L7 Test statistics

Null Hypothesis H0

This represents the initial belief or assumption about the population parameter. Always includes equality component and include a specific point or value. We begin hypothesis testing by assuming the statement written in the null hypothesis is true and then check.

Alternative hypothesis

This states that the belief or assumption in H0 (null hypothesis) is incorrect and is another alternative to that. Sometimes defined first as the conclusion the researcher hopes to support (the research question). Involves terms like increases, stricter or greater.

Alternative hypothesis defines whether the test is one-sided or two-sided:

- Lower Tailed Test: The rejection region is in the left tail.

- Upper Tailed Test: The rejection region is in the right tail.

- Two Tailed Test: The rejection region is in both tails.

When deciding, two types of errors are possible:

- Type I error (alpha)

- Type II error (beta)

Type I error (alpha)

- Rejecting the null hypothesis when it is actually true.

o If a person is innocent, sentencing them to jail is a Type I error.

Type II error (beta)

- Accepting the null hypothesis when it is actually false.

o If a person is guilty, releasing them is a Type II error.

The significance level (alpha):

Is the size of the rejection region and denotes the probability of committing a Type I error.

- Commonly used are: 0.01, 0.05, and 0.1.

- Important trade-off: Decreasing the probability of a Type I error increases the probability of a Type II error, and vice versa (when other factors are held fixed).

To decide if the sample data is consistent with the null hypothesis (H0) we use a:

Test statistic with a known distribution (like Z or T) is calculated using sample data. The conclusion is reached using two equivalent methods:

After defining the hypothesis, setting the significance level, and computing the test statistic (Steps 1-3), the decision is made using two equivalent methods that always result in the same conclusion.

Critical values aand p-values

Critical values

1. Find the critical value(s) corresponding to the specified significance level.

The null hypothesis (H0) is rejected if the computed test statistic falls beyond the positive or negative critical value(s) (i.e., within the rejection region).

Rejection rule for critical values:

Reject H0 if the computed test statistic:

- Exceeds the positive CV (for an upper tailed test).

- Falls below the negative CV (for a lower tailed test).

- Meets either of the above conditions (for a two tailed test)

P-value

1. Is calculated from the realized test statistic. The calculation method depends on whether the test is lower-tailed, upper-tailed, or two-tailed.

a. It is the observed value or a more extreme value—meaning further away from the hypothesized value

Rejection rule for P-value:

If the p-value is smaller than the significance level (alpha), we reject H0; otherwise, we fail to reject $H_0$.