Hypothesis testing

Question 1

What is a statistical hypotesis?

Answer 1

A claim about one or more population parameters.

Question 2

What are the null hypothesis (Ho) and the alternative hypothesis (Ha/H1)?

Answer 2

The null hypothesis is the one we are trying to reject or nullify while the alternative one is thealternative.

Question 3

In hypothesis testing the null hypothesis Ho and the alternative hypothesis Ha can be represented as which equality?

Answer 3

θ = θo
Where θ is a population parameter and θo is a fixed value.

There are then three alternative hypotheses that can exist:

θ > θo upper-tailed test (one-tailed tests)
θ < θo lower-tailed test (one-tailed tests)
θ ̸= θo two-tailed tests

Question 4

What are the 2 types of error that can be made when reaching a conclusion in hypothesis testing?

Answer 4

Since the statements about the population parameters are made based on sample statistics and probability theory there is a possibility of being wrong:

• Type I error: rejecting the true null hypothesis.
• Type II error: accepting the false null hypothesis

Question 5

What is a test procedure?

Answer 5

A rule based on sample data for deciding weather or not rejecting H0.

It consists of a test statistic and a reject region which is the set of all test statistic values for which H0 is rejected.

The shape of the region depends on the alternative
hypothesis.

Question 6

How should the cutoff (or critical) values for the rejection region be chosen?

Answer 6

to use probability theory and the
knowledge of the distribution of the test sample statistic.

Question 7

What does the choice of particular cutoff values for the rejection region influence?

Answer 7

the probabilities of type I error (α) and type II errors (β).

Unlike Type I error, which is controlled by the researcher through the choice of significance level, Type II error depends on the true value of the parameter being tested and the sample size.

The null hypothesis typically specifies a particular value for θ, while the alternative hypothesis (Ha) allows for a range of values for θ.

For each possible value of θ consistent with the alternative hypothesis, there’s a corresponding Type II error rate (β). Since we don’t know the true value of θ, we often can’t calculate β exactly. Instead, we may estimate β for specific values of θ or use statistical power analysis to assess the probability of detecting a difference or effect of a given size.

Question 8

What is the significance level α?

Answer 8

the probability we allow ourselves to make a type I error: i.e. to reject the null hypothesis H0 while it is actually true.