Statistics Test

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Lecture Slides

Elementary Statistics Eleventh Edition

and the Triola Statistics Series

by Mario F. Triola

Chapter 8

Hypothesis Testing

8-1 Review and Preview

8-2 Basics of Hypothesis Testing

8-3 Testing a Claim about a Proportion

8-4 Testing a Claim About a Mean: σ Known

8-5 Testing a Claim About a Mean: σ Not Known

Section 8-1

Review and Preview

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

Methods of inferential statistics use sample data to make an inference or conclusion about a population. The two main activities of inferential statistics are using sample data to (1) estimate a population parameter (such as estimating a population parameter with a confidence interval), and (2) test a hypothesis or claim about a population parameter.

In Chapter 7 we presented methods for estimating a population parameter with a confidence interval, and in this chapter we present the method of hypothesis testing.

Definitions

In statistics, a hypothesis is a claim or statement about a property of a population.

A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population.

Main Objective

The main objective of this chapter is to

develop the ability to conduct hypothesis

tests for claims made about a:

- population proportion p,

- a population mean ,

- or a population standard deviation .

Examples of Hypotheses that can be Tested

• Genetics: The Genetics & IVF Institute claims that its XSORT method allows couples to increase the probability of having a baby girl.

• Business: A newspaper headline makes the claim that most workers get their jobs through networking.

• Medicine: Medical researchers claim that when people with colds are treated with echinacea, the treatment has no effect.

Examples of Hypotheses that can be Tested

• Aircraft Safety: The Federal Aviation Administration claims that the mean weight of an airline passenger (including carry-on baggage) is greater than 185 lb, which it was 20 years ago.

• Quality Control: When new equipment is used to manufacture aircraft altimeters, the new altimeters are better because the variation in the errors is reduced so that the readings are more consistent. (In many industries, the quality of goods and services can often be improved by reducing variation.)

Caution

When conducting hypothesis tests as

described in this chapter and the

following chapters, instead of jumping

directly to procedures and calculations,

be sure to consider the context of the

data, the source of the data, and the

sampling method used to obtain the

sample data.

Section 8-2

Basics of Hypothesis

Testing

Key Concept

This section presents individual

components of a hypothesis test. We should

know and understand the following:

• How to identify the null hypothesis and alternative hypothesis from a given claim, and how to express both in symbolic form

• How to calculate the value of the test statistic, given a claim and sample data

• How to identify the critical value(s), given a significance level

• How to identify the P-value, given a value of the test statistic

• How to state the conclusion about a claim in simple and nontechnical terms

Part 1:

The Basics of Hypothesis Testing

Rare Event Rule for

Inferential Statistics

If, under a given assumption, the

probability of a particular observed event

is exceptionally small, we conclude that

the assumption is probably not correct.

Components of a

Formal Hypothesis

Test

Null Hypothesis:

• The null hypothesis (denoted by H0) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.

• We test the null hypothesis directly.

• Either reject H0 or fail to reject H0.

Alternative Hypothesis:

• The alternative hypothesis (denoted by H1 or Ha or HA) is the statement that the parameter has a value that somehow differs from the null hypothesis.

• The symbolic form of the alternative hypothesis must use one of these symbols: , <, >.

Note about Forming Your

Own Claims (Hypotheses)

If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.

Note about Identifying

H0 and H1

Figure 8-2

Example:

Consider the claim that the mean weight of

airline passengers (including carry-on

baggage) is at most 195 lb (the current value

used by the Federal Aviation Administration).

Follow the three-step procedure outlined in

Figure 8-2 to identify the null hypothesis and

the alternative hypothesis.

Example:

Step 1: Express the given claim in symbolic

form. The claim that the mean is at

most 195 lb is expressed in symbolic

form as  ≤ 195 lb.

Step 2: If  ≤ 195 lb is false, then  > 195 lb

must be true.

Example:

Step 3: Of the two symbolic expressions

 ≤ 195 lb and  > 195 lb,

we see that  > 195 lb does not contain

equality, so we let the alternative

hypothesis H1 be  > 195 lb.

