+

The Practice of Statistics, 4th edition – For AP*

STARNES, YATES, MOORE

Chapter 9: Testing a Claim

Section 9.1

Significance Tests: The Basics

+ Chapter 9

Testing a Claim

9.1 Significance Tests: The Basics

9.2 Tests about a Population Proportion

9.3 Tests about a Population Mean

+ Section 9.1

Significance Tests: The Basics

After this section, you should be able to…

STATE correct hypotheses for a significance test about a population

proportion or mean.

INTERPRET P-values in context.

INTERPRET a Type I error and a Type II error in context, and give

the consequences of each.

DESCRIBE the relationship between the significance level of a test,

P(Type II error), and power.

Learning Objectives

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Introduction

Confidence intervals are one of the two most common types of

statistical inference. Use a confidence interval when your goal is

to estimate a population parameter. The second common type of

inference, called significance tests, has a different goal: to assess

the evidence provided by data about some claim concerning a

population.

A significance test is a formal procedure for comparing observed

data with a claim (also called a hypothesis) whose truth we want

to assess. The claim is a statement about a parameter, like the

population proportion p or the population mean µ. We express the

results of a significance test in terms of a probability that

measures how well the data and the claim agree.

In this chapter, we’ll learn the underlying logic of statistical tests,

how to perform tests about population proportions and population

means, and how tests are connected to confidence intervals.

+ The Reasoning of Significance Tests

Statistical tests deal with claims about a population. Tests ask if sample

data give good evidence against a claim. A test might say, “If we took

many random samples and the claim were true, we would rarely get a

result like this.” To get a numerical measure of how strong the sample evidence is, replace the vague term “rarely” by a probability.

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We can use software to simulate 400 sets of 50 shots

assuming that the player is really an 80% shooter.

The observed statistic is so unlikely if the

actual parameter value is p = 0.80 that it

gives convincing evidence that the player’s

claim is not true.

You can say how strong the evidence

against the player’s claim is by giving the

probability that he would make as few as

32 out of 50 free throws if he really makes

80% in the long run.

Suppose a basketball player claimed to be an 80% free-throw shooter.

To test this claim, we have him attempt 50 free-throws. He makes 32 of

them. His sample proportion of made shots is 32/50 = 0.64.

What can we conclude about the claim based on this sample data?

+ The Reasoning of Significance Tests

Based on the evidence, we might conclude the player’s claim is

incorrect.

In reality, there are two possible explanations for the fact that he made only 64% of his free throws.

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1) The player’s claim is correct (p = 0.8), and by bad luck, a

very unlikely outcome occurred.

2) The population proportion is actually less than 0.8, so the

sample result is not an unlikely outcome.

An outcome that would

rarely happen if a claim

were true is good evidence

that the claim is not true.

Basic Idea

+ Alternate Example – Can you be confident of victory?

Heading into the mayoral election, Jack is feeling fairly confident that he will be

elected by obtaining more than 50% of the vote. Suppose that a random

sample of 100 voters shows that 56 will vote for Jack. Is his confidence

warranted? Maybe. There are actually two possible explanations for why the

majority of voters in the sample seem to favor Jack.

At most 50% of the voters actually support Jack and the favorable sample

proportion was due to sampling variability. If this explanation is plausible, then

Jack should not be confident of victory.

The sample proportion is above 50% because more than 50% of the voters

actually support Jack. Jack should only believe this explanation and be

confident of victory if explanation #1 can safely be ruled out.

We used Fathom software to simulate 400 samples of size 100 from a

population in which exactly 50% of voters support Jack. Each dot represents

the proportion of voters who prefer Jack in the simulated sample.

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SampleProportion

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Collection 1 Dot Plot

In 13.75% of the simulated samples (marked in

red), the sample proportion of voters who

supported Jack was at least 0.56. This means

that it is plausible that at most 50% of the

voters prefer Jack and the favorable sample proportion was due to sampling variability. Since it is plausible that the majority of

voters do not favor Jack, he should not be confident of victory.

