

 = 0.05

Finding z2 for a 95%

Confidence Level - cont

Definition

When data from a simple random sample are

used to estimate a population proportion p, the

margin of error, denoted by E, is the maximum

likely difference (with probability 1 – , such as

0.95) between the observed proportion and

the true value of the population proportion p.

The margin of error E is also called the

maximum error of the estimate and can be found

by multiplying the critical value and the standard

deviation of the sample proportions:

p̂

Margin of Error for

Proportions

ˆ ˆpq E z

n =

p = population proportion

Confidence Interval for Estimating

a Population Proportion p

= sample proportion

n = number of sample values

E = margin of error

z/2 = z score separating an area of /2 in the right tail of the standard normal distribution

p̂

Confidence Interval for Estimating

a Population Proportion p

1. The sample is a simple random sample.

2. The conditions for the binomial distribution

are satisfied: there is a fixed number of

trials, the trials are independent, there are

two categories of outcomes, and the

probabilities remain constant for each trial.

3. There are at least 5 successes and 5

failures.

Confidence Interval for Estimating

a Population Proportion p

p – E < < + Eˆ p̂p

where

ˆ ˆpq E z

n =

p – E < < + E

p + E

ppˆ ˆ

Confidence Interval for Estimating

a Population Proportion p

(p – E, p + E)ˆ ˆ

Round-Off Rule for

Confidence Interval Estimates of p

Round the confidence interval limits

for p to

three significant digits.

1. Verify that the required assumptions are satisfied.

(The sample is a simple random sample, the

conditions for the binomial distribution are satisfied,

and the normal distribution can be used to

approximate the distribution of sample proportions

because np  5, and nq  5 are both satisfied.)

2. Refer to Table A-2 and find the critical value z /2 that

corresponds to the desired confidence level.

3. Evaluate the margin of error

Procedure for Constructing

a Confidence Interval for p

2 ˆ ˆE z pq n=

4. Using the value of the calculated margin of error, E

and the value of the sample proportion, p, find the

values of p – E and p + E. Substitute those values

in the general format for the confidence interval:

ˆ ˆ ˆ

p – E < p < p + Eˆ ˆ

5. Round the resulting confidence interval limits to

three significant digits.

Procedure for Constructing

a Confidence Interval for p - cont

Example:

a. Find the margin of error E that corresponds to a

95% confidence level.

b. Find the 95% confidence interval estimate of the

population proportion p.

c. Based on the results, can we safely conclude that

the majority of adults believe in global warming?

d. Assuming that you are a newspaper reporter, write

a brief statement that accurately describes the

results and includes all of the relevant information.

In the Chapter Problem we noted that a Pew Research

Center poll of 1501 randomly selected U.S. adults

showed that 70% of the respondents believe in global

warming. The sample results are n = 1501, and ˆ 0.70p =

Requirement check: simple random sample; fixed number of trials, 1501; trials are independent; two categories of outcomes (believes or does not); probability remains constant. Note: number of successes and failures are both at least 5.

Example:

a) Use the formula to find the margin of error.

( )( ) 2

0 70 0 30 1 96

1501

0 023183

ˆ ˆ . . .

pq E z

= =

b) The 95% confidence interval:

Example:

ˆ ˆp E p p E−   +

0.70 − 0.023183  p  0.70 + 0.023183

0.677  p  0.723

c) Based on the confidence interval

obtained in part (b), it does appear that

the proportion of adults who believe in

global warming is greater than 0.5 (or

50%), so we can safely conclude that the

majority of adults believe in global

warming. Because the limits of 0.677 and

0.723 are likely to contain the true

population proportion, it appears that the

population proportion is a value greater

than 0.5.

Example:

d) Here is one statement that summarizes

the results: 70% of United States adults

believe that the earth is getting warmer.

That percentage is based on a Pew

Research Center poll of 1501 randomly

selected adults in the United States. In

theory, in 95% of such polls, the

percentage should differ by no more than

2.3 percentage points in either direction

from the percentage that would be found

by interviewing all adults in the United

States.

Example:

Analyzing Polls

When analyzing polls consider:

1. The sample should be a simple random sample,

not an inappropriate sample (such as a voluntary

response sample).

2. The confidence level should be provided. (It is

often 95%, but media reports often neglect to

identify it.)