Also, the null hypothesis must be a

statement that the mean equals 195 lb, so

we let H0 be  = 195 lb.

Note that the original claim that the mean is at most

195 lb is neither the alternative hypothesis nor the null

hypothesis.

(However, we would be able to address the original claim

upon completion of a hypothesis test.)

The test statistic is a value used in making

a decision about the null hypothesis, and is

found by converting the sample statistic to

a score with the assumption that the null

hypothesis is true.

Test Statistic

Test Statistic - Formulas

Test statistic for

proportion

z = p̂ − p

Test statistic

for mean z = x − 



or t = x − 

Example:

Let’s again consider the claim that the XSORT

method of gender selection increases the

likelihood of having a baby girl.

Preliminary results from a test of the XSORT

method of gender selection involved 14 couples

who gave birth to 13 girls and 1 boy. Use the given

claim and the preliminary results to calculate the

value of the test statistic.

Use the format of the test statistic given above, so

that a normal distribution is used to approximate a

binomial distribution.

(There are other exact methods that do not use the

normal approximation.)

Example:

The claim that the XSORT method of gender

selection increases the likelihood of having a

baby girl results in the following null and

alternative hypotheses H0: p = 0.5 and

H1: p > 0.5.

We work under the assumption that the null

hypothesis is true with p = 0.5.

The sample proportion of 13 girls in 14 births

results in . Using p = 0.5,

and n = 14,

we find the value of the test statistic as follows:

p̂ = 13 14 = 0.929 p̂ = 0.929

Example:

We know from previous chapters that a z score of

3.21 is “unusual” (because it is greater than 2).

It appears that in addition to being greater than

0.5, the sample proportion of 13/14 or 0.929 is

significantly greater than 0.5.

The figure on the next slide shows that the sample

proportion of 0.929 does fall within the range of

values considered to be significant because they

are so far above 0.5 that they are not likely to

occur by chance (assuming that the population

proportion is p = 0.5).

z = p̂ − p

= 0.929 − 0.5

0.5( ) 0.5( ) 14

= 3.21

Example:

Sample proportion of: or Test Statistic z = 3.21

p̂ = 0.929

Fail to reject H0 Reject H0

Critical Region

The critical region (or rejection region) is the

set of all values of the test statistic that

cause us to reject the null hypothesis. For

example, see the red-shaded region in the

previous figure.

Significance Level

The significance level (denoted by ) is the

probability that the test statistic will fall in the

critical region when the null hypothesis is

actually true. This is the same  introduced

in Section 7-2. Common choices for  are

0.05, 0.01, and 0.10.

Critical Value

A critical value is any value that separates the

critical region (where we reject the null

hypothesis) from the values of the test

statistic that do not lead to rejection of the null

hypothesis. The critical values depend on the

nature of the null hypothesis, the sampling

distribution that applies, and the significance

level . See the previous figure where the

critical value of z = 1.645 corresponds to a

significance level of  = 0.05.

P-Value

The P-value (or p-value or probability value)

is the probability of getting a value of the test

statistic that is at least as extreme as the one

representing the sample data, assuming that

the null hypothesis is true.

Critical region

in the left tail:

Critical region

in the right tail:

Critical region

in two tails:

P-value = area to the left of

the test statistic

P-value = area to the right of

the test statistic

P-value = twice the area in the

tail beyond the test statistic

P-Value

The null hypothesis is rejected if the P-value

is very small, such as 0.05 or less.

Here is a memory tool useful for interpreting

the P-value:

If the P is low, the null must go.

If the P is high, the null will fly.

Procedure for Finding P-Values

Figure 8-5

H1:>H1:<

H1:≠

Caution

Don’t confuse a P-value with a proportion p.

Know this distinction:

P-value = probability of getting a test

statistic at least as extreme as

the one representing sample

data

p = population proportion

Example Consider the claim that with the XSORT method

of gender selection, the likelihood of having a

baby girl is different from p = 0.5, and use the

test statistic z = 3.21 found from 13 girls in 14

births.