+ Stating Hypotheses

A significance test starts with a careful statement of the claims we want to

compare. The first claim is called the null hypothesis. Usually, the null

hypothesis is a statement of “no difference.” The claim we hope or

suspect to be true instead of the null hypothesis is called the alternative

hypothesis.

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Definition:

The claim tested by a statistical test is called the null hypothesis (H0).

The test is designed to assess the strength of the evidence against the

null hypothesis. Often the null hypothesis is a statement of “no

difference.”

The claim about the population that we are trying to find evidence for is the alternative hypothesis (Ha).

In the free-throw shooter example, our hypotheses are

H0 : p = 0.80

Ha : p < 0.80

where p is the long-run proportion of made free throws.

+ Stating Hypotheses

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Definition:

The alternative hypothesis is one-sided if it states that a parameter is

larger than the null hypothesis value or if it states that the parameter is

smaller than the null value.

It is two-sided if it states that the parameter is different from the null hypothesis value (it could be either larger or smaller).

Hypotheses always refer to a population, not to a sample. Be sure

to state H0 and Ha in terms of population parameters.

It is never correct to write a hypothesis about a sample statistic,

such as

ˆ p 0.64 or x 85.

In any significance test, the null hypothesis has the form

H0 : parameter = value

The alternative hypothesis has one of the forms

Ha : parameter < value

Ha : parameter > value

Ha : parameter ≠ value

To determine the correct form of Ha, read the problem carefully.

+ Example: Studying Job Satisfaction

Does the job satisfaction of assembly-line workers differ when their work is machine-

paced rather than self-paced? One study chose 18 subjects at random from a

company with over 200 workers who assembled electronic devices. Half of the

workers were assigned at random to each of two groups. Both groups did similar

assembly work, but one group was allowed to pace themselves while the other

group used an assembly line that moved at a fixed pace. After two weeks, all the

workers took a test of job satisfaction. Then they switched work setups and took

the test again after two more weeks. The response variable is the difference in

satisfaction scores, self-paced minus machine-paced.

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a) Describe the parameter of interest in this setting.

b) State appropriate hypotheses for performing a significance test.

The parameter of interest is the mean µ of the differences (self-paced

minus machine-paced) in job satisfaction scores in the population of all

assembly-line workers at this company.

Because the initial question asked whether job satisfaction differs, the

alternative hypothesis is two-sided; that is, either µ < 0 or µ > 0. For

simplicity, we write this as µ ≠ 0. That is,

H0: µ = 0

Ha: µ ≠ 0

+ Alternate Example – A better golf club?

Mike is an avid golfer who would like to improve his play. A friend suggests

getting new clubs and lets Mike try out his 7-iron. Based on years of

experience, Mike has established that the mean distance that balls travel when

hit with his old 7-iron is = 175 yards with a standard deviation of = 15 yards.

He is hoping that this new club will make his shots with a 7-iron more

consistent (less variable), so he goes to the driving range and hits 50 shots

with the new 7-iron.

Problem:

(a) Describe the parameter of interest in this setting.

Mike is interested in being more consistent, the parameter of interest is the

standard deviation of the distance he hits the ball when using the new 7-iron.

(b) State appropriate hypotheses for performing a significance test.

Because Mike wants to be more consistent, he wants the standard deviation of

the distance he hits the ball to be smaller than 15 yards.

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15:

15:0

aH

H

+ Interpreting P-Values

The null hypothesis H0 states the claim that we are seeking evidence

against. The probability that measures the strength of the evidence

against a null hypothesis is called a P-value.

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Definition:

The probability, computed assuming H0 is true, that the statistic would

take a value as extreme as or more extreme than the one actually

observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against H0 provided by the data.

Small P-values are evidence against H0 because they say that the

observed result is unlikely to occur when H0 is true.

Large P-values fail to give convincing evidence against H0 because

they say that the observed result is likely to occur by chance when H0

is true.

