Hypothesis testing is used to make decisions concerning the value of a parameter.

Null Hypothesis: H0

a working hypothesis about the population parameter in

question

The value specified in the null hypothesis is often:

• a historical value

• a claim

• a 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: = 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: < 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

Do Not Reject Reject

True

False

H0 is

Our Choices:

Errors in Hypothesis TestingErrors in Hypothesis Testing

Do Not Reject Reject

True

False

H0 is

Our Choices:

Type I error

Errors in Hypothesis TestingErrors in Hypothesis Testing

Do Not Reject Reject

True

False

H0 is

Our Choices:

Type I error

Type II error

Errors in Hypothesis TestingErrors in Hypothesis Testing

Do Not Reject Reject

True

False

H0 is

Our Choices:

Correct decision

Type I error

Type II error

Correct decision

Level of Significance, Alpha ()

the probability with which we are willing to risk a type I error

Type II Error = beta =probability of a type II error (failing to reject a false hypothesis)

small is normally is associated with a (relatively) large , 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 – .

• The power of the test increases as the level of significance () 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

Our Decision

Do not reject H0 Reject H0

H0 is true

H0 is false

Probabilities Associated with a Hypothesis Test

Our Decision

Do not reject H0 Reject H0

H0 is true Correct decisionwith probability

1 - H0 is false

Probabilities Associated with a Hypothesis Test

Our Decision

Do not reject H0 Reject H0

H0 is true Correct decisionwith probability

1 -

Type I errorwith probability

H0 is false

Probabilities Associated with a Hypothesis Test

Our Decision

Do not reject H0 Reject H0

H0 is true Correct decisionwith probability

1 -

Type I errorwith probability

H0 is false Type II error

with probability

Probabilities Associated with a Hypothesis Test

Our Decision

Do not reject H0 Reject H0

H0 is true Correct decisionwith probability

1 -

Type I errorwith probability

H0 is false Type II error

with probability

Correct decisionwith probability

1 -

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 has not been proven.

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 =

0.01,

… average age of its job applicants is fifteen years. We

suspect that the true age is lower than 15.

H0: = 15

H1: < 15

Describe Type I and Type II errors.

H0: = 15 H1: < 15 = 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: = 15 H1: < 15

= 0.01A 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 Tests

• When the alternate hypothesis contains the “not equal to” symbol ( ), perform a two-tailed test..

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

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

Two-Tailed Test

H0: = k

H1: k

Two-Tailed Test

– z 0 z

If test statistic is at or near the

claimed mean, we

do not reject the Null

Hypothesis

If test statistic is in either taileither tail - the critical region -

of the distribution, we reject the Null Hypothesis.

H0: = k

H1: k

Right-Tailed TestRight-Tailed Test

H0: = k

H1: > k

Right-Tailed TestRight-Tailed Test

0 z

If test statistic is at, near, or below the

claimed mean, we do not reject the Null

Hypothesis

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

the distribution, we reject the Null Hypothesis.

H0: = k

H1: > k

Left-Tailed TestLeft-Tailed Test

z 0

If test statistic is at, near, or above the

claimed mean, we do not reject the Null

Hypothesis

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

distribution, we reject the Null Hypothesis.

H0: = k

H1: < k