First determine whether the given conditions

result in a critical region in the right tail, left tail,

or two tails, then use Figure 8-5 to find the P-

value. Interpret the P-value.

H0: p=0.5

H1: p≠0.5

Example

The claim that the likelihood of having a baby

girl is different from p = 0.5 can be expressed as

p ≠ 0.5 so the critical region is in two tails.

Using Figure 8-5 to find the P-value for a two-

tailed test, we see that the P-value is twice the

area to the right of the test statistic z = 3.21.

We refer to Table A-2 (or use technology) to find

that the area to the right of z = 3.21 is 0.0007.

In this case, the P-value is twice the area to the

right of the test statistic, so we have:

P-value = 2  0.0007 = 0.0014

Example

The P-value is 0.0014 (or 0.0013 if greater

precision is used for the calculations). The small

P-value of 0.0014 shows that there is a very

small chance of getting the sample results that

led to a test statistic of z = 3.21. This suggests

that with the XSORT method of gender

selection, the likelihood of having a baby girl is

different from 0.5.

Types of Hypothesis Tests:

Two-tailed, Left-tailed, Right-tailed

The tails in a distribution are the extreme

regions bounded by critical values.

Determinations of P-values and critical values

are affected by whether a critical region is in

two tails, the left tail, or the right tail. It

therefore becomes important to correctly

characterize a hypothesis test as two-tailed,

left-tailed, or right-tailed.

Two-tailed Test

H0: =

H1: 

 is divided equally between the two tails of the critical

region

Means less than or greater than

Left-tailed Test

H0: =

H1: <

Points Left

 the left tail

Right-tailed Test

H0: =

H1: >

Points Right

 the Right tail

Conclusions

in Hypothesis Testing

We always test the null hypothesis.

The initial conclusion will always be

one of the following:

1. Reject the null hypothesis.

2. Fail to reject the null hypothesis.

P-value method:

Using the significance level :

If P-value   , reject H0.

If P-value >  , fail to reject H0.

Decision Criterion

Traditional method:

If the test statistic falls within the

critical region, reject H0.

If the test statistic does not fall

within the critical region, fail to

reject H0.

Decision Criterion

Another option:

Instead of using a significance

level such as 0.05, simply identify

the P-value and leave the decision

to the reader.

Decision Criterion

Confidence Intervals:

A confidence interval estimate of a

population parameter contains the

likely values of that parameter.

If a confidence interval does not

include a claimed value of a

population parameter, reject that

claim.

Wording of Final Conclusion

Figure 8-7

Caution

Never conclude a hypothesis test with a

statement of “reject the null hypothesis”

or “fail to reject the null hypothesis.”

Always make sense of the conclusion

with a statement that uses simple

nontechnical wording that addresses the

original claim.

Accept Versus Fail to Reject

• Some texts use “accept the null hypothesis.”

• We are not proving the null hypothesis.

• Fail to reject says more correctly

• The available evidence is not strong enough to warrant rejection of the null hypothesis (such as not enough evidence to convict a suspect).

Type I Error

• A Type I error is the mistake of

rejecting the null hypothesis when it

is actually true.

• The symbol  (alpha) is used to

represent the probability of a type I

error.

Type II Error

• A Type II error is the mistake of failing

to reject the null hypothesis when it is

actually false.

• The symbol  (beta) is used to

represent the probability of a type II

error.

Type I and Type II Errors

Example:

a) Identify a type I error.

b) Identify a type II error.

Assume that we are conducting a hypothesis

test of the claim that a method of gender

selection increases the likelihood of a baby

girl, so that the probability of a baby girls is p >

0.5. Here are the null and alternative

hypotheses: H0: p = 0.5, and H1: p > 0.5.

Example:

a) A type I error is the mistake of rejecting a

true null hypothesis, so this is a type I error:

Conclude that there is sufficient evidence to

support p > 0.5, when in reality p = 0.5.

b) A type II error is the mistake of failing to

reject the null hypothesis when it is false, so

this is a type II error: Fail to reject p = 0.5

(and therefore fail to support p > 0.5) when in

reality p > 0.5.