+

Alternate Example – Can you be confident of

victory

In the alternate example on page 530, when we

assumed that exactly 50% of voters supported Jack,

13.75% of the 400 simulated samples gave values of at

least as high as the observed value of = 0.56. In

other words, the P-value is P( ≥ 0.56 | p = 0.5) ≈

0.1375. So, if exactly 50% of the voters in the

population support Jack, there is about a 14% chance

that in a sample of 100 voters, 56% or more would

support Jack. This is not a small probability, so this

does not provide convincing evidence to support the

alternative hypothesis that p > 0.5.

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p̂p̂

+ Example: Studying Job Satisfaction

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a) Explain what it means for the null hypothesis to be true in this setting.

b) Interpret the P-value in context.

In this setting, H0: µ = 0 says that the mean difference in satisfaction

scores (self-paced - machine-paced) for the entire population of

assembly-line workers at the company is 0. If H0 is true, then the workers

don’t favor one work environment over the other, on average.

For the job satisfaction study, the hypotheses are

H0: µ = 0

Ha: µ ≠ 0

Data from the 18 workers gave x 17 and sx 60. That is, these workers rated the

self -paced environment, on average, 17 points higher. Researchers performed a

significance test using the sample data and found a P - value of 0.2302.

The P-value is the probability of observing a sample result as extreme or more

extreme in the direction specified by Ha just by chance when H0 is actually true.

Because the alternative hypothesis is two- sided, the P - value is the probability of

getting a value of x as far from 0 in either direction as the observed x 17 when

H0 is true. That is, an average difference of 17 or more points between the two

work environments would happen 23% of the time just by chance in random

samples of 18 assembly- line workers when the true population mean is = 0.

An outcome that would occur so often just by chance

(almost 1 in every 4 random samples of 18 workers) when

H0 is true is not convincing evidence against H0.

We fail to reject H0: µ = 0.

+ Alternate Example: A better golf club?

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When Mike was testing a new 7-iron, the hypotheses were:

Where σ = the true standard deviation of the distances Mike hits golf balls using the

new 7-iron. Based on 50 shots with the new 7-iron, the standard deviation was

SX = 10.9 yards.

Problem: A significance test using the sample data produced a P-value of 0.002.

(a) Interpret the P-value in this context.

If the true standard deviation is 15 yards, then there is an approximate

probability of 0.002 that the sample standard deviation would be 10.9 yards or

lower because of random chance alone.

(b) Do the data provide convincing evidence against the null hypothesis?

Explain.

Yes. Since the P-value is very small, random chance is not a plausible

explanation for why the sample standard deviation was lower than 15 yards.

Thus, there is convincing evidence that the true standard deviation with the new

7-iron is smaller.

15:

15:0

aH

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+ Statistical Significance

The final step in performing a significance test is to draw a conclusion

about the competing claims you were testing. We will make one of two

decisions based on the strength of the evidence against the null

hypothesis (and in favor of the alternative hypothesis) -- reject H0 or fail

to reject H0.

If our sample result is too unlikely to have happened by chance

assuming H0 is true, then we’ll reject H0.

Otherwise, we will fail to reject H0.

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Note: A fail-to-reject H0 decision in a significance test doesn’t mean

that H0 is true. For that reason, you should never “accept H0” or use

language implying that you believe H0 is true.

In a nutshell, our conclusion in a significance test comes down to

P-value small → reject H0 → conclude Ha (in context)

P-value large → fail to reject H0 → cannot conclude Ha (in context)

+

Alternate Example – Elections and golf clubs

In the mayoral election example, the estimated P-value

was 0.1375. Since the P-value is large, we fail to reject

H0 : p = 0.5 and cannot conclude that more than 50% of

all voters favor Jack.

In the 7-iron example, the estimated P-value was 0.002.

Since the P-value is small, we reject H0 : = 15 and

conclude that Mike is more consistent when using the

new 7-iron (i.e. the true standard deviation of distances

when using the new 7-iron is less than 15 yards).

