Statistics Quiz on Hypothesis Testing

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Questions and Answers

What is the null hypothesis (H0) regarding the agreement on banning cigarette smoking?

Males and females have different levels of agreement on banning smoking.

There is no evidence to support the association between gender and smoking bans.

Males have a higher level of agreement than females on banning smoking.

Males and females do not differ in their levels of agreement on banning smoking. (correct)

If the p-value is greater than the level of significance of 0.05, what should be concluded about the null hypothesis?

Accept the alternative hypothesis.

Re-evaluate the significance level.

Do not reject the null hypothesis. (correct)

Reject the null hypothesis.

What does the Chi-square Test of Association examine?

The variance within a single sample.

The means of two independent samples.

The correlation between two quantitative variables.

The relationship between two qualitative variables. (correct)

What is a condition for applying the Chi-square Test of Association?

Each observation must belong to one category from each variable. Signup and view all the answers

What does the alternative hypothesis (Ha) indicate in the context of the Chi-square Test of Association?

The two variables are associated. Signup and view all the answers

What is the null hypothesis (H0) regarding the fall status and lifestyle changes?

There is no association between fall status and lifestyle changes because of fear of falling. Signup and view all the answers

What level of significance is set for the hypothesis testing in this study?

0.05 Signup and view all the answers

Which test statistic is appropriate for testing the association between fall status and lifestyle changes?

Chi-square Test of Association Signup and view all the answers

In the hypothesis testing steps, what is the purpose of determining the critical region?

To decide whether to reject the null hypothesis Signup and view all the answers

How many fallers made lifestyle changes due to the fear of falling?

131 Signup and view all the answers

What is the formula for the Chi Square test statistic?

$rac{(O - E)^2}{E}$ Signup and view all the answers

How is expected frequency (E) calculated?

$rac{row otal imes column otal}{grand otal}$ Signup and view all the answers

What does the degrees of freedom (df) equal in a Chi Square test?

(r - 1)(c - 1) Signup and view all the answers

What does 'O' represent in the Chi Square formula?

Observed frequency Signup and view all the answers

What is a characteristic of the Chi Square test of association?

It assesses the relationship between two categorical variables. Signup and view all the answers

What is the purpose of calculating expected frequencies in a Chi Square test?

To test the null hypothesis of independence Signup and view all the answers

Which statement about the Chi Square test of independence is true?

It is used to determine if two categorical variables are associated. Signup and view all the answers

What happens if the null hypothesis is true in a Chi Square test?

Observed frequencies will equal expected frequencies. Signup and view all the answers

What is the purpose of the Chi Square Test of Homogeneity?

To assess whether two or more populations share the same proportions across categories Signup and view all the answers

In the context of the Chi Square Test of Homogeneity, what does a contingency table typically represent?

Observed frequencies categorized by two different variables Signup and view all the answers

Which step is NOT part of the hypothesis testing process?

Calculate the mean and median Signup and view all the answers

What is the basis for calculating expected frequencies in the Chi Square Test of Homogeneity?

It is based on the assumption of homogeneity in the population as stated in the null hypothesis Signup and view all the answers

Which of the following steps should be performed first in hypothesis testing?

State the null and alternative hypotheses Signup and view all the answers

What are the 'marginals' in a contingency table?

The entries that represent the total counts for each category Signup and view all the answers

Which of the following statements about sample selection in the Chi Square Test of Homogeneity is most accurate?

Independent samples are drawn from specific populations identified in advance Signup and view all the answers

When can the Chi Square Test of Homogeneity be used interchangeably with the z-test?

When two populations are involved and the variable has two categories Signup and view all the answers

What is the null hypothesis in the study regarding smoking status and type of school?

There is no association between smoking status and type of school. Signup and view all the answers

What is the level of significance used in the study on smoking cessation?

0.01 Signup and view all the answers

What type of statistical test is employed to determine the association between smoking status and type of school?

Chi-square Test of Association Signup and view all the answers

What is the expected frequency of Large Cell Nonkeratinizing for the age group 30-39?

19.59 Signup and view all the answers

What conclusion can be drawn if the p-value is less than the significance level of 0.01?

Reject the null hypothesis. Signup and view all the answers

In the example given, what is the total number of high school students surveyed?

834 Signup and view all the answers

How many patients fall within the age group 50-59?

144 Signup and view all the answers

What is the calculated value of 𝜒2 based on the observed and expected frequencies provided?

4.444 Signup and view all the answers

Why is it important to calculate expected cell frequencies in this study?

To assess independence of the two variables. Signup and view all the answers

Which cell type had the lowest expected frequency among all age groups?

Keratinizing Cell Type in 50-59 Signup and view all the answers

What can be inferred if the alternative hypothesis is supported in a study?