Controlling Type I and

Type II Errors

• For any fixed , an increase in the sample size n will cause a decrease in 

• For any fixed sample size n, a decrease in  will cause an increase in . Conversely, an increase in  will cause a decrease in .

• To decrease both  and , increase the sample size.

Comprehensive

Hypothesis Test –

P-Value Method

Comprehensive

Hypothesis Test –

Traditional Method

Comprehensive

Hypothesis Test - cont

A confidence interval estimate of a population

parameter contains the likely values of that

parameter. We should therefore reject a claim

that the population parameter has a value that

is not included in the confidence interval.

In some cases, a conclusion based on a

confidence interval may be different

from a conclusion based on a

hypothesis test. See the comments in

the individual sections.

Caution

Part 2:

Beyond the Basics of Hypothesis Testing:

The Power of a Test

Definition

The power of a hypothesis test is the

probability (1 –  ) of rejecting a false null

hypothesis. The value of the power is

computed by using a particular significance

level  and a particular value of the

population parameter that is an alternative to

the value assumed true in the null hypothesis.

That is, the power of the hypothesis test is the

probability of supporting an alternative

hypothesis that is true.

Power and the

Design of Experiments

Just as 0.05 is a common choice for a significance

level, a power of at least 0.80 is a common

requirement for determining that a hypothesis test is

effective. (Some statisticians argue that the power

should be higher, such as 0.85 or 0.90.) When

designing an experiment, we might consider how

much of a difference between the claimed value of a

parameter and its true value is an important amount

of difference. When designing an experiment, a goal

of having a power value of at least 0.80 can often be

used to determine the minimum required sample

size.

Recap

In this section we have discussed:

❖ Null and alternative hypotheses.

❖ Test statistics.

❖ Significance levels.

❖ P-values.

❖ Decision criteria.

❖ Type I and II errors.

❖ Power of a hypothesis test.

Section 8-3

Testing a Claim About a

Proportion

Key Concept

This section presents complete procedures

for testing a hypothesis (or claim) made about

a population proportion.

This section uses the components introduced

in the previous section for the P-value

method, the traditional method or the use of

confidence intervals.

Key Concept

Two common methods for testing a claim

about a population proportion are (1) to use a

normal distribution as an approximation to

the binomial distribution, and (2) to use an

exact method based on the binomial

probability distribution.

Part 1 of this section uses the approximate

method with the normal distribution, and Part

2 of this section briefly describes the exact

method.

Part 1:

Basic Methods of Testing Claims about a Population Proportion p

Notation

p = population proportion (used in the

null hypothesis)

q = 1 – p



n = number of trials

p = (sample proportion) x n

1) The sample observations are a simple random sample.

2) The conditions for a binomial distribution are satisfied.

3) The conditions np  5 and nq  5 are both satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution with µ = np and  = npq . Note: p is the assumed proportion not the sample proportion.

Requirements for Testing Claims

About a Population Proportion p

p – p

pq n

z =



Test Statistic for Testing

a Claim About a Proportion

P-values:

Critical Values:

Use the standard normal

distribution (Table A-2) and refer to

Figure 8-5

Use the standard normal

distribution (Table A-2).

Caution

Don’t confuse a P-value with a proportion p.

P-value = probability of getting a test

statistic at least as extreme as

the one representing sample

data

p = population proportion

P-Value Method:

Use the same method as described

in Section 8-2 and in Figure 8-8.

Use the standard normal

distribution (Table A-2).

Traditional Method

Use the same method as described

in Section 8-2 and in Figure 8-9.

Confidence Interval Method

Use the same method as described

in Section 8-2 and in Table 8-2.

CAUTION When testing claims about a population proportion,

the traditional method and the P-value method are

equivalent and will yield the same result since they

use the same standard deviation based on the claimed

proportion p.

