chapter thirteen part ii hypothesis testing: means

Chapter Thirteen Part II

Hypothesis Testing: Hypothesis Testing:

MeansMeans

Where does this come from

• Suppose Tide is contemplating a new ad campaign. They feel that mean consumer attitude to the old campaign is 2 on a [1 (dislike very much) -5 (like very much) scale]

• Past studies have confirmed that the average variability in the target audience is 1.30

• Tide wants to verify this feeling so they survey 500 consumers and measure consumer attitudes to the old campaign. The mean attitudes in the sample are 3.

• Should Tide continue with the old campaign (and save money) or make a new campaign?

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses

Is this a one-tailed or two-tailed test?

Ho: = hypothesized value of population

Ha: hypothesized value of population

Our problem: What is Ho and Ha?

Which test to apply?

• Is the population standard deviation known?

• If ‘yes’ – use the Z test• If ‘no’ use the t test

• Our problem: Which test do we use?

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula T = sample mean – population mean / standard error

of the meanZ = sample mean – population mean / standard error

of the meanWhere, standard error of the mean = population

standard deviation / sq.root of N, (z test) ORSample standard deviation / sq. root of N (t test)

Our problem: What is the standard error of the mean?

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula3) Select significance level

1%, 5% or 10%

What is the usual significance level in the social sciences?

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic

Our problem: What is the observed test statistic?

xs

Xt

)( where sx = s

n

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)

Our problem: What are the degrees of freedom?

d.f. = n-1

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table

Our problem: What is the critical value from the table?

Hypothesis Testing About a Single Mean - Step-by-Step1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table7) Make decision regarding the Null-

hypothesisOur problem: What would you advise Tide?

One-tailed: if tts > t then reject Ho

Two-tailed: if |tts| > t/2 then reject Ho

Hypothesis Testing About two means from independent samples• Suppose your survey of brand preferences

towards Coke revealed that the sample drawn from Charlotte had a mean preference score of 3.5 and the sample drawn from Columbia had a mean preference score of 3.0.

• Is this difference statistically significant?• Do the two samples originate from two

different populations (two different cities) or are they from a single population (the US South East)

Hypothesis Testing About two means from independent samples

1) Formulate Hypotheses

Ho: X 1 = X 2

Ha: X 1 X 2

Standard Error of the Differences Between Means• (to recap) Same logic as the standard error of

the mean– Very large number of samples with replacement

possible (each sample will have a mean)– This generates a hypothetical distribution of means– standard deviation of the distribution of means

• We have two sets of means (since we have two independent samples)– Here we have two distributions of means– Take the difference between the means and we have

the third distribution of difference between means– Standard deviation of the differences between

means


1) Formulate Hypotheses2) Select appropriate formula

Note: This assumes that variance are equal across the two independent samples

2121 nn

11S XX

where sp

Where sp2

(n1 – 1) s12 + (n2 – 1) s2

2

n1 + n2 -2

21ts

X ) - 0(Xt

21 XXS

1) Formulate Hypotheses2) Select appropriate formula3) Select significance level


1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate t statistic


2121 nn

11S XX

where sp

Where sp2

(n1 – 1) s12 + (n2 – 1) s2

2

n1 + n2 -2

21ts

X ) - 0(Xt

21 XXS

1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)

d.f. = n1 + n2 -2


1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table


1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table7) Make decision regarding the Null-

hypothesis




Hypothesis Testing About Differences in Means (dependent samples)1) Formulate Hypotheses

Ho: X 1 - X 2 = 0

Ha: X 1- X 2 0

Hypothesis Testing About Differences in Means (dependent samples)1) Formulate Hypotheses2) Select appropriate formula

where

Where

ts

d )(Dt

s

D/n

D =1n Di

i = 1

n

s

D2

=1

n - 1Di

2 – nD2)i = 1

n

1) Formulate Hypotheses2) Select appropriate formula3) Select significance level

Hypothesis Testing About Differences in Means (dependent samples)

1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate t statistic


where

Where

ts

d )(Dt

s

D/n

D =1n Di

i = 1

n

s

D2

=1

n - 1Di

2 – nD2)i = 1

n

1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)

d.f. = n - 1


1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table


1) Formulate Hypotheses2) Select appropriate formula3) Select significance level4) Calculate z or t statistic5) Calculate degrees of freedom (for t-test)6) Obtain critical value from table7) Make decision regarding the Null-

hypothesis




Example: Test of Difference in Means

Before:

5 2 3 3 4 2 5 4

After: 6 4 3 6 3 4 5 6

D: 1 2 0 3 -1 2 0 2

D2: 1 4 0 9 1 4 0 4

Rules to remember

1. State Ho (Ho is the hypothesis of no difference / no relationship between the sample and population or two samples)

2. State Ha (Ha is the mirror image of Ho)3. Identify critical region (this the region beyond the

critical value on either side. Ho is rejected if observed value falls in the critical region)

4. Decide distribution (if ơ is known use Z distribution, if not, use t distribution)

5. Identify the test (1 sample = one sample z or t; two independent samples = z or t; two dependent samples = z or t on differences.

6. Decide one / two tailed (choose tail according to direction of Ha)

chapter thirteen part ii hypothesis testing: means

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