One-Sample Tests of Hypothesis

Chapter 10

LEARNING OBJECTIVES LO 10-1 Define a hypothesis.

LO 10-2 Explain the five-step hypothesis-testing procedure.

LO 10-3 Define Type I and Type II errors.

LO 10-4 Define the term test statistic and explain how it is used.

LO 10-5 Distinguish between a one-tailed and a two-tailed hypothesis.

LO 10-6 Conduct a test of hypothesis about a population mean.

LO 10-7 Compute and interpret a p-value.

LO 10-8 Conduct a test of hypothesis about a population proportion.

10-2

Hypothesis and Hypothesis Testing

LO 10-1 Define a hypothesis. LO 10-2 Explain the five-step hypothesis-testing procedure.

10-3

In the legal system, a person is presumed innocent until proven guilty.

A jury hypothesized that a person is innocent and then reviews the evidence to

assess if there is enough evidence to claim that the person is not innocent or guilty.

→ Like this, in statistical analysis, we make a claim-that is, state a hypothesis-

collect data and then use the data to test the claim.

HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing.

Hypothesis and Hypothesis Testing

10-4

test a statement to determine whether the sample does or does not support the statement concerning the population

LO 10-1

A hypothesis is a claim (assumption) about a population parameter:

• population mean

Example: The mean monthly cell phone bill in this city is μ = $42

• population proportion

Example: The proportion of adults in this city who own I-phones is π = 0.02

Hypothesis and Hypothesis Testing

HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.

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LO 10-1

The Null and Alternate Hypotheses

ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.

NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence.

LO 10-2

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• The null hypothesis (H0) will always state that the parameter equals the value • The alternative hypothesis (H1) will challenge the claim

The Null and Alternate Hypotheses

LO 10-2

10-7

•A criminal trial is an example of hypothesis testing without the

statistics.

• In a trial a jury must decide between two hypotheses. The null

hypothesis is

H0: The defendant is innocent

• The alternative hypothesis or research hypothesis is

H1: The defendant is guilty

•To begin with, the person is assumed innocent.

•The jury must make a decision on the basis of evidence presented.

•The prosecutor presents evidence, trying to convince the jury to reject the

original assumption of innocence, and conclude that the person is guilty.

LO 10-2

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Example: The average number of TV sets in U.S. homes is equal to three

( H0 : μ = 3)

The Null and Alternate Hypotheses

Null Hypothesis, H0 States the claim or assertion to be tested

Alternate Hypothesis, H1 Is the opposite of the null hypothesis

Example: The average number of TV sets in U.S. homes is not equal to 3

( H1: μ ≠ 3 )

→ Challenges the null hypothesis

The Null Hypotheses

LO 10-2

10-9

• States the claim or assertion to be tested

Example: The average number of TV sets in U.S.

homes is equal to three ( H0 : μ = 3)

• Is always about a population parameter, not about a

sample statistic

Begin with the assumption that the null hypothesis is true.

Similar to the notion of innocent until proven guilty

Important Things to Remember about H0 and H1

H0: null hypothesis and H1: alternate hypothesis.

H0 and H1 are mutually exclusive and collectively exhaustive.

H0 is always presumed to be true.

H1 has the burden of proof.

A random sample (n) is used to “reject H0”.

If we conclude “do not reject H0”, this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H0; rejecting the null hypothesis then suggests that the alternative hypothesis may be true.

Equality is always part of H0 (e.g. “=” , “≥” , “≤”).

“≠”, “<”, and “>” are always part of H1.

LO 10-2

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The Null and Alternate Hypotheses

LO 10-2

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The critical concepts are these:

1. There are two hypotheses, the null (H0) and the alternative

hypotheses (H1).

2. The procedure begins with the assumption that the null

hypothesis is true.

3. The goal is to determine whether there is enough evidence

to infer that the alternative hypothesis is true.

4. There are two possible decisions/conclusions: • There is enough evidence to support the alternative hypothesis.

• There is not enough evidence to support the alternative hypothesis.

The Null and Alternate Hypotheses

LO 10-2

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• There are two possible decisions that can be made:

→ Conclude that there is enough evidence to support the alternative hypothesis (reject the null hypothesis in favor of the alternative) → Conclude that there is not enough evidence to support the alternative hypothesis (do not reject the null hypothesis in favor of the alternative)

NOTE: we do not say that we accept the null hypothesis…

Decisions and Errors in Hypothesis Testing

LO 10-3 Define the Type I and Type II errors.

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• There are two possible errors.

• A Type I error occurs when we reject a true null hypothesis.

That is, a Type I error occurs when the jury convicts an innocent

person. • A Type II error occurs when we fail to reject a false null

hypothesis. That occurs when a guilty defendant is acquitted.

Type I and Type II errors cannot happen at the same time

• A Type I error can only occur if H0 is true

- Type I error: Reject a true null hypothesis

• A Type II error can only occur if H0 is false

- Type II error: Fail to reject a false null hypothesis

Decisions and Errors in Hypothesis Testing

Type I Error

Defined as the probability of rejecting the null hypothesis when it is actually true.

This is denoted by the Greek letter .

Also known as the significance level of a test.

set by researcher in advance

Type II Error

Defined as the probability of failing to reject the null hypothesis when it is actually false.

This is denoted by the Greek letter β.

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LO 10-3

Decisions and Errors in Hypothesis Testing

The confidence coefficient (1‐α) is the probability of not rejecting H0

when it is true.

The confidence level of a hypothesis test is (1‐α)*100%.

The power of a statistical test (1‐β) is the probability of rejecting H0

when it is false.

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LO 10-3

※ The two probabilities, α and β, are inversely related.

Decreasing one increases the other.