This is 2/2 of my EXTENSIVE notes for Test 1 for the pre-master course Inferential Statistics on the University of Twente. With these notes, and the first part, you have a good knowledge to pass the first exam. I passed my exam with a 8.2! These notes consists of lectures 4 - 6. For lectures 1 - 3,...
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Lecture 4: Inference about two means ........................................................................................................ 2
4.2: Exercises two independent samples.................................................................................................... 13
Lecture 4B: Related samples ..................................................................................................................... 27
Two related samples: paired t-test .................................................................................................................... 27
Canvas 4B: Related samples ...................................................................................................................... 31
Lecture 5: Association and Correlation ...................................................................................................... 40
Canvas 5.1: Correlation, Inference for correlation and Spearman’s rho ....................................................... 56
5.2: Association for qualitative measurement levels .................................................................................. 68
Lecture 6; Linear Regression – Inference for regression .............................................................................. 70
,Lecture 4: Inference about two means
Inference about two means
- Comparing two groups; you have strictly taken two populations
- Two sample problem = within your sample you make sub-samples (e.g. distinction
between male and female, Enschede and Groningen, etc.)
Theory test for comparing two means; two sample t-test
STEP1: RETHINK THE PROBLEM.
Assumptions and conditions for difference of means test
The more complex the test is, the more detailed the assumptions will be
1. Independence assumption (independent observations)
- Are both samples selected randomly from the population?
- Are both samples less than 10% of the population?
2. Normal population assumption
= can the distribution of both populations be considered nearly normal?
3. Independent groups assumption
= the selection of the two groups (group 1 and group 2) must be independent from
each other
(e.g. groups males and females are from the UT so related, BUT it is about the
selection that must be independent: which male or female is in the sample is
independent from each other. However, research between male and female based on
couples, we have related samples in one unit. Avoid this!)
4. Equal variance assumption: (sigma1^2 = sigma2^2)
(about sd of sampling distribution model)
- If sigma1 = sigma2 → equal variance method → pooled t-test for two independent
samples
(Method to estimate SE; Spread around the mean in both groups are more or less the
same. If one of the two is very skewed, the method is less reliable)
- If sigma1 = not equal to sigma2 → no equal variance → general t-test for two
independent samples
(method to estimate SE; if one of the two is very skewed, the method is still
applicable. Spread around the mean of group 1 is completely different compared to
group 2)
, Advantage of the pooled method → power of the test is higher
(= chance that you will find the difference between two means in both groups is bigger when
using pooled t-test instead of general t-test)
Equal variance assumption
Pooled t-test (PREFERED!)
- Sigma1^2 = Sigma2^2
→ variance of group 1 is more or less the same compared to variance of group 2
General t-test
- Sigma1^2 = not equal to Sigma2^2
→ if both distributions are different, we have a problem because they are not equal so
we have to use the general t-test
How to investigate equal variance assumption? (sigma1^2 = sigma2^2)
- Answer I: Ratio between the biggest (b) and the smallest (s) variance
o Rule of thumb:
o Since population variances are most often unknown…. We use sample
variances
o Less than 2? We accept equal variance assumed. More than 2? Then we
say it is not the same and we must go to the general t-test.
- Answer II: Levene’s test
o A more formal test: Levene’s test for equivale variance (only via SPSS)
STEP 2. FORMULATE H0 AND HA AND DEFINE ALPHA
Hypotheses (comparing two means)
There are three possible hypotheses
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