Comparing two means

Question 1

Paired design and paired T test

Answer 1

Paired design
- Makes a comparison of measures made from units with shared features – in space or time
- Is a good way of controlling for other sources of variation that contribute to a measurement than the treatment in question

eg. take sample of plots and clear cut a randomly chosen half of each plot and leave other half untouched (reduces effect of extraneous variation among sampling units)

1. Work out the difference between each pair of values and take an average.
2. Work out t value (t=Mean difference- difference under null / Standard error of difference)
3. Use the table to work out P value from this t value

Question 2

Two sample design and two sample T test

Answer 2

• Compares measures made from independent sampling units.
• It is useful when it is difficult/not possible to use a paired‐design

eg. take sample of plots and randomly assign each to be clear cut or not clear cut

1. Work out standard error of means using the pooled sample variance.
2. Work out degrees of freedom (n1 + n2 - 2)
3. Work t value (t=Y1-Y2/ SE Y1- Y2)
4. Work out confidence interval using all of these values

Question 3

Assumptons for a paired T test

Answer 3

• Sampling units are randomly sampled from the population
• Paired differences have normal distribution in the population

Question 4

Falacy of indirect comparison

Answer 4

Can’t compare two group’s significance against a null and then say that this means groups must be different – instead must calculate difference in two groups then test against null (eg. mother / father resemblance example)

Comparisons between two groups should always be made directly, not indirectly by comparing both to the same null hypothesised value

Question 5

Interpreting over lap in confidence intervals

Answer 5

1. There is no overlap = reject null of no difference in the 2 groups
2. Confidence intervals of at least one overlaps sample mean of other = cannot reject null of no difference
3. If confidence intervals alone overlap (eg. neither intervals overlap the sample mean of the other) -> can’t be sure of what results of a hypothesis test would be

Question 6

Assumptions of two sample T test

Answer 6

1) Each of the two samples is a random sample from its population
2) The numerical variable (i.e. the response variable) is normally
distributed in each population
3) The standard deviation (and variance) of the numerical variable is the same in both populations.

If it is not normally disitrbuted use the non parametric test-> mann Whitney U test

If the variances are not the same use Welch’s T test.
- Welch’s T test works out standard error and degrees of freedom in a different way but uses the same formula for T.

Question 7

How to test for equality of variances (two sample T test)

Answer 7

F test
null: variance 1= variance 2

F= variance of group 1/ variance of group 2

• Use the statistical table to find the critical F value (need to use the degrees of free -> n-1 for each)
• If F> F from table then reject the null hypothesis that the variances are equal.

Levene’s test
- Levene’s test for homogeneity of variances is more
common because it is more robust than the F‐test
- Use a statistical program to calculate this