Quantitative Methods Chapter 5

The rejection region

Critical value which is the value that divides the rejection region from the rest of the distribution. Two‐tailed tests have both upper and lower critical values, while one‐tailed tests have either a lower or upper critical value.

Null hypothesis

Existing theory

Hypothesis Formulation

Hypothesis testing begins by defining two alternative, mutually exclusive propositions about one or more population parameters. The null hypothesis represents an existing theory or belief that is accepted as correct in the absence of contradictory data. The alternative hypothesis must be true if we reject the null hypothesis.

One‐Sample Tests for Means

T-Test Statistic

One-tailed Tests of Hypothesis

Types of hypothesis which specifies a direction of relationship where H0 is either ≤ or ≥. In this case, the rejection region occurs only in one tail of the distribution.

Levene Test

Used when you suspect unequal sample sizes.

Lower tail rejection region

If H1 is stated as < the rejection region is in the lower tail (just think of the inequality as an arrow pointing to the proper tail direction!).

Two-tailed Test of Hypothesis

If the null hypothesis is structured as "=" and the alternative hypothesis as "Z," then we would reject H0 if the test statistic is either significantly high or low. In this case, the rejection region will occur in both the upper and lower tails of the distribution Because the probability that the test statistic falls into the rejection region, given that H0 is true, the combined area of both tails must be α.

Rejecting null hypothesis an lower one-tailed test

If the test statistic is < the critical value, we would reject the null hypothesis.

Rejecting null hypothesis an upper one-tailed test

If the test statistic is > the critical value, the decision would be to reject the null hypothesis.

Rejecting null hypothesis a two-tailed test

If the test statistic is either > the upper critical value or < the lower critical value, the decision would be to reject the null hypothesis.

Statistical inference

Process of drawing conclusions about populations from sample data.

Power of the Test

1 - β represents the probability of correctly rejecting the null hypothesis when it is indeed false, or P(Rejecting H0|H0 is false).

Hypothesis test steps

1. Hypothesis Formulation 2. Selecting a level of significance, which defines the risk of drawing an incorrect conclusion about the assumed hypothesis that is actually true 3. Determining a decision rule on which to base a conclusion 4. Collecting data and calculating a test statistic 5. Applying the decision rule to the test statistic and drawing a conclusion

Four possible outcomes of Hypothesis testing

1. The null hypothesis is actually true, and the test correctly fails to reject it. 2. The null hypothesis is actually false, and the hypothesis test correctly reaches this conclusion. 3. The null hypothesis is actually true, but the hypothesis test incorrectly rejects it (called Type I error). 4. The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it (called Type II error).

Assumptions of ANOVA

ANOVA requires assumptions that the m groups or factor levels being studied represent populations whose outcome measures 1. Are randomly and independently obtained (easily validated with random samples are chosen) 2. Are normally distributed 3. Have equal variances (Required in order to pool the variances within groups. If sample sizes are equal, this assumption does not have serious effects on the statistical conclusions; however, with unequal sample sizes, it can. When you suspect this, you can use the Levene test to investigate the hypothesis)

Hypothesis testing

Allows you to draw inferences about two contrasting propositions (hypotheses) relating to the value of a population parameter, such as a mean, proportion, standard deviation, or variance.

β

The probability of a Type II error, P(Not rejecting H0|H0 is false). Unlike α, this cannot be specified in advance but depends on the true value of the (unknown) population parameter.

Level of Significance of the Test

The probability of making a Type I error, P(Rejecting H0|H0 is true), is generally denoted by α

T‐distribution

The sampling distribution is divided into two parts, a rejection region and a non-rejection region.

Alternative hypothesis

Theory based on new information provided by sample data

Upper tail rejection region

if H1 is stated as > the rejection region is in the upper tail (just think of the inequality as an arrow pointing to the proper tail direction!).