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+ Statistical Significance

There is no rule for how small a P-value we should require in order to reject

H0 — it’s a matter of judgment and depends on the specific

circumstances. But we can compare the P-value with a fixed value that we regard as decisive, called the significance level. We write it as α,

the Greek letter alpha. When our P-value is less than the chosen α, we

say that the result is statistically significant.

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Definition:

If the P-value is smaller than alpha, we say that the data are statistically significant at level α. In that case, we reject the null

hypothesis H0 and conclude that there is convincing evidence in favor of the alternative hypothesis Ha.

When we use a fixed level of significance to draw a conclusion in a

significance test,

P-value < α → reject H0 → conclude Ha (in context)

P-value ≥ α → fail to reject H0 → cannot conclude Ha (in context)

+ Example: Better Batteries

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a) What conclusion can you make for the significance level α = 0.05?

b) What conclusion can you make for the significance level α = 0.01?

Since the P-value, 0.0276, is less than α = 0.05, the sample result is

statistically significant at the 5% level. We have sufficient evidence to

reject H0 and conclude that the company’s deluxe AAA batteries last

longer than 30 hours, on average.

A company has developed a new deluxe AAA battery that is supposed to last longer

than its regular AAA battery. However, these new batteries are more expensive to

produce, so the company would like to be convinced that they really do last longer.

Based on years of experience, the company knows that its regular AAA batteries last

for 30 hours of continuous use, on average. The company selects an SRS of 15 new

batteries and uses them continuously until they are completely drained. A significance

test is performed using the hypotheses

H0 : µ = 30 hours

Ha : µ > 30 hours

where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P-

value is 0.0276.

Since the P-value, 0.0276, is greater than α = 0.01, the sample result is

not statistically significant at the 1% level. We do not have enough

evidence to reject H0 in this case. therefore, we cannot conclude that the

deluxe AAA batteries last longer than 30 hours, on average.

+ Alternate Example: Tasty chips

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a) What conclusion would you make for the significance level α = 0.01?

b) What conclusion can you make for the significance level α = 0.001?

Since the P-value is less than σ (0.0055 < 0.01), we reject H0 and

conclude that students at Zenon’s school prefer name-brand chips.

For his second semester project in AP Statistics, Zenon decided to

investigate if students at his school prefer name-brand potato chips to

generic potato chips. He randomly selected 50 students and had each

student try both types of chips, in random order. Overall, 34 of the 50

students preferred the name-brand chips. Zenon performed a significance

test using the hypotheses:

H0 : p = 0.5

Ha : p > 0.5

Where p is the true proportion of students at his school that prefer name-

brand chips. The resulting P-value was 0.0055.

Since the P-value is greater than σ (0.0055 > 0.001), we fail to reject H0

and cannot conclude that students at Zenon’s school prefer name-brand

chips.

+ Type I and Type II Errors

When we draw a conclusion from a significance test, we hope our

conclusion will be correct. But sometimes it will be wrong. There are two

types of mistakes we can make. We can reject the null hypothesis when

it’s actually true, known as a Type I error, or we can fail to reject a false

null hypothesis, which is a Type II error.

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Definition:

If we reject H0 when H0 is true, we have committed a Type I error.

If we fail to reject H0 when H0 is false, we have committed a Type II error.

Truth about the population

H0 true H0 false

(Ha true)

Conclusion

based on

sample

Reject H0 Type I error Correct

conclusion

Fail to reject

H0

Correct

conclusion Type II error

+ Example: Perfect Potatoes

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Describe a Type I and a Type II error in this setting, and explain the

consequences of each.

• A Type I error would occur if the producer concludes that the proportion of

potatoes with blemishes is greater than 0.08 when the actual proportion is

0.08 (or less). Consequence: The potato-chip producer sends the truckload

of acceptable potatoes away, which may result in lost revenue for the

supplier.

• A Type II error would occur if the producer does not send the truck away

when more than 8% of the potatoes in the shipment have blemishes.

Consequence: The producer uses the truckload of potatoes to make potato

chips. More chips will be made with blemished potatoes, which may upset

consumers.

