By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
The Chi-Square Goodness-of-Fit Test is a statistical method used to determine how well observed data fit a theoretical distribution or model. It is used to test the hypothesis that the observed frequencies are consistent with the expected frequencies under a specific distribution.
The Chi-Square Goodness-of-Fit Test is widely used in various fields such as medicine, social sciences, and engineering to evaluate the fit of a theoretical model to observed data. For example, in medical research, it can be used to determine if the observed distribution of a disease in a population matches the expected distribution based on a theoretical model.
Observed frequencies are the actual counts of data points in each category or bin.
Expected frequencies are the theoretical counts of data points in each category or bin based on a specific distribution or model.
The Chi-Square statistic is a measure of the difference between observed and expected frequencies. It is calculated as the sum of the squared differences between observed and expected frequencies divided by the expected frequencies.
$$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$
where $O_i$ is the observed frequency and $E_i$ is the expected frequency for the $i^{th}$ category.
The degrees of freedom is the number of categories minus one. It is used to determine the critical value of the Chi-Square distribution.
State the null and alternative hypotheses.
Calculate the expected frequencies based on a specific distribution or model.
Calculate the Chi-Square statistic using the observed and expected frequencies.
Determine the degrees of freedom based on the number of categories.
Find the critical value of the Chi-Square distribution using the degrees of freedom and a chosen significance level.
Compare the calculated Chi-Square statistic to the critical value to determine if the observed frequencies are consistent with the expected frequencies.
Problem Statement: A researcher wants to determine if the observed distribution of a disease in a population matches the expected distribution based on a theoretical model. The observed frequencies are: 10, 15, 20, 25, 30. The expected frequencies are: 12, 18, 24, 30, 36.
Solution:
$$\chi^2 = \frac{(10-12)^2}{12} + \frac{(15-18)^2}{18} + \frac{(20-24)^2}{24} + \frac{(25-30)^2}{30} + \frac{(30-36)^2}{36}$$
$$\chi^2 = 0.33 + 0.61 + 1.25 + 2.22 + 3.89$$
$$\chi^2 = 8.3$$
The degrees of freedom is 4. The critical value of the Chi-Square distribution is 9.488 for a significance level of 0.05.
Answer: The calculated Chi-Square statistic (8.3) is less than the critical value (9.488), so the observed frequencies are consistent with the expected frequencies.
Problem Statement: A researcher wants to determine if the observed distribution of a disease in a population matches the expected distribution based on a theoretical model. The observed frequencies are: 20, 25, 30, 35, 40, 45. The expected frequencies are: 18, 22, 26, 30, 34, 38.
$$\chi^2 = \frac{(20-18)^2}{18} + \frac{(25-22)^2}{22} + \frac{(30-26)^2}{26} + \frac{(35-30)^2}{30} + \frac{(40-34)^2}{34} + \frac{(45-38)^2}{38}$$
$$\chi^2 = 0.22 + 0.59 + 1.15 + 2.22 + 3.53 + 4.97$$
$$\chi^2 = 12.68$$
The degrees of freedom is 5. The critical value of the Chi-Square distribution is 11.07 for a significance level of 0.05.
Answer: The calculated Chi-Square statistic (12.68) is greater than the critical value (11.07), so the observed frequencies are not consistent with the expected frequencies.
Problem Statement: A marketing researcher wants to determine if the observed distribution of customer preferences for a new product matches the expected distribution based on a theoretical model. The observed frequencies are: 150, 200, 250, 300, 350. The expected frequencies are: 120, 180, 240, 300, 360.
$$\chi^2 = \frac{(150-120)^2}{120} + \frac{(200-180)^2}{180} + \frac{(250-240)^2}{240} + \frac{(300-300)^2}{300} + \frac{(350-360)^2}{360}$$
$$\chi^2 = 1.67 + 1.11 + 0.42 + 0 + 1.39$$
$$\chi^2 = 4.59$$
Answer: The calculated Chi-Square statistic (4.59) is less than the critical value (9.488), so the observed frequencies are consistent with the expected frequencies.
Make sure to calculate the expected frequencies correctly based on the theoretical distribution or model.
Make sure to calculate the Chi-Square statistic correctly using the observed and expected frequencies.
Make sure to determine the degrees of freedom correctly based on the number of categories.
Make sure to compare the calculated Chi-Square statistic to the critical value correctly to determine if the observed frequencies are consistent with the expected frequencies.
Double-check your calculations to ensure accuracy.
Use a calculator or software to calculate the Chi-Square statistic and determine the critical value.
Understand the theoretical distribution or model used to calculate the expected frequencies.
Use graphing calculators to calculate the Chi-Square statistic and determine the critical value.
Use statistical software to calculate the Chi-Square statistic and determine the critical value.
Use symbolic math tools to calculate the Chi-Square statistic and determine the critical value.
Use the Chi-Square Goodness-of-Fit Test to determine if the observed distribution of a disease in a population matches the expected distribution based on a theoretical model.
Use the Chi-Square Goodness-of-Fit Test to determine if the observed distribution of customer preferences for a new product matches the expected distribution based on a theoretical model.
Use the Chi-Square Goodness-of-Fit Test to determine if the observed distribution of defects in a manufacturing process matches the expected distribution based on a theoretical model.
What is the purpose of the Chi-Square Goodness-of-Fit Test?
A) To determine if two variables are correlated B) To determine if the observed distribution of a variable matches the expected distribution based on a theoretical model C) To determine if a variable is normally distributed D) To determine if a variable is independent of another variable
Correct Answer: B) To determine if the observed distribution of a variable matches the expected distribution based on a theoretical model
What is the formula for the Chi-Square statistic?
A) $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ B) $\chi^2 = \sum \frac{(O_i - E_i)^2}{O_i}$ C) $\chi^2 = \sum \frac{(O_i - E_i)^2}{O_i + E_i}$ D) $\chi^2 = \sum \frac{(O_i - E_i)^2}{O_i - E_i}$
Correct Answer: A) $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$
What is the critical value of the Chi-Square distribution used to determine if the observed frequencies are consistent with the expected frequencies?
A) 9.488 B) 11.07 C) 12.68 D) 15.07
Correct Answer: A) 9.488
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