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One Sample T-test

Question: Has the mean preference for teenagers before entering the theme park changed from the previous survey?

Step 1: Formulate null and alternative hypotheses

H0: The mean preference for teenagers before entering the theme park did not change from the previous survey

H1: The mean preference for teenagers before entering the theme park changed from the previous survey

Let’s say we  know  that  last year the  mean  preference  was  5.  Therefore the above hypotheses can be rewritten analytically as:

H0: μ = 5

H1: μ ≠ 5

Step 2: Test null hypothesis

Look at One-Sample t-test

and compare the significance (p-value) of t statistic with α=0.05. In our case: 0.177>0.05 => Do NOT reject null hypothesis.

Conclusion: The mean preference for teenagers before entering the theme park did not change from the previous survey. (Though this year the mean preference is 5.5 it is not significantly different from 5.0).

Note: If  significance  of  t  statistic  would  be  lower  than  α=0.05  (for  instance, 0.03<0.05), then the null hypothesis would be rejected and the conclusion would be opposite.

Independent Samples T-test

Question: Do teenagers have different preference than adults before entering the park?

Step 1: Formulate null and alternative hypotheses

H0: Teenagers have the same mean preference before entering park as do adults

H1: Teenagers have different mean preference before entering park as do adults

The same hypotheses but written analytically:

H0: μ1  = μ2

H1: μ1  ≠ μ2

Step 2: Test null hypothesis

Look at Independent Samples t-test

First we have to choose which t-test to use: t-test with equal variances or t-test with not equal variances.

Step 2.1: Test the equality of variances between two populations

H0: σ1(2) = σ2(2) (equal variances assumed)

H1: σ1(2) ≠ σ2(2) (equal variances not assumed)

To test this hypothesis we look at significance of F statistic, which is 0.703 in our case.  As  0.703>0.05  =>  we  cannot  reject  null  hypothesis  about  equality  of variances.

Conclusion: variances of two populations are the same and we can use the first t-test (with equal variances assumed). This test is located in the first row of the Independent Samples Test table.

Note: If this significance would be lower than 0.05 (for instance, 0.02<0.05), then we would reject null hypothesis and would use the second row of the table for t- test.

Step 2.2: Test the equality of means hypothesis

H0: μ1  = μ2

H1: μ1 ≠ μ2

To test this hypothesis we compare the significance (p-value) of t statistic with α=0.05.  In  our  case  0.006<0.05 =>  we  reject  null  hypothesis  of  equality  of means.

Conclusion: Teenagers have different mean preference before entering park as do adults (teenagers have mean preference 5.5 and adults have 4.0, and this difference is significant).

Note:  The  above  example  involved  two-tailed  test.   However,  one  could  be interested to use one-tailed test.

One tailed test:

H0: μ1  ≤ μ2 (teenagers have lower mean preference than adults)

H1: μ1  > μ2 (teenagers have higher mean preference than adults)

If you use one-tailed test you have to adjust p-value by dividing it by 2. In our case for one-tailed test the significance  p=0.006/2=0.003<0.05 =>  Reject  null hypothesis.

Conclusion: Teenagers have higher mean preference before entering park than do adults (teenagers have mean preference 5.5 and adults have 4.0).

Paired Samples T-test

Question: Is there a difference in the preference before and after visiting the Disney theme park for teenagers?

Step 1: Formulate null and alternative hypotheses

H0: Teenagers have the same preference before and after visiting Disney park

H1: Teenagers have different preferences before and after visiting Disney park

The same hypotheses but written analytically (D=Preference After – Preference Before):

H0: μD = 0

H1: μD ≠ 0

Step 2: Test null hypothesis

Look at Paired Samples t-test

and compare the significance (p-value) of t statistic with α=0.05. In our case: 0.000<0.05 => reject null hypothesis.

Conclusion: The  preference  for  teenagers after visiting the theme park  is different from preference before entering park. (After=7.9  ≠  Before=5.5).

Note: If you would like to use one-tailed test, the hypotheses will look as follows

H0: μD ≤ 0

H1: μD > 0

p=0.000/2=0.000<0.05 => Reject null hypothesis

Conclusion: The preference for teenagers after visiting the theme park is higher than preference before entering park. (After=7.9   >   Before=5.5).

ANOVA

Question: Do  the  various  promotion  campaigns  differ in  terms  of generated sales?

Step 1: Formulate null and alternative hypotheses

H0: All promotions have the same influences on sales

H1: At least one promotion level has different influence on sales

The same hypotheses but written analytically:

H0: μ1  = μ2  = μ3

H1: at least one mean is different

Step 2: Test null hypothesis

Look at SPSS output

and compare the significance (p-value) of F statistic with α=0.05. In our case: 0.001<0.05 => reject null hypothesis.

Conclusion: Promotions  have influence   on   sales (the  means  of  sales  for different levels of promotion are significantly different from each other).

Step 3: Determine the strength of effect

To determine the strength of effect we have to calculate eta-squared: η2 = SSX/SSY = 70 / 98 = 0.714 .  It  means  that  promotions  have  strong  effect  on sales (recall that η2  ∈ (0,1) ).

Step 4: Interpret the pattern of the relationship between variables

More intensive campaign leads to higher sales.