Lecture 7 - Hypothesis Testing for Nominal and Ordinal Variables (Chi Square)

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Questions and Answers

What is the purpose of defining the critical region in hypothesis testing?

To ensure that all research hypotheses are accepted

To establish a boundary for rejecting the null hypothesis (correct)

To identify all possible sample outcomes

To select the sample size for the test

Which of the following alpha levels is most commonly used in hypothesis testing?

0.10

0.001

0.01

0.05 (correct)

What does the p value represent in hypothesis testing?

The significance level chosen by the researcher

The probability that the sample statistic falls into the critical region (correct)

The proportion of the critical region in the sampling distribution

The likelihood of making a Type I error

What is the critical value in hypothesis testing?

The value marking the beginning of the critical region Signup and view all the answers

What is a Type I error in the context of hypothesis testing?

Rejecting the null hypothesis when it is true Signup and view all the answers

How is the size of the critical region determined?

By the arbitrary choice of the researcher using alpha level Signup and view all the answers

What remains true about the alpha level and the p value in hypothesis testing?

They represent different concepts but both are probabilities Signup and view all the answers

What is an example of an alpha level in hypothesis testing?

0.005 Signup and view all the answers

What does the alpha level indicate in hypothesis testing?

The probability of a Type I error Signup and view all the answers

How does decreasing the alpha level affect the critical region?

It decreases the size of the critical region Signup and view all the answers

Which of the following is true about Type II errors?

They occur when failing to reject a false null hypothesis Signup and view all the answers

What is the relationship between alpha level and the probability of Type I and Type II errors?

As alpha decreases, Type I error probability decreases; Type II error probability increases. Signup and view all the answers

How can Type I errors be minimized in hypothesis testing?

By using very small values for alpha Signup and view all the answers

What is the critical region in hypothesis testing?

The set of outcomes that leads to rejecting the null hypothesis Signup and view all the answers

What best describes a Type I error?

Rejecting a null hypothesis that is actually true Signup and view all the answers

What happens to the non-critical region when the alpha level is decreased?

It becomes larger in size Signup and view all the answers

What does it indicate when the test statistic falls in the critical region?

Reject the null hypothesis. Signup and view all the answers

What happens if the p-value is greater than the alpha level?

There is no significance in the results. Signup and view all the answers

What does the alpha level define in hypothesis testing?

The critical region for decision making. Signup and view all the answers

Which of the following best explains a Type I error?

Rejecting a true null hypothesis. Signup and view all the answers

Which statement is correct regarding the power of a test?

It is the probability of rejecting the null hypothesis when it is false. Signup and view all the answers

If the null hypothesis is false, which scenario describes a Type II error?

Failing to reject the null hypothesis. Signup and view all the answers

How does increasing the alpha level affect the decision-making process in hypothesis testing?

Increases the critical region. Signup and view all the answers

In hypothesis testing, what does failing to reject the null hypothesis imply?

There is insufficient evidence to support the alternative hypothesis. Signup and view all the answers

Study Notes

Hypothesis Testing with Nominal and Ordinal Variables: Chi Square

Hypothesis testing is an inferential statistical procedure used to determine if a relationship exists between variables or if there's a difference between groups.
The logic involves comparing observed data from a sample to the expected data if there were no relationship between the variables.
Hypothesis testing procedures are used when two categorical variables (nominal or ordinal) are analysed, assessing association between them in the context of bivariate tables and column percentages.
A null hypothesis ("H₀") is a statement of "no difference" or "no relationship".
A research hypothesis ("H₁") is the belief by the researcher that a difference exists between the groups or variables.
The alpha level (α) represents the probability of rejecting the null hypothesis when it is actually true (a type I error).
The five-step model is a systematic framework for conducting hypothesis tests in various contexts regardless of the unique characteristics of the test. The five steps are as follows:
- Step 1: Make assumptions and satisfy all test requirements.
- Step 2: Outline the null hypothesis.
- Step 3: Select the sampling distribution and the critical region.
- Step 4: Compute the test statistic (obtained) value.
- Step 5: Make a decision, reject the null hypothesis or fail to reject it, and interpret results.
Sample size affects the probability of rejecting the null hypothesis. Larger samples are more accurate.
Statistical significance does not equal practical or theoretical importance.
Chi-square tests measure the relationship between categorical variables in bivariate tables.
- It's nonparametric and doesn't rely on assumptions about the shape of the population distribution.
- It's applied to nominal or ordinal variables to find if there is a relationship.
- Expected frequencies are calculated given that the null hypothesis is true
- The observed frequencies are compared against expected frequencies
- The difference between these frequencies yields the test statistic. The greater the difference, the more statistically significant the relationship is.

Computing Chi Square

To compute chi-square, the test statistic (x²) is calculated from the sample data.
The value of the obtained chi-square (x²) is compared with the critical value (x²) from the chi-square table for the specified alpha level and degrees of freedom.
Degrees of freedom is calculated using the formula df=(r-1)(c-1), where r is the number of rows and c is the number of columns in the bivariate table.
A computing table (e.g, Table 7.7) facilitates the multiple steps involved in calculating the chi-square test statistic.

Selecting an Alpha Level

Researchers select an alpha level to define what constitutes an “unlikely” sample outcome, affecting the decision on the null hypothesis
The alpha level (e.g., 0.05) refers to the probability.
Lower alpha levels (e.g., 0.01) yield a smaller critical region and reduce type I errors (false positive).
Higher alpha levels (e.g., 0.10) increase the likelihood of rejecting the null hypothesis even if there is no true relationship (increasing type II error- false negative).

The Five-Step Model for Hypothesis Testing: Application to Bivariate Tables

Using the five-step model, ensure the sample data represents the population perfectly, and consider random sampling error as a potential factor.
The Chi-Square Test for Independence is a nonparametric statistical test that determines if differences between two categorical variables in bivariate tables are significant
This test does not establish the causality between the variables.