3. The sample size should be provided. (It is usually

provided by the media, but not always.)

4. Except for relatively rare cases, the quality of the

poll results depends on the sampling method and

the size of the sample, but the size of the

population is usually not a factor.

Caution

Never follow the common misconception

that poll results are unreliable if the

sample size is a small percentage of the

population size. The population size is

usually not a factor in determining the

reliability of a poll.

Sample Size

Suppose we want to collect sample

data in order to estimate some

population proportion. The question is

how many sample items must be

obtained?

Determining Sample Size

(solve for n by algebra)

( )2 ˆp q  2Z n = ˆ E 2

  2zE = p qˆ ˆ n

Sample Size for Estimating

Proportion p

When an estimate of p is known: ˆ

ˆ( )2 p qn = ˆ E 2

  2z

When no estimate of p is known:

( )2 0.25n = E 2

  2 z ˆ

Round-Off Rule for Determining

Sample Size

If the computed sample size n is not

a whole number, round the value of n

up to the next larger whole number.

Example:

The Internet is affecting us all in many

different ways, so there are many reasons for

estimating the proportion of adults who use

it. Assume that a manager for E-Bay wants to

determine the current percentage of U.S.

adults who now use the Internet. How many

adults must be surveyed in order to be 95%

confident that the sample percentage is in

error by no more than three percentage

points?

a. In 2006, 73% of adults used the Internet.

b. No known possible value of the proportion.

a) Use

To be 95% confident that our sample percentage is within three percentage points of the true percentage for all adults, we should obtain a simple random sample of 842 adults.

Example:

ˆ ˆ ˆ0.73 and 1 0.27

0.05 so 1.96

0.03

p q p



= = − =

= =

( )

( ) ( )( )

( )

ˆ ˆ

1.96 0.73 0.27

0.03

841.3104

842

z pq n

 =

b) Use

To be 95% confident that our sample percentage is within three percentage points of the true percentage for all adults, we should obtain a simple random sample of 1068 adults.

Example:

 = 0.05 so z  2

= 1.96

E = 0.03

( )

0.25

1.96 0.25

0.03

1067.1111

1068

z n

 =

Finding the Point Estimate and E

from a Confidence Interval

Margin of Error:

E = (upper confidence limit) — (lower confidence limit)

Point estimate of p:

p = (upper confidence limit) + (lower confidence limit)

2 ˆ

Recap

In this section we have discussed:

❖ Point estimates.

❖ Confidence intervals.

❖ Confidence levels.

❖ Critical values.

❖ Margin of error.

❖ Determining sample sizes.

Section 7-3

Estimating a Population

Mean:  Known

Key Concept

This section presents methods for

estimating a population mean. In

addition to knowing the values of the

sample data or statistics, we must also

know the value of the population

standard deviation, .

Here are three key concepts that

should be learned in this section:

Key Concept

1. We should know that the sample mean

is the best point estimate of the

population mean .

2. We should learn how to use sample data

to construct a confidence interval for

estimating the value of a population mean,

and we should know how to interpret such

confidence intervals.

3. We should develop the ability to determine

the sample size necessary to estimate a

population mean.

Point Estimate of the

Population Mean

The sample mean x is the best point estimate

of the population mean µ.

Confidence Interval for

Estimating a Population Mean

(with  Known)

 = population mean

 = population standard deviation

= sample mean

n = number of sample values

E = margin of error

z/2 = z score separating an area of /2 in the

right tail of the standard normal

distribution

Confidence Interval for

Estimating a Population Mean

(with  Known)

1. The sample is a simple random sample.

(All samples of the same size have an

equal chance of being selected.)

2. The value of the population standard

deviation  is known.

3. Either or both of these conditions is

satisfied: The population is normally

distributed or n > 30.

Confidence Interval for

Estimating a Population Mean

(with  Known)

x − E    x + E where E = z

 2 



or x  E

or x − E,x + E( )

Definition

The two values x – E and x + E are

called confidence interval limits.

1. For all populations, the sample mean x is an unbiased estimator of the population mean , meaning that the distribution of sample means tends to center about the value of the population mean .

2. For many populations, the distribution of sample means x tends to be more consistent (with less variation) than the distributions of other sample statistics.

