Chapter 8 Hypothesis

Testing

Section 8.1 Introduction to Statistical Tests

It is used to make decisions concerning the

value of a parameter.

Hypothesis testing

It is used to make decisions concerning the value of a

parameter.

Hypothesis testing

Null Hypothesis: H0

A working hypothesis about the population

parameter in question.

The value specified in the null hypothesis is often:

Øa historical value Øa claim Øa production specification

Alternate Hypothesis: H1

any hypothesis that differs from the null

hypothesis

An alternate hypothesis is constructed in such a way that it is the one to be accepted when the

null hypothesis must be rejected.

A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.

A manufacturer claims that their light bulbs burn for an

average of 1000 hours. ...

The null hypothesis (the claim) is that the true average life is

1000 hours.

H0: µ = 1000

… A manufacturer claims that their light bulbs burn for an

average of 1000 hours. We have reason to believe that the bulbs

do not last that long. ...

If we reject the manufacturer’s claim, we must accept the alternate

hypothesis that the light bulbs do not last as long as 1000 hours.

H1: µ < 1000

Type I Error

Rejecting a null hypothesis which

is in fact true

Type II Error

Not rejecting a null hypothesis which

is in fact false

Options in Hypothesis Testing

H0 is

Our Choices:

Do Not Reject Reject

True

False

Type I Error

Correct DecisionType II Error

Correct Decision

Level of Significance, Alpha

The probability with which we are

willing to risk a type I error

α( )

Type II Error➡β = beta = probability of a type II

error (failing to reject a false hypothesis)

➡Α small α is normally is associated with a (relatively) large β, and vice-versa.

➡Choices should be made according to which error is more serious.

Power of the Test 1 – Beta

➡The probability of rejecting H0 when it is in fact false = 1 – β.

➡The power of the test increases as the level of significance (α) increases.

➡Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

Probabilities Associated with a Hypothesis Test

Do Not Reject H0

Reject H0

H0 is True

H0 is False

Type I Error with probability α

Correct Decision with probability

of 1 - α

Type II Error with probability of β

Correct Decision with probability

of 1 - β

Reject or ...➡ When the sample evidence is not

strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis.

➡ Use of the term “accept the null hypothesis” should be avoided.

➡ When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

Fail to Reject H0

There is not enough evidence to reject H0. The null hypothesis is retained but not

proved.

Reject H0 There is enough

evidence to reject H0. Choose the alternate hypothesis with the

understanding that it has NOT been proven.

A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15.

We wish to test the claim with a level of significance of α = 0.01.

… average age of its job applicants is fifteen years. We suspect that the true

age is lower than 15.H0: µ = 15 H1: µ < 15

Describe Type I and Type II errors.

H0: µ = 15 H1: µ < 15 α = 0.01

A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

H0: µ = 15 H1: µ < 15 α= 0.01

A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.

Types of TestsüWhen the alternate hypothesis

contains the “not equal to” symbol ( ≠ ), perform a two-tailed test.

ü When the alternate hypothesis contains the “greater than” symbol ( > ), perform a right-tailed test.

ü When the alternate hypothesis contains the “less than” symbol ( < ), perform a left-tailed test.

Two-Tailed Test

H0: µ = k

H1: µ ≠ k

Two-Tailed Test

If test statistic is in either tail - the critical region - of the distribution,

we REJECT the Null Hypothesis.

H0: µ = k H1: µ ≠ k

If test statistic is at or near the claimed

mean, we DO NOT REJECT

the Null Hypothesis

–z 0 z

Right-Tailed Test

H0: µ = k

H1: µ > k

Right-Tailed Test

If test statistic is in the right tail - the critical region - of the

distribution, weREJECT the Null Hypothesis.

H0: µ = k H1: µ > k

If test statistic is at, near, or

below the claimed

mean, we DO NOT REJECT

the Null Hypothesis

0 z

Left-Tailed Test

H0: µ = k

H1: µ < k

Left-Tailed Test

If test statistic is in the left tail - the critical region - of the distribution,

weREJECT the Null Hypothesis.

H0: µ = k H1: µ < k

If test statistic is at, near, or

above the claimed

mean, we DO NOT REJECT

the Null Hypothesis

-z 0

END OFSECTION

8.1