However, the confidence interval uses an estimated

standard deviation based upon the sample proportion p.

Consequently, it is possible that the traditional and P-

value methods may yield a different conclusion than the

confidence interval method.

A good strategy is to use a confidence interval to

estimate a population proportion, but use the P-value

or traditional method for testing a claim about the

proportion.



Example:

The text refers to a study in which 57 out of

104 pregnant women correctly guessed the

sex of their babies.

Use these sample data to test the claim that

the success rate of such guesses is the 50%

success rate expected with random chance

guesses.

Use a 0.05 significance level.

Example:

Requirements are satisfied: simple random

sample; fixed number of trials (104) with two

categories (guess correctly or do not); np =

(104)(0.5) = 52 ≥ 5 and nq = (104)(0.5) = 52 ≥ 5

Step 1: original claim is that the success rate

is no different from 50%: p = 0.50

Step 2: opposite of original claim is p ≠ 0.50

Step 3: p ≠ 0.50 does not contain equality so

it is H1.

H0: p = 0.50 null hypothesis and original claim

H1: p ≠ 0.50 alternative hypothesis

Example:

Step 4: significance level is  = 0.05

Step 5: sample involves proportion so the

relevant statistic is the sample

proportion,

Step 6: calculate z:

p̂

z = p̂ − p

104 − 0.50

0.50( ) 0.50( ) 104

= 0.98

two-tailed test, P-value is twice the

area to the right of test statistic

Example:

Table A-2: z = 0.98 has an area of 0.8365 to its

left, so area to the right is 1 – 0.8365 = 0.1635,

doubles yields 0.3270 (technology provides a

more accurate P-value of 0.3268

Step 7: the P-value of 0.3270 is greater than

the significance level of 0.05, so fail to

reject the null hypothesis

Here is the correct conclusion: There is not

sufficient evidence to warrant rejection of the

claim that women who guess the sex of their

babies have a success rate equal to 50%.

(determining the sample proportion of households with cable TV)

p = = = 0.64 x n

(96+54)



and 54 do not” is calculated using

p sometimes must be calculated “96 surveyed households have cable TV



 p sometimes is given directly

“10% of the observed sports cars are red” is expressed as

p = 0.10 

Obtaining P 

Part 2:

Exact Method for Testing Claims about a Proportion p

Testing Claims We can get exact results by using the binomial probability distribution.

Binomial probabilities are a nuisance to calculate manually, but technology makes this approach quite simple.

Also, this exact approach does not require that np ≥ 5 and nq ≥ 5 so we have a method that applies when that requirement is not satisfied.

To test hypotheses using the exact binomial distribution, use the binomial probability distribution with the P-value method, use the value of p assumed in the null hypothesis, and find P- values as follows:

Testing Claims

Left-tailed test:

The P-value is the probability of getting x or fewer successes among n trials.

Right-tailed test:

The P-value is the probability of getting x or more successes among n trials.

p p̂

Testing Claims

Two-tailed test:

the P-value is twice the probability of getting x or more successes

the P-value is twice the probability of getting x or fewer successes

If p̂  p,

If p̂  p,

p p ^

Recap

In this section we have discussed:

❖ Test statistics for claims about a proportion.

❖ P-value method.

❖ Confidence interval method.

❖ Obtaining p. 

Section 8-4

Testing a Claim About a

Mean:  Known

Key Concept

This section presents methods for testing a

claim about a population mean, given that the

population standard deviation is a known

value.

This section uses the normal distribution with

the same components of hypothesis tests

that were introduced in Section 8-2.

Notation

n = sample size

= sample mean

= population mean of all sample means from samples of size n

 = known value of the population standard deviation

x

Requirements for Testing Claims About

a Population Mean (with  Known)

1) The sample is a simple random sample.

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.

Test Statistic for Testing a Claim About a Mean (with  Known)

x – µxz = 

Example: People have died in boat accidents because an

obsolete estimate of the mean weight of men was

used.

Using the weights of the simple random sample of

men, we obtain these sample statistics:

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.