Sample Mean

Procedure for Constructing a

Confidence Interval for µ (with Known )

1. Verify that the requirements are satisfied.

2. Refer to Table A-2 or use technology to find the critical value z2 that corresponds to the desired confidence level.

3. Evaluate the margin of error

5. Round using the confidence intervals round-off

rules.

4. Find the values of Substitute

those values in the general format of the confidence interval:

2E z n =

x − E and x + E.

x − E    x + E

1. When using the original set of data, round the

confidence interval limits to one more decimal

place than used in original set of data.

2. When the original set of data is unknown and only

the summary statistics (n, x, s) are used, round the

confidence interval limits to the same number of

decimal places used for the sample mean.

Round-Off Rule for Confidence

Intervals Used to Estimate µ

Example:

People have died in boat and aircraft accidents because an obsolete estimate of the mean weight of men was used. In recent decades, the mean weight of men has increased considerably, so we need to update our estimate of that mean so that boats, aircraft, elevators, and other such devices do not become dangerously overloaded. Using the weights of men from Data Set 1 in Appendix B, we obtain these sample statistics for the simple random sample: n = 40 and = 172.55 lb. Research from several other sources suggests that the population of weights of men has a standard deviation given by  = 26 lb.

Example:

a. Find the best point estimate of the mean weight of the population of all men.

b. Construct a 95% confidence interval estimate of the mean weight of all men.

c. What do the results suggest about the mean weight of 166.3 lb that was used to determine the safe passenger capacity of water vessels in 1960 (as given in the National Transportation and Safety Board safety recommendation M-04-04)?

Example:

a. The sample mean of 172.55 lb is the best point estimate of the mean weight of the population of all men.

X – E <  < x + E

b. A 95% confidence interval or 0.95 implies = 0.05, so z/2 = 1.96. Calculate the margin of error.

Construct the confidence interval.

x − E    x − E

172.55 − 8.0574835    172.55 + 8.0574835

164.49    180.61

E = z/2 n

σ =1.96

26 =8.0574835

Example:

c. Based on the confidence interval, it is possible that the mean weight of 166.3 lb used in 1960 could be the mean weight of men today. However, the best point estimate of 172.55 lb suggests that the mean weight of men is now considerably greater than 166.3 lb. Considering that an underestimate of the mean weight of men could result in lives lost through overloaded boats and aircraft, these results strongly suggest that additional data should be collected. (Additional data have been collected, and the assumed mean weight of men has been increased.)

Finding a Sample Size for

Estimating a Population Mean

(z/2) •  n =

 = population mean

σ = population standard deviation

= sample mean

E = desired margin of error

zα/2 = z score separating an area of /2 in the right tail of

the standard normal distribution

Round-Off Rule for Sample Size n

If the computed sample size n is not a

whole number, round the value of n up

to the next larger whole number.

Finding the Sample Size n When  is Unknown

1. Use the range rule of thumb (see Section 3-3) to estimate the standard deviation as follows:   range/4.

2. Start the sample collection process without knowing  and, using the first several values, calculate the sample standard deviation s and use it in place of  . The estimated value of  can then be improved as more sample data are obtained, and the sample size can be refined accordingly.

3. Estimate the value of  by using the results of some other study that was done earlier.

Example:

 = 0.05

 /2 = 0.025

z / 2 = 1.96

E = 3

 = 15

n = 1.96 • 15 = 96.04 = 97

With a simple random sample of only

97 statistics students, we will be 95%

confident that the sample mean is

within 3 IQ points of the true

population mean .

Assume that we want to estimate the mean IQ score for

the population of statistics students. How many

statistics students must be randomly selected for IQ

tests if we want 95% confidence that the sample mean

is within 3 IQ points of the population mean, and

population standard deviation is 15 ?

Recap

In this section we have discussed:

❖ Margin of error.

❖ Confidence interval estimate of the

population mean with σ known.

❖ Round off rules.

❖ Sample size for estimating the mean μ.

Section 7-4

Estimating a Population

Mean:  Not Known

Key Concept

This section presents methods for estimating

a population mean when the population

standard deviation is not known. With σ

unknown, we use the Student t distribution

assuming that the relevant requirements are

satisfied.

The sample mean is the best point

estimate of the population mean.

