Workgroup notes with Inferential Statistics at the Leiden University - 2018/2019

The answers can be found in the corresponding download file (right column)

Week 1

Preparatory assignment 1.0

Theory

1. What symbols are used for the population mean and the sample mean?

2. What is the difference between the H₀ and the H_a?

3. What is the definition of a p-value?

4. What is the definition of the rejection criterion α?

Application

Select the correct response(s). (More than one may be correct.) The p-value for testing H₀ : µ = 100 against H_a : µ is not 100 is p = .001. This indicates that:

1. There is strong evidence that µ = 100.

2. There is strong evidence that µ is not 100.

3. There is strong evidence that µ > 100.

4. There is strong evidence that µ < 100.

5. If µ were equal to 100, it would be unusual to obtain data such as those observed.

Workgroup tips

It is important that you understand the difference between a sample and a population, and that you understand the difference between the symbols for them. A sample is the one test result set that a research or investigation delivers, or one group of people from one specific experiment, while a population is an overview of anybody who fits to the requirements. Below is a table of different symbols for the two different situations.

Week 2

Preparatory assignment 2.0

Theory

1. What is meant by events A and B being independent?

2. What is meant by events A and B being disjoint?

3. Can disjoint events be independent from each other?

4. What are the formulas for the:

   1. General additive rule

   2. Additive rule for disjoint events

   3. Complement rule

   4. General multiplicative rule

   5. Multiplicative rule for independent events

5. What is the definition a random variable?

6. What is the expected value of a random variable?

7. What is the expected value of X?

8. What is the variance of a random variable?

9. What is the expected value of the sum of two random variables?

10. What is the variance of the sum of two random variables?

Workgroup tips

There are three types of relationships between probability and variables. First there is Marginal, which means there is only a single variable involved. Then there is joint, when there are two or more variables, dependently or independently involved and last, there is conditional, which means that for instance event A needs to happen before it is even possible for even B to occur.

The thing about H₀ and H_a is that they are about the truth of the parameters of the population.

Week 3

Preparatory assignment 3.0

Theory

1. How do numerical data differ from categorical data?

2. When are two categorical variables independent?

3. What are the three different research designs for chi-square tests?

4. How does one determine the degrees of freedom for a χ²-test of independence?

5. How does one determine the degrees of freedom for a goodness of fit χ²-test?

6. In a χ²-test of independence, what is the formula for the expected cell frequency f_e?

7. In a χ²-test of independence, what is the formula for the χ² test statistic?

Workgroup tips

When deciding when to use which chi-squared test, there is a certain scheme you can follow that can help you. First, look at how many populations you have. If this number is more that just one, you know right away that you need to use the test for homogeneity. It the answer is one population, look at how many variables you have. If this is also just one, take the goodness of fit test. If you have more than one variable, use the test of independence.

The thing about chi-squared tests is that they are about the difference between two or more populations.

The difference between dependent and independent samples is that dependent are mostly before and after, husband and wife, so related statistics. Independent

Week 4

Preparatory assignment 4.0

Theory

1. What is the general H₀ for a one-sample z- or t- test?

2. What is the standard deviation of the distribution of sample means if σ is unknown?

3. What is the standard deviation of the distribution of sample means if σ is known?

4. What is the formula for the z test statistic?

5. What is the formula for the t test statistic?

6. When is the p value multiplied by 2: For one-sided or two-sided hypothesis tests?

Workgroup tips

Keep in mind that the observed results are always rounded numbers because, well, you can’t really observe half people, now can you? The expected, or usual results can, however, have decimals, because they can be an average.

Also, if the definition of the degrees of freedom sounds hard, look at it like this; df = number of rows – 1/ number of columns – 1

The thing with T and Z test is that they are about the difference between the sample and the population

Week 5

Preparatory assignment 5.0

Theory

1. What is the general H₀ and H_a for a two-sample t test for dependent samples?

2. What is the general H₀ and H_a for a two-sample t test for independent samples?

3. What is the number of degrees of freedom of a t test when:

   1. 2 paired variables with each n values?

   2. 2 independent samples with n₁ and n₂ values and σ²₁ , σ²₂?