Use these results to test the claim that men have a

mean weight greater than 166.3 lb, which was the

weight in the National Transportation and Safety

Board’s recommendation.

Use a 0.05 significance level, and use the P-value

method outlined in Figure 8-8.

Example:

Requirements are satisfied: simple random

sample,  is known (26 lb), sample size is 40

(n > 30)

Step 1: Express claim as  > 166.3 lb

Step 2: alternative to claim is  ≤ 166.3 lb

Step 3:  > 166.3 lb does not contain equality,

it is the alternative hypothesis:

H0:  = 166.3 lb null hypothesis

H1:  > 166.3 lb alternative hypothesis and

original claim

Example:

Step 4: significance level is  = 0.05

Step 5: claim is about the population mean,

so the relevant statistic is the sample

mean (172.55 lb),  is known (26 lb),

sample size greater than 30

Step 6: calculate z

z = x − 



= 172.55 − 166.3

= 1.52

right-tailed test, so P-value is the area

to the right of z = 1.52;

Example:

Table A-2: area to the left of z = 1.52

is 0.9357, so the area to the right is

1 – 0.9357 = 0.0643.

The P-value is 0.0643

Step 7: The P-value of 0.0643 is greater than

the significance level of  = 0.05, we

fail to reject the null hypothesis.

x = 172.55

 = 166.3 or

z = 0 or z = 1.52

P-value = 0.0643

Example:

The P-value of 0.0643 tells us that if men have a

mean weight given by  = 166.3 lb, there is a

good chance (0.0643) of getting a sample mean

of 172.55 lb.

A sample mean such as 172.55 lb could easily

occur by chance.

There is not sufficient evidence to support a

conclusion that the population mean is greater

than 166.3 lb, as in the National Transportation

and Safety Board’s recommendation.

Example:

The traditional method: Use z = 1.645 instead of

finding the P-value. Since z = 1.52 does not fall

in the critical region, again fail to reject the null

hypothesis.

Confidence Interval method: Use a one-tailed

test with  = 0.05, so construct a 90%

confidence interval:

165.8 <  < 179.3

The confidence interval contains 166.3 lb, we

cannot support a claim that  is greater than

166.3. Again, fail to reject the null hypothesis.

Underlying Rationale of

Hypothesis Testing

If, under a given assumption, there is an

extremely small probability of getting sample

results at least as extreme as the results that

were obtained, we conclude that the

assumption is probably not correct.

When testing a claim, we make an

assumption (null hypothesis) of equality. We

then compare the assumption and the

sample results and we form one of the

following conclusions:

• If the sample results (or more extreme results)

can easily occur when the assumption (null

hypothesis) is true, we attribute the relatively

small discrepancy between the assumption and

the sample results to chance.

• If the sample results cannot easily occur when

that assumption (null hypothesis) is true, we

explain the relatively large discrepancy between

the assumption and the sample results by

concluding that the assumption is not true, so

we reject the assumption.

Underlying Rationale of

Hypotheses Testing - cont

Recap

In this section we have discussed:

❖ Requirements for testing claims about

population means, σ known.

❖ P-value method.

❖ Traditional method.

❖ Confidence interval method.

❖ Rationale for hypothesis testing.

Section 8-5

Testing a Claim About a

Mean:  Not Known

Key Concept

This section presents methods for testing a

claim about a population mean when we do

not know the value of σ.

The methods of this section use the Student t

distribution introduced earlier.

Notation

n = sample size

= sample mean

= population mean of all sample means from samples of size n

x

Requirements for Testing Claims

About a Population

Mean (with  Not Known)

1) The sample is a simple random sample.

2) The value of the population standard deviation  is not known.

3) Either or both of these conditions is satisfied: The population is normally distributed or n > 30.

Test Statistic for Testing a Claim About a Mean (with  Not Known)

P-values and Critical Values ❖Found in Table A-3

❖Degrees of freedom (df) = n – 1

x – µxt = s n

Important Properties of the

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

sample sizes (see Figure 7-5 in Section 7-4).

