Chapter Thirteen Part II Hypothesis Testing: Means

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-Step 1) Formulate Hypotheses Is this a one-tailed or two-tailed test? H o :  = hypothesized value of population H a :   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-Step 1) Formulate Hypotheses 2) Select appropriate formula T = sample mean – population mean / standard error of the mean Z = sample mean – population mean / standard error of the mean Where, standard error of the mean = population standard deviation / sq.root of N, (z test) OR Sample 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-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) 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-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic Our problem: What is the observed test statistic? where s x = s n

Hypothesis Testing About a Single Mean - Step-by-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) 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-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) 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-Step 1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table 7) Make decision regarding the Null-hypothesis Our problem: What would you advise Tide? One-tailed: if t ts > t  then reject H o Two-tailed: if |t ts | > t  /2 then reject H o

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 H o : X 1 = X 2 H a : 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

Hypothesis Testing About two means from independent samples 1) Formulate Hypotheses 2) Select appropriate formula Note: This assumes that variance are equal across the two independent samples nn 1 1 S XX    where spsp Where sp2sp2  (n 1 – 1) s (n 2 – 1) s 2 2 n 1 + n ts X ) - 0 (X t   2 1 XX S 

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level  Hypothesis Testing About two means from independent samples

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate t statistic Hypothesis Testing About two means from independent samples nn 1 1 S XX    where spsp Where sp2sp2  (n 1 – 1) s (n 2 – 1) s 2 2 n 1 + n ts X ) - 0 (X t   2 1 XX S 

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) d.f. = n 1 + n 2 -2 Hypothesis Testing About two means from independent samples

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table Hypothesis Testing About two means from independent samples

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table 7) Make decision regarding the Null-hypothesis Hypothesis Testing About two means from independent samples One-tailed: if t ts > t  then reject H o Two-tailed: if |t ts | > t  /2 then reject H o

Hypothesis Testing About Differences in Means (dependent samples) 1) Formulate Hypotheses H o : X 1 - X 2 = 0 H a : X 1 - X 2  0

Hypothesis Testing About Differences in Means (dependent samples) 1) Formulate Hypotheses 2) Select appropriate formula where Where ts d ) (D t   s D/s D/ n D = 1 n  D i i = 1 n s D2s D2 = 1 n - 1  D i 2 – nD 2 ) i = 1 n

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level  Hypothesis Testing About Differences in Means (dependent samples)

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate t statistic Hypothesis Testing About Differences in Means (dependent samples) where Where ts d ) (D t   s D/s D/ n D = 1 n  D i i = 1 n s D2s D2 = 1 n - 1  D i 2 – nD 2 ) i = 1 n

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) d.f. = n - 1 Hypothesis Testing About Differences in Means (dependent samples)

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table Hypothesis Testing About Differences in Means (dependent samples)

1) Formulate Hypotheses 2) Select appropriate formula 3) Select significance level 4) Calculate z or t statistic 5) Calculate degrees of freedom (for t-test) 6) Obtain critical value from table 7) Make decision regarding the Null-hypothesis Hypothesis Testing About Differences in Means (dependent samples) One-tailed: if t ts > t  then reject H o Two-tailed: if |t ts | > t  /2 then reject H o

Example: Test of Difference in Means

Rules to remember 1.State H o (H o is the hypothesis of no difference / no relationship between the sample and population or two samples) 2.State H a (H a is the mirror image of H o ) 3.Identify critical region (this the region beyond the critical value on either side. H o 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 H a )