There is a significant association between the studied variables. Signup and view all the answers

In the age group 60-69, how many patients were classified as Keratinizing Cell Type?

25 Signup and view all the answers

Which statement accurately reflects a potential misconception about the p-value in hypothesis testing?

A low p-value guarantees a true effect. Signup and view all the answers

Which age group has the highest total number of patients?

50-59 Signup and view all the answers

What is the expected frequency for Small Cell Nonkeratinizing in the age group 40-49?

25.27 Signup and view all the answers

Which option correctly represents the O-E (Observed minus Expected) values for Small Cell in the age group 50-59?

23.49 Signup and view all the answers

What was the total number of patients across all age groups?

380 Signup and view all the answers

In terms of expected frequencies, which cell type had the highest value for the age group 60-69?

Large Cell Nonkeratinizing Signup and view all the answers

Study Notes

Chi-Square Tests

Used when analyzing qualitative data with mutually exclusive and exhaustive categories.
Quantitative data are frequencies associated with each category.
Compares observed frequencies to expected frequencies under the null hypothesis.
Large differences indicate rejection of the null hypothesis.

Learning Outcomes

Students will be able to describe the characteristics of chi-square distributions.
Students will be able to differentiate between tests of homogeneity of proportions and tests of association.
Students will be able to interpret computed chi-square test values.
Students will be able to identify the requirements for valid use of a chi-square test.

Chi-Square Test Statistics

The test uses frequencies associated with categories of qualitative variables.
It compares observed frequencies of elements in various categories with expected frequencies assuming the null hypothesis is true.
A significant difference between observed and expected frequencies signals rejection of the null hypothesis.

Types of Chi-Square Tests

Goodness of Fit
Test of Homogeneity
Test of Association

Chi-Square Distribution: Characteristics

The shape of the distribution changes with degrees of freedom (df).
Lower df results in a more positively skewed distribution.
Higher df leads to a more symmetrical and normal distribution.
The mean of a chi-square distribution equals its degrees of freedom.
The total area under the curve of any given chi-square distribution is 1.

Applicability of Chi-Square Tests

Data in contingency tables, particularly 2x2 tables, require expected frequencies of 5 or more in each cell for appropriate chi-square application.
For larger tables, each expected frequency should be at least 1, and no more than 20% of cells can have an expected frequency below 5.
If these conditions aren't met, alternative methods (e.g., Fisher's Exact Test) are necessary or cells should be combined.

Chi-Square Test of Homogeneity

Used to determine if two or more populations have the same proportions for the different categories of a categorical variable.
When dealing with only two populations and a two-category variable, homogeneity testing is interchangeable with the z-test for two proportions.
Data is presented in a contingency table, with rows for one variable and columns for another variable.

Chi-Square Homogeneity: Characteristics

This test identifies if two or more populations have the same proportions.
Calculations depend on a pooled estimate of the sample probability.
Statements are made in terms of population homogeneity (of groups or categories).

Chi-Square Test Statistic

Formula for calculating the chi-square test statistic:

x² = Σ [(O - E)² / E]

Where: O = Observed frequency E = Expected frequency

Hypothesis Testing Steps

State null and alternative hypotheses.
Specify the significance level (α).
Select an appropriate test statistic.
Determine the critical region based on the α level and degrees of freedom.
Calculate the test statistic.
Make a decision (reject or fail to reject null hypothesis).
Draw a conclusion based on the decision.

Example Scenarios/Exercises (Chi-Square Applications)

Several examples are provided in the slides, illustrating applications of chi-square tests to different scenarios involving categorical data.

Fisher's Exact Test

Used for small sample sizes or when expected frequencies fail to meet minimum requirements for a chi-square test.
A 2x2 contingency table is typical.
Data must be discrete, from random samples.
The test focuses on the exact probability relating to the observed values.

A 2x2 Contingency Table (Fisher's Exact Test)

Presents a visual structure for the data.
Shows the relationship between two variables with two categories each.

Example: Smoking Cessation Program

Demonstrates application of a chi-square test for association to identify association between school type and smoking status
Highlights the need to assess if the expected frequency requirements are met to use the test properly.

Chi-Square Test of Association

Determines if there's a relationship between two categorical variables within a single population.

Example: Exercise Preference

Illustrates the usage of this test to determine whether a specific activity preference is related to participant gender.
Displays the importance of verifying conditions for the test before application.

Sample Size Requirements

For 2x2 tables, expected frequencies should be at least 5 in each cell
For larger tables, there are also requirements about expected cell counts

References

Several sources are given in the slides, allowing further exploration for students.

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Description

This quiz focuses on key concepts in hypothesis testing, specifically as they relate to Chi-square tests and null hypotheses in social studies. It covers the interpretation of p-values, test statistics, and the conditions for conducting these tests. Test your understanding of the application of statistical methods in research.