2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected when s is used to estimate  .

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  = 1).

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

Choosing between the Normal and Student t Distributions when Testing a

Claim about a Population Mean µ

Use the Student t distribution when  is

not known and either or both of these

conditions is satisfied:

The population is normally distributed or

n > 30.

Example: People have died in boat accidents because an

obsolete estimate of the mean weight of men was

used.

Using the weights of the simple random sample

of men from Data Set 1 in Appendix B, we obtain

these sample statistics: n = 40 and = 172.55 lb,

and s= 26.33 lb. Do not assume that the value of 

is known.

Use these results to test the claim that men have a

mean weight greater than 166.3 lb, which was the

weight in the National Transportation and Safety

Board’s recommendation M-04-04.

Use a 0.05 significance level, and the traditional

method outlined in Figure 8-9.

Example:

Requirements are satisfied: simple random

sample, population standard deviation is not

known, sample size is 40 (n > 30)

Step 1: Express claim as  > 166.3 lb

Step 2: alternative to claim is  ≤ 166.3 lb

Step 3:  > 166.3 lb does not contain equality,

it is the alternative hypothesis:

H0:  = 166.3 lb null hypothesis

H1:  > 166.3 lb alternative hypothesis and

original claim

Example:

Step 4: significance level is  = 0.05

Step 5: claim is about the population mean,

so the relevant statistic is the sample

mean, 172.55 lb

Step 6: calculate t

t = x − 

= 172.55 − 166.3

26.33

= 1.501

df = n – 1 = 39, area of 0.05, one-tail

yields t = 1.685;

Example:

Step 7: t = 1.501 does not fall in the critical

region bounded by t = 1.685, we fail

to reject the null hypothesis.

 = 166.3 or

z = 0 x = 172.55

or t = 1.52

Critical value t = 1.685

Example:

Because we fail to reject the null hypothesis, we

conclude that there is not sufficient evidence to

support a conclusion that the population mean

is greater than 166.3 lb, as in the National

Transportation and Safety Board’s

recommendation.

The critical value in the preceding example

was t = 1.782, but if the normal distribution

were being used, the critical value would have

been z = 1.645.

The Student t critical value is larger (farther to

the right), showing that with the Student t

distribution, the sample evidence must be

more extreme before we can consider it to be

significant.

Normal Distribution Versus

Student t Distribution

P-Value Method

❖ Use software or a TI-83/84 Plus calculator.

❖ If technology is not available, use Table A-3 to identify a range of P-values.

a) In a left-tailed hypothesis test, the sample size

is n = 12, and the test statistic is t = –2.007.

b) In a right-tailed hypothesis test, the sample size

is n = 12, and the test statistic is t = 1.222.

c) In a two-tailed hypothesis test, the sample size

is n = 12, and the test statistic is t = –3.456.

Example: Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the P-value corresponding to the given results.

a) The test is a left-tailed test with test

statistic t = –2.007, so the P-value is the

area to the left of –2.007. Because of the

symmetry of the t distribution, that is the

same as the area to the right of +2.007. Any

test statistic between 2.201 and 1.796 has a

right-tailed P-value that is between 0.025

and 0.05. We conclude that

0.025 < P-value < 0.05.

Example: Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the P-value corresponding to the given results.

b) The test is a right-tailed test with test

statistic t = 1.222, so the P-value is the

area to the right of 1.222. Any test

statistic less than 1.363 has a right-tailed

P-value that is greater than 0.10. We

conclude that P-value > 0.10.

c) The test is a two-tailed test with test statistic

t = –3.456. The P-value is twice the area to

the left of –3.456 (= right of +3.456). Any test

statistic greater than 3.106 has a two-

tailed P-value that is less than 0.01. We

conclude that

P-value < 0.01.

Example: Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the P-value corresponding to the given results.

Recap

In this section we have discussed:

❖ Assumptions for testing claims about

population means, σ unknown.

❖ Student t distribution.

❖ P-value method.