CHAPTER 9

STATISTICAL HYPOTHESIS TESTING:HYPOTHESIS TESTS FOR A

POPULATION MEAN

The Nature of Hypothesis Testing

• In hypothesis testing, a statement—call it a hypothesis— is made about some characteristic of a particular population. A sample is then taken in an effort to establish whether or not the statement is true.

• If the sample produces a result that would be highly

unlikely under an assumption that the statement is true, then we’ll conclude that the statement is false.

Choosing the Null Hypothesis

• The status quo or “if-it’s-not-broken-don’t-fix-it” approach

• The good sport approach

• The skeptic’s “show me” approach

Standard Forms for the Null and Alternative Hypotheses

H0: > A (The population mean is greater than or equal to A)

Ha: < A (The population mean is less than A)

H0: < A (The population mean is less than or equal to A)

Ha: > A (The population mean is greater than A)

H0: = A (The population mean is equal to A)

Ha: ≠ A (The population mean is not equal to A)

Figure 9.1 The Sampling Distribution of the

Sample Mean

n

x

x

=

Figure 9.2 The “Null” Sampling Distribution = 5000

5000

x

Figure 9.3 Likely and Unlikely Sample Means in the “Null” Sampling Distribution

5000

x

xxx = 5 = 4997

= 4998

Significance Level ()

A significance level defines what we mean by unlikely sample results under an assumption that the null hypothesis is true (as an equality).

Figure 9.4 Setting the Boundary on the “Null” Sampling Distribution

z

5000

0zc = -1.65

= .05

REJECT H0

pounds scale

z scalex

Figure 9.5 Showing zstat on the Null Sampling Distribution

x5000

0zc = -1.65

= 4912

REJECT H0

pounds scale

z scale

x

zstat = -2.49

z

The Four Steps of Hypothesis Testing

Step 1: State the null and alternative hypotheses.

Step 2: Choose a test statistic and use the

significance level to establish a decision rule.

Step 3: Compute the value of the test statistic.

Step 4: Apply the decision rule and make your

decision.

Figure 9.6 Showing the Boundary, c, on the Null Sampling Distribution

5000

REJECT H0

zzc = -1.65

c = 4941.6

pounds scale

z scale

=.05

x

p-value

The p-value measures the probability that, if the null hypothesis were true, we would randomly produce a sample result at least as unlikely as the sample result we actually produced.

Figure 9.7 p-value for a Sample Mean of 4962

(pounds)

z -1.07

4962

p-value = .1423

0

x

p-value Decision Rule

If the p-value is less than , reject the null hypothesis.

Figure 9.8 Using the p-value to Make a Decision

(pounds)

z

zstat =-1.07

4962

p-value = .1423

4941.6

zc = -1.65

REJECT H0

x

Error Possibilities in Hypothesis Testing

TYPE I Error: Rejecting a true null hypothesis.

TYPE II Error: Accepting a false null hypothesis.

and the Risk of Type I Error

measures the maximum probability of making a Type I Error.

Figure 9.9 A Two-tailed Hypothesis Test

REJECT H0

zzcl

/2

zcu

REJECT H0

2

x

Figure 9.10 A Two-tailed Hypothesis Test for Montclair Motors

5000

REJECT H0

z-1.96

= .025

+1.96

REJECT H0

= .025

cL = 4930.7 cu = 5069.3 x

Test Statistic When s Replaces (9.2)

ns

x

/

tstat =

Figure 9.11 Testing with the t

Distribution

t

REJECT H0

tc

tstat