A potato chip producer and its main supplier agree that each shipment of potatoes

must meet certain quality standards. If the producer determines that more than 8% of

the potatoes in the shipment have “blemishes,” the truck will be sent away to get

another load of potatoes from the supplier. Otherwise, the entire truckload will be

used to make potato chips. To make the decision, a supervisor will inspect a random

sample of potatoes from the shipment. The producer will then perform a significance

test using the hypotheses

H0 : p = 0.08

Ha : p > 0.08

where p is the actual proportion of potatoes with blemishes in a given truckload.

+

Alternate Example: Faster fast food? S

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Describe a Type I and a Type II error in this setting, and explain the

consequences of each.

• A Type I error would occur if the manager decides that the true proportion of

drive-through customers that have to wait at least 2 minutes has been

reduced, when it really hasn’t been reduced. A consequence is that the

manager will have to pay unnecessarily for an additional employee.

• A Type II error would occur if the manager didn’t decide that the true

proportion of customers that had to wait at least 2 minutes had been reduced,

when it really had been reduced. A consequence is that the restaurant would

not have an additional employee helping with the drive-through, so they aren’t

providing faster service when they could.

The manager of a fast-food restaurant want to reduce the proportion of drive-through

customers who have to wait more than 2 minutes to receive their food once their

order is placed. Based on store records, the proportion of customers who had to wait

at least 2 minutes was p = 0.63. To reduce this proportion, the manager assigns an

additional employee to assist with drive-through orders. During the next month the

manager will collect a random sample of drive-through times and test the following

hypotheses:

H0 : p = 0.63

Ha : p < 0.63

where p is the true proportion of drive-through customers who have to wait more than

2 minutes after their order is placed to receive their food.

+ Error Probabilities

We can assess the performance of a significance test by looking at the

probabilities of the two types of error. That’s because statistical inference

is based on asking, “What would happen if I did this many times?”

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For the truckload of potatoes in the previous example, we were testing

H0 : p = 0.08

Ha : p > 0.08

where p is the actual proportion of potatoes with blemishes. Suppose that the

potato-chip producer decides to carry out this test based on a random sample of

500 potatoes using a 5% significance level (α = 0.05).

Assuming H0 : p 0.08 is true, the sampling distribution of ̂ p will have :

Shape : Approximately Normal because 500(0.08)= 40 and

500(0.92)= 460 are both at least 10.

Center : ̂p p 0.08

Spread: ̂p p(1 p)

n0.08(0.92)

500 0.0121

The shaded area in the right tail is 5%.

Sample proportion values to the right of

the green line at 0.0999 will cause us to

reject H0 even though H0 is true. This will

happen in 5% of all possible samples.

That is, P(making a Type I error) = 0.05.

+ Alternate Example – Faster fast food?

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For the fast food alternate example, we were testing the following hypotheses:

H0 : p = 0..63

Ha : p < 0.63

where p = the true proportion of drive-through customers who have to wait

more than 2 minutes after their order is placed to receive their food. Suppose

that the manager decided to carry out this test using a random sample of 250

orders and a significance level of σ = 0.10. What is the probability of a

making a Type I error?

p

P(Type I error) =α = 0.10

0.63

Values of

Reject Ho Fail to Reject H0

To make a Type I error means

that we reject H0 when H0 is

actually true. In this case, a Type

I error occurs when the true

proportion of customers that

have to wait at least 2 minutes

remains p = 0.63, but we get a

value of small enough that the

P-value is less than 0.10. When

H0 is true, this will happen 10%

of the time. In other words,

P(Type I error) = σ. p

+ Error Probabilities

The probability of a Type I error is the probability of rejecting H0 when it is

really true. As we can see from the previous example, this is exactly the

significance level of the test.

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The significance level α of any fixed level test is the probability of a Type I

error. That is, α is the probability that the test will reject the null

hypothesis H0 when H0 is in fact true. Consider the consequences of a

Type I error before choosing a significance level.

