Hypothesis Testing for the Mean and Variance of a Population

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Hypothesis Testing on the Mean

• Null hypothesis (Ho): A statement concerning a population parameter.

• Alternative hypothesis: A statement in contradiction of the null hypothesis

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Type I and Type II ErrorsActual Situation

Ho True | Ho FalseConclusion

Fail to Reject Ho

Reject Ho

Correct decision

Correct decisionType I error

Type II error

= probability of rejecting the Ho when Ho is true (Type I error) = probability of failing to rejecting the Ho when Ho is false (Type II error)

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Hypothesis Testing Process

• Determine the Ho and Ha

• Determine the significance level

• Compare the sample mean (variance) to the hypothesized mean (variance).

• Decide whether to Fail to reject or Reject Ho

• Determine what the decision means in reference to the problem.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Example

• Ho: = 5.9

• Ha: 5.9

= .05 = P(rejecting Ho when Ho is true) critical value = ± 1.96

reject H0 ifX 5.9

/ n> 1.96

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Decision Rule

Figure 8.1

Figure 8.2Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

ZX 5.9

/ n

X 5.9

s/ n

5.76 5.9

.48 / 75 2.53 Z*

Because - 2.53 < - 1.96, we reject Ho thus weconclude that the average population male height is not equal to 5.9

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Hypothesis Testing5 Step Procedure

• Set up the null and alternative hypothesis

• Define the test statistic

ZX 0

/ n

X 0

s/ n

Define the rejection region Calculate the test statisticGive a conclusion in terms of the problem

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Example 8.2

• Define the hypothesesHo: = 400 Ha: 400

• Define the test statistic

• Define the rejection region (fig8.5)– reject Ho if Z > 1.645 or Z < -1.645

Fig 8.5

ZX 400

/ n

X 400

s/ n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

• Calculate the value of the test statistic

• State the conclusion– There is sufficient evidence to conclude that the

average lifetime of Everglo bulbs is not 400 hours

Z* 411 400

42.5/ 100

11

4.252.59

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Power of a Statistical Test

= P(fail to reject Ho when Ho is false)

• 1- = P(rejecting Ho when Ho is false)

• 1- = the Power of the test

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

One Tailed Test for

Example 8.4

Ho: 32.5Ha: < 32.5

Fig 8.8

reject H0 if ZX 32.5

s / n 1.645

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Results support the claim that the average mileagefor the Bullet is less than 32.5 mpg.

Z* 30.4 32.5

5.3 / 50 2.80

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

p-Value

• The p-value is the value of at which the hypothesis test procedure changes conclusions based on a given set of data. It is the largest value of for which you will fail to reject Ho

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Procedure for Finding the p-Value

• For Ha: o

p = 2 * (area outside Z*)

• For Ha: > o

p = area to the right of Z*

• For Ha: < o

p = area to the left of Z* Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Fir 8.14

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Interpreting the p-Value

• Classical Approach– reject Ho if p-value <

– fail to reject Ho is p-value

• General rule of thumb– reject Ho if p-value is small (p < .01)

– fail to reject Ho is p-value is large (p > .1)

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Hypothesis Testing on the Mean of a Normal Population: Small Sample

• Normal population

• Population standard deviation unknown

• Small sample

• Student t distribution

• Nonparametric procedure

t X 0

s/ n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Example 8.8

Ho: = 10Ha: 10

reject Ho if |t | > t.025,17 = 2.11

t X 10

s/ n

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Because 1.83 < 2.11 we fail to reject Ho

Insufficient evidence to reject the Ho that the average output voltage is different from 10 volts

t* 10.331 10

.767 / 181.83

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Small-Sample Tests on aNormal Population Mean

Two-tailed testHo: = o Ha: o

reject Ho if |t*| > t/2, n-1

One tail testHo: = O Ho: = O Ha: > O Ha: < O

reject Ho if t* > t, n-1 reject Ho if t* < t, n-1

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing