Problem 2 - Basic Statistics for Business and Economics with Student

Question

In a particular chi-square goodness-of-fit test there are six categories and 500 observations. Use the .01 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. 5 degrees of freedom. b. Critical value is approximately 15.086.

Step 1: Determine the Number of Degrees of Freedom

The degrees of freedom (df) in a chi-square goodness-of-fit test is given by the formula \( df = n - 1 \), where \( n \) is the number of categories. For this exercise, there are 6 categories. Thus, the degrees of freedom is: \[ df = 6 - 1 = 5 \]

Step 2: Find the Critical Value of Chi-Square

To find the critical value of chi-square, we use the chi-square distribution table or a calculator. The critical value depends on the degrees of freedom and the significance level. For \( df = 5 \) and \( \alpha = 0.01 \), locate or calculate the critical value in the chi-square distribution table:The critical value is approximately \( 15.086 \).

Key Concepts

Degrees of Freedom

In the context of a chi-square goodness-of-fit test, the degrees of freedom are an essential element. But what exactly are they? Degrees of freedom refer to the number of independent values or quantities that can be assigned to a statistical distribution. It is a measure that helps determine the shape of the chi-square distribution.
To calculate the degrees of freedom in a chi-square goodness-of-fit test, you use the formula: \( df = n - 1 \), where \( n \) is the number of categories being analyzed. For instance, if there are six categories, as in our example, the degrees of freedom would be \( df = 6 - 1 = 5 \). This calculation helps us understand how many of the observed frequency counts can vary independently.
Degrees of freedom are important because they account for variability. The higher the degrees of freedom, the more data we have to estimate variability, resulting in a more accurate reflection of our sample when conducting hypothesis tests.

Significance Level

The significance level is a vital concept in hypothesis testing, including the chi-square goodness-of-fit test. It is denoted by \( \alpha \) and indicates the probability of rejecting a true null hypothesis. This is often referred to as the Type I error.
In simple terms, the significance level defines how confident you want to be in your hypothesis test results. Commonly used significance levels include 0.05, 0.01, and 0.10, but in our specific chi-square test, the significance level is set at 0.01. This means there is a 1% risk of rejecting the null hypothesis when it is actually true.
Choosing a lower significance level means you require stronger evidence before rejecting the null hypothesis, which decreases the likelihood of making a mistake in your hypothesis testing. However, it's crucial to balance this with the practicalities of your analysis and the consequences of potential errors.

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution that plays a critical role in various statistical tests, including the chi-square goodness-of-fit test. It is asymmetric and becomes more symmetric as the degrees of freedom increase.
This distribution is used to determine how well an observed distribution fits an expected distribution. When performing a chi-square goodness-of-fit test, the test statistic is calculated and then compared to a critical value from the chi-square distribution table. This comparison helps to decide whether to reject the null hypothesis.
For example, if you have 5 degrees of freedom and a 0.01 significance level, you'll look up a critical value in the chi-square table, which is approximately \( 15.086 \). If your calculated test statistic exceeds this critical value, you would reject the null hypothesis, indicating that the observed data do not fit the expected distribution well. Understanding how to use the chi-square distribution helps in identifying patterns and differences in categorical data, making it a powerful tool for statistical analysis.