Significance and Type I Error

What about Type II errors? A significance test makes a Type II error when it

fails to reject a null hypothesis that really is false. There are many values of

the parameter that satisfy the alternative hypothesis, so we concentrate on

one value. We can calculate the probability that a test does reject H0 when

an alternative is true. This probability is called the power of the test against

that specific alternative.

Definition:

The power of a test against a specific alternative is the probability that

the test will reject H0 at a chosen significance level α when the specified alternative value of the parameter is true.

+ Error Probabilities

The potato-chip producer wonders whether the significance test of H0 : p = 0.08

versus Ha : p > 0.08 based on a random sample of 500 potatoes has enough

power to detect a shipment with, say, 11% blemished potatoes. In this case, a

particular Type II error is to fail to reject H0 : p = 0.08 when p = 0.11.

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Earlier, we decided to reject

H0 at α = 0.05 if our sample

yielded a sample proportion

to the right of the green line.

What if p = 0.11?

Since we reject H0 at α= 0.05

if our sample yields a

proportion > 0.0999, we’d

correctly reject the shipment

about 75% of the time.

The power of a test against any alternative is

1 minus the probability of a Type II error

for that alternative; that is, power = 1 - β.

Power and Type II Error

( ˆ p 0.0999)

+ Planning Studies: The Power of a Statistical Test

How large a sample should we take when we plan to carry out a

significance test? The answer depends on what alternative values of the

parameter are important to detect.

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Here are the questions we must answer to decide how many

observations we need:

1.Significance level. How much protection do we want against a

Type I error — getting a significant result from our sample when

H0 is actually true?

2.Practical importance. How large a difference between the

hypothesized parameter value and the actual parameter value

is important in practice?

3.Power. How confident do we want to be that our study will

detect a difference of the size we think is important?

Summary of influences on the question “How many

observations do I need?”

•If you insist on a smaller significance level (such as 1% rather

than 5%), you have to take a larger sample. A smaller

significance level requires stronger evidence to reject the null

hypothesis.

• If you insist on higher power (such as 99% rather than 90%),

you will need a larger sample. Higher power gives a better

chance of detecting a difference when it is really there.

• At any significance level and desired power, detecting a small

difference requires a larger sample than detecting a large

difference.

+ Section 9.1

Significance Tests: The Basics

In this section, we learned that…

A significance test assesses the evidence provided by data against a null hypothesis H0 in favor of an alternative hypothesis Ha.

The hypotheses are stated in terms of population parameters. Often, H0 is a statement of no change or no difference. Ha says that a parameter differs from its null hypothesis value in a specific direction (one-sided alternative) or in either direction (two-sided alternative).

The reasoning of a significance test is as follows. Suppose that the null hypothesis is true. If we repeated our data production many times, would we often get data as inconsistent with H0 as the data we actually have? If the data are unlikely when H0 is true, they provide evidence against H0 .

The P-value of a test is the probability, computed supposing H0 to be true, that the statistic will take a value at least as extreme as that actually observed in the direction specified by Ha .

Summary

+ Section 9.1

Significance Tests: The Basics

Small P-values indicate strong evidence against H0 . To calculate a P-value,

we must know the sampling distribution of the test statistic when H0 is true.

There is no universal rule for how small a P-value in a significance test

provides convincing evidence against the null hypothesis.

If the P-value is smaller than a specified value α (called the significance

level), the data are statistically significant at level α. In that case, we can

reject H0 . If the P-value is greater than or equal to α, we fail to reject H0 .

A Type I error occurs if we reject H0 when it is in fact true. A Type II error

occurs if we fail to reject H0 when it is actually false. In a fixed level α

significance test, the probability of a Type I error is the significance level α.

The power of a significance test against a specific alternative is the

probability that the test will reject H0 when the alternative is true. Power

measures the ability of the test to detect an alternative value of the

parameter. For a specific alternative, P(Type II error) = 1 - power.

Summary

+ Looking Ahead…

We’ll learn how to test a claim about a population proportion.

We’ll learn about

Carrying out a significance test

The one-sample z test for a proportion

Two-sided tests

Why confidence intervals give more information

than significance tests

In the next Section…