Problem 1 - Basic Statistics for Business and Economics with Student

Question

In a particular chi-square goodness-of-fit test there are four categories and 200 observations. Use the .05 significance level. a. How many degrees of freedom are there? b. What is the critical value of chi-square?

Step-by-Step Solution

Answer

a. 3 degrees of freedom; b. Critical value is approximately 7.81.

Step 1: Understanding Degrees of Freedom

In a chi-square goodness-of-fit test, the degrees of freedom (\( df \)) are calculated using the formula:\[ df = k - 1 \]where \( k \) is the number of categories. Here, we are given 4 categories.

Step 2: Calculating Degrees of Freedom

Using the formula \( df = k - 1 \) and substituting \( k = 4 \), the degrees of freedom are:\[ df = 4 - 1 = 3 \]

Step 3: Finding the Critical Value

To find the critical value for chi-square (\( \chi^2 \)) at a 0.05 significance level, we refer to the chi-square distribution table. Given our degrees of freedom (\( df = 3 \)), we look up the critical value for \( \chi^2 \) at the 0.05 significance level.

Step 4: Reading the Critical Value

For \( df = 3 \) and a significance level of 0.05, the critical value of \( \chi^2 \) can be found in standard chi-square tables, which is approximately 7.81.

Key Concepts

Degrees of Freedom

Degrees of freedom (often abbreviated as df) in a chi-square goodness-of-fit test is a crucial concept. It determines the number of values that are free to vary in your calculation, essentially telling us how many independent comparisons we can make without exhausting the data's variability.
To calculate the degrees of freedom for a chi-square test, you use the simple formula:

• For goodness-of-fit tests: \( df = k - 1 \)

where \( k \) is the number of categories.
In our exercise, with 4 categories, the degrees of freedom are calculated as \(3\), implying we have that many independent comparisons available.

Critical Value

The critical value in a chi-square test acts as a threshold. It helps us decide whether to reject or not reject the null hypothesis.
When performing this test, you'd look up a critical value in a chi-square distribution table. The critical value depends on both the degrees of freedom and the chosen significance level (more on that later).
Here's how the search for critical value can be broken down:

• Find the row for your degrees of freedom (df). In our case, df is 3.
• Identify the column for your significance level, which is 0.05.
• The intersection of this row and column gives the critical value.

For df = 3, and a 0.05 significance level, the critical value found in standard tables is usually around 7.81.

Significance Level

The significance level, denoted by \( \alpha \), is a probability threshold we set before conducting a statistical test. It's a way to control for Type I errors—rejecting a true null hypothesis.
The significance level is typically set at \( \alpha = 0.05 \) or 5% for many statistical tests, including this one. This means that if the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample is less than 0.05, you reject the null hypothesis.
Some key points about significance level:

• A smaller significance level means stricter requirements for rejecting the null hypothesis (e.g., \( \alpha = 0.01 \)).
• The chosen \( \alpha \) should reflect the acceptable risk of a wrong rejection, often informed by the context of the research.

So, for our chi-square test, a significance level of 0.05 frames the assessment of evidence against the null hypothesis.

Goodness-of-Fit Test

The goodness-of-fit test is a specific application of the chi-square test that evaluates how well a theoretical distribution fits observed data.
This is particularly useful when you want to see if your data distribution matches an expected one, for example, a hypothesized proportion of categories.
Here’s how it typically works:

• Formulate the null hypothesis, stating that there’s no difference between observed and expected frequencies.
• Calculate the chi-square statistic using the formula:\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]where \( O_i \) is the observed frequency, and \( E_i \) is the expected frequency for each category.
• Compare the calculated statistic with the critical value found in the chi-square table.
• If the statistic exceeds the critical value, you reject the null hypothesis, indicating a significant difference.

By applying the goodness-of-fit test with a proper understanding, you confirm or refute assumptions about your data’s distribution.