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Transcript of What is a Hypothesis?mba. 2013. 2. 14.آ  What is a Hypothesis? A hypothesis is a claim...

• What is a Hypothesis?

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

population mean

 population mean

 population proportion

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

Example: The proportion of adults in this city with cell phones is π = 0.68

• The Null Hypothesis, H0

 States the claim or assertion to be tested

Example: The average number of TV sets in

U.S. Homes is equal to three ( )3μ:H0 

 Is always about a population parameter, not about a sample statistic

3μ:H0  3X:H0 

• The Null Hypothesis, H0

 Begin with the assumption that the null hypothesis is true

 Similar to the notion of innocent until proven guilty

(continued)

proven guilty

 Refers to the status quo or historical value

 Always contains “=” , “≤” or “” sign

 May or may not be rejected

• The Alternative Hypothesis, H1

 Is the opposite of the null hypothesis

 e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 )

 Challenges the status quo

Challenges the status quo

 Never contains the “=” , “≤” or “” sign

 May or may not be proven

 Is generally the hypothesis that the researcher is trying to prove

• The Hypothesis Testing Process

 Claim: The population mean age is 50.  H0: μ = 50, H1: μ ≠ 50

 Sample the population and find sample mean.

Population

Sample

• The Hypothesis Testing Process

Sampling Distribution of X

(continued)

μ = 50 If H0 is true

If it is unlikely that you would get a sample mean of this value ...

... then you reject the null hypothesis

that μ = 50.

20

... When in fact this were the population mean…

X

• The Test Statistic and Critical Values

 If the sample mean is close to the assumed population mean, the null hypothesis is not rejected.

 If the sample mean is far from the assumed

 If the sample mean is far from the assumed population mean, the null hypothesis is rejected.

 How far is “far enough” to reject H0?

 The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough.

• The Test Statistic and Critical Values

Sampling Distribution of the test statistic

Region of Rejection

Region of Rejection

Region of

Critical Values

“Too Far Away” From Mean of Sampling Distribution

Rejection Region of

Non-Rejection

• Possible Errors in Hypothesis Test Decision Making

 Type I Error

 Reject a true null hypothesis

 Considered a serious type of error

 The probability of a Type I Error is 

 Called level of significance of the test

 Set by researcher in advance

 Type II Error

 Failure to reject false null hypothesis

 The probability of a Type II Error is β

• Possible Errors in Hypothesis Test Decision Making

Possible Hypothesis Test Outcomes

Actual Situation

(continued)

Decision H0 True H0 False

Do Not Reject H0

No Error

Probability 1 - α

Type II Error

Probability β

Reject H0 Type I Error

Probability α

No Error

Probability 1 - β

• Possible Errors in Hypothesis Test Decision Making

 The confidence coefficient (1-α) is the probability of not rejecting H0 when it is true.

 The confidence level of a hypothesis test is

(continued)

 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.

• Type I & II Error Relationship

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

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

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

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

If Type I error probability (  ) , then

Type II error probability ( β )

• Factors Affecting Type II Error

 All else equal,

 β when the difference between

hypothesized parameter and its true value

 β when 

 β when σ

 β when n

• Level of Significance and the Rejection Region

Level of significance = aH0: μ = 3

H1: μ ≠ 3

/2a/2a

This is a two-tail test because there is a rejection region in both tails

Critical values

Rejection Region

0

• 2 Test of Independence

 Similar to the 2 test for equality of more than two proportions, but extends the concept to contingency tables with r rows and c columns

H0: The two categorical variables are independent

(i.e., there is no relationship between them)

H1: The two categorical variables are dependent

(i.e., there is a relationship between them)

• Wilcoxon Rank-Sum Test: Hypothesis and Decision Rule

H : M = M H : M  MH : M  M Two-Tail Test Left-Tail Test Right-Tail Test

M1 = median of population 1; M2 = median of population 2 Test statistic = T1 (Sum of ranks from smaller sample)

H0: M1 = M2 H1: M1  M2

H0: M1  M2 H1: M1 > M2

H0: M1  M2 H1: M1 < M2

Reject

T1L T1U

RejectDo Not Reject

Reject

T1L

Do Not Reject

T1U

RejectDo Not Reject

Reject H0 if T1 ≤ T1L or if T1 ≥ T1U

Reject H0 if T1 ≤ T1L Reject H0 if T1 ≥ T1U

• Sample data are collected on the capacity rates

(% of capacity) for two factories.

Are the median operating rates for two factories

the same?

Wilcoxon Rank-Sum Test: Small Sample Example

the same?

 For factory A, the rates are 71, 82, 77, 94, 88

 For factory B, the rates are 85, 82, 92, 97

Test for equality of the population medians

at the 0.05 significance level

• Wilcoxon Rank-Sum Test: Small Sample Example

Capacity Rank

Factory A Factory B Factory A Factory B

71 1

77 2

82 3.5Tie in 3rd

Ranked Capacit

y values:

(continued)

82 3.5

82 3.5

85 5

88 6

92 7

94 8

97 9

Rank Sums: 20.5 24.5

Tie in 3rd

and 4th

places

• Wilcoxon Signed Ranks Test

 A nonparametric test for two related populations

 Steps:

1. For each of n sample items, compute the difference, Di, between two measurements

Di, between two measurements

2. Ignore + and – signs and find the absolute values, |Di|

3. Omit zero differences, so sample size is n’

4. Assign ranks Ri from 1 to n’ (give average rank to

ties)

5. Reassign + and – signs to the ranks Ri 6. Compute the Wilcoxon test statistic W as the sum of

the positive ranks

• Wilcoxon Signed Ranks Test Statistic

 The Wilcoxon signed ranks test statistic is the sum of the positive ranks:

  n'

)(RW

 For small samples (n’ < 20), use Table E.9 for

the critical value of W

 

 1i

)( iRW

• Wilcoxon Signed Ranks Test Statistic

 For samples of n’ > 20, W is approximately

normally distributed with

1)(n'n' 