Sample Mean

If the distribution of a population is essentially

normal, then the distribution of

is a Student t Distribution for all samples of size n. It is often referred to as a t distribution and is used to find critical values denoted by t/2.

t = x - µ

s n

Student t Distribution

degrees of freedom = n – 1

in this section.

Definition

The number of degrees of freedom for a

collection of sample data is the number of

sample values that can vary after certain

restrictions have been imposed on all data

values. The degree of freedom is often

abbreviated df.

Margin of Error E for Estimate of 

(With σ Not Known)

Formula 7-6

where t2 has n – 1 degrees of freedom.

n s

E = t 2

Table A-3 lists values for tα/2

 = population mean

= sample mean

s = sample standard deviation

n = number of sample values

E = margin of error

t/2 = critical t value separating an area of /2

in the right tail of the t distribution

Notation

where E = t/2 n s

x – E < µ < x + E

t/2 found in Table A-3

Confidence Interval for the

Estimate of μ (With σ Not Known)

df = n – 1

2. Using n – 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t2 that corresponds to the desired confidence level.

Procedure for Constructing a

Confidence Interval for µ

(With σ Unknown)

1. Verify that the requirements are satisfied.

3. Evaluate the margin of error E = t2 • s / n .

4. Find the values of Substitute those values in the general format for the confidence interval:

5. Round the resulting confidence interval limits.

x − E and x + E.

x − E    x + E

Example:

A common claim is that garlic lowers cholesterol

levels. In a test of the effectiveness of garlic, 49

subjects were treated with doses of raw garlic, and

their cholesterol levels were measured before and

after the treatment. The changes in their levels of LDL

cholesterol (in mg/dL) have a mean of 0.4 and a

standard deviation of 21.0. Use the sample statistics of

n = 49, = 0.4 and s = 21.0 to construct a 95%

confidence interval estimate of the mean net change in

LDL cholesterol after the garlic treatment. What does

the confidence interval suggest about the

effectiveness of garlic in reducing LDL cholesterol?

Example:

Requirements are satisfied: simple random

sample and n = 49 (i.e., n > 30).

21 0 2 009 6 027

. . .E t

n 

 = = =

95% implies α = 0.05.

With n = 49, the df = 49 – 1 = 48

Closest df is 50, two tails, so t/2 = 2.009

Using t/2 = 2.009, s = 21.0 and n = 49 the

margin of error is:

Example:

Construct the confidence

interval: x = 0.4, E = 6.027

We are 95% confident that the limits of –5.6 and 6.4

actually do contain the value of , the mean of the

changes in LDL cholesterol for the population. Because

the confidence interval limits contain the value of 0, it is

very possible that the mean of the changes in LDL

cholesterol is equal to 0, suggesting that the garlic

treatment did not affect the LDL cholesterol levels. It

does not appear that the garlic treatment is effective in

lowering LDL cholesterol.

x − E    x + E

0.4 − 6.027    0.4 + 6.027

−5.6    6.4

Important Properties of the

Student t Distribution 1. The Student t distribution is different for different sample sizes

(see the following slide, for the cases n = 3 and n = 12).

2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.

3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).

4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a  = 1).

5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.

Student t Distributions for

n = 3 and n = 12

Figure 7-5

Choosing the Appropriate Distribution

Use the normal (z) distribution

 known and normally distributed population or  known and n > 30

Use t distribution  not known and normally distributed population or  not known and n > 30

Use a nonparametric method or bootstrapping

Population is not normally distributed and n ≤ 30

Point estimate of µ:

x = (upper confidence limit) + (lower confidence limit)

Margin of Error:

E = (upper confidence limit) – (lower confidence limit)

Finding the Point Estimate

and E from a Confidence Interval

Confidence Intervals for

Comparing Data

As in Sections 7-2 and 7-3, confidence

intervals can be used informally to

compare different data sets, but the

overlapping of confidence intervals should

not be used for making formal and final

conclusions about equality of means.

Recap

In this section we have discussed:

❖ Student t distribution.

❖ Degrees of freedom.

❖ Margin of error.

❖ Confidence intervals for μ with σ unknown.

❖ Choosing the appropriate distribution.

❖ Point estimates.

❖ Using confidence intervals to compare data.