   3. 2 independent samples with n₁ and n₂ values and σ²₁ = σ²₂?

4. What is the formula for the test statistic for a two-sample t test for independent samples with unequal variances?

5. Why can we leave µ₁ − µ₂ from the formula most of the time when using the test statistic for a two-sample t test of independent samples ?

6. What is the formula for the test statistic for a two-sample t test of dependent samples?

Workgroup tips

The thing with matched pairs/paired sample/dependant samples is that they are about the difference between each other. It is a one sample test, tested twice. Like, for instance, the difference between the man and the woman of a couple, or a person’s right and left arm.

Also, when you have a large t-test statistic, this means there is a smaller p-value. This, in turn, means that H₀ is more likely to be rejected, because p can fall into the rejection area more quickly.

Standard Error is the standard deviation of a sampling distribution

Week 6

Preparatory assignment 6.0

Theory

1. What is the definition of a 95% confidence interval of µ?

2. What is the formula for it?

3. What is the formula of the 95% confidence interval of µ₁ and µ₂?

4. What is the formula of the effect size η²?

5. What is the formula of the effect size Cohen’s d?

Application

Which of the following interpretations of a 95% confidence interval (CI) is/are correct?

1. A 95% CI contains 95% of the data in the population.

2. The probability that the population parameter is in this particular 95% CI is 95%.

3. 95% of all CIs that are calculated in this manner will capture the population parameter.

4. I am 95% sure that the mean of a sample will fall within the 95% CI for the mean.

Workgroup tips

To consider variances equal, the ration of the different standard deviations have to be at least equal or smaller that 2 (variance with a ratio of µ). This is calculated by S_bigger/S_smaller. If the ratio is indeed two or less, use the following formula: SE_{x̄1 - x̄2} = √( s²_p/n₁ + s²_p/n₂)

In which case S²_p= (n₁ – 1) × s²₁ + (n₂ – 1) × s²₂ / n₁ + n₂ – 2.

If the ratio is larger than 2: SE_{x̄1 - x̄2} = √( s²₁/n₁ + s²₂/n₂)

Week 7

Preparatory assignment 7.0

Theory

1. What is a nonparametric test?

2. In which two situations do you conduct a nonparametric test (instead of a parametric one)?

3. Which Wilcoxon test is suitable for paired data? Which one for independent samples?

4. What do you do when two or more observations have equal scores?

5. What are the formulas for the expected value of the sum and for the standard deviation of the Wilcoxon signed-rank test?

6. What are the formulas for the expected value of the sum and for the standard deviation of the Wilcoxon rank-sum test?

7. What are the formula’s for the z and t calculations.

Application

Use the information from Howell 18.4

Analyse the data by hand with a Wilcoxon rank-sum test. Assume that the test statistic has a normal distribution (Howell 18.3).

1. Rank the counts.

2. Add the rank numbers for one of the groups.

3. Calculate the expected value of the sum (under H₀).

4. Calculate the corresponding standard deviation.

5. Calculate the value of the test statistic z.

6. What is the probability p on the result (or even more extreme) found?

7. What is your statistical and substantive conclusion at α = .05

Workgroup Tips

When n is large enough (20 - 30 or more) it doesn’t matter what your original data looked like, the sampling distribution will be normal.

When can you say that there is a systematic difference between two small samples? When P is smaller than the critical value.

When dealing with a rank-sum test, you’ll need to use a z-test. When dealing with a signed rank test, you’ll need to use t. When one of the data in a signed rank test is 0, it disappears from the equation and you n becomes smaller.

Week 8

Workgroup Tips

They won’t ask what type of chi-squared test you are dealing with. Read the guidelines for the SPSS test in the workbook, they’re important!

On a chi-squared, you never double P of half α. In a 2 by x table, φ and Cramer’s V are always the same, because k-1 turns into 1.

A one-way chi-squared test is just one row of data, such as skittles. A two-way test is dealing with two or more variables

Expected frequency is row total x column total / by all.

Preparatory assignment 8.0

Theory

1. What is the formula for the effect measure φ for a χ²-test for 2-by-2 contingency tables?

2. What is the formula for the effect measure Cramer’s V for a χ²-test?