Question

Find the tabled values of \(q_{\alpha}(k, d f)\) using the information given. $$ \alpha=.05, k=3, d f=9 $$

Short answer

Answer: To find the tabled value of \(q_{\alpha}(k, d f)\) for a Chi-square distribution, follow these steps: 1. Understand the Chi-square distribution table, which shows critical values for different levels of significance (α) and degrees of freedom (d f). 2. Locate the given degrees of freedom (d f = 9) and significance level (α = 0.05) in the table. 3. Read the critical value at the intersection of the row with d f = 9 and the column with α = 0.05. 4. Report the tabled value of \(q_{\alpha}(k, d f)\) found in the previous step. Note that the actual value might vary depending on the specific table used.

Step-by-step solution

Understanding the Chi-square distribution table

The Chi-square distribution table shows the critical values for different levels of significance (α) and degrees of freedom (d f). Each cell in the table represents a combination of α and d f, and the value inside is the critical value or the quantile \(q_{\alpha}\) at that level of significance and degrees of freedom.

Locate the given degrees of freedom and significance level in the table

First, we have to locate the given degrees of freedom (d f = 9) in the table. The degrees of freedom are usually listed in the leftmost column of the table. Once we find the row with the given d f, we move along the row to find the column with the given significance level (α = 0.05).

Read the critical value at the intersection of the row and column

Once we have located the correct row and column in the Chi-square distribution table, we read the value inside the cell where the row and column intersect. The value inside this cell is the tabled value of \(q_{\alpha}(k, d f)\) that we are looking for.

Report the tabled value of \(q_{\alpha}(k, d f)\)

After locating and reading the critical value from the table, we can report the tabled value of \(q_{\alpha}(k, d f)\) for the given α and d f. Please note that the actual value might vary slightly depending on the specific Chi-square distribution table you are using. In general, you should use the most accurate and up-to-date table provided by your textbook or other resources. Since we are not including the actual table, we can't provide the exact value for \(q_{\alpha}(k, d f)\), but following these steps will yield the result using the appropriate Chi-square distribution table.

Key concepts

Degrees of Freedom

In the context of statistical analysis, the term degrees of freedom (df) refers to the number of independent pieces of information used to calculate a statistic. The concept is particularly crucial when dealing with various hypothesis testing techniques, including the Chi-square distribution which is commonly used for tests of independence and goodness-of-fit.

For a Chi-square distribution, the degrees of freedom are determined by the size of the data set or the number of categories minus any constraints necessary to calculate the statistic. For example, if you are testing the fit of a die roll with six possible outcomes (categories), the degrees of freedom would be five (that is, six minus one).

Understanding the degrees of freedom is important because it impacts the shape of the Chi-square distribution. Higher degrees of freedom generally lead to a distribution that is more spread out and resembles a normal distribution. When you're performing a Chi-square test, accurately identifying the degrees of freedom is essential to finding the correct critical values from the Chi-square distribution table. As such, when given a problem like the exercise, recognizing the degrees of freedom (df=9) allows the correct identification of the row in the Chi-square table.

Critical Values

A critical value in statistics is a point on the test distribution that is compared to the test statistic to decide whether to reject the null hypothesis. If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis.

In a Chi-square test, the critical value is determined based on the significance level (α) and the degrees of freedom. It marks the boundary of the rejection region for the test. The critical value is a key element in hypothesis testing because it establishes the threshold for statistical significance.

To find a critical value using the Chi-square distribution table as in the provided exercise, you locate the intersecting value of degrees of freedom and the significance level. This is done by first finding the row of the table corresponding to the degrees of freedom and then locating the column representing the chosen significance level. The exercise illustrates this by directing you to find the critical value for a test with df=9 and α=0.05. Since critical values vary for different tests and significance levels, using the correct table and interpreting its values accurately is of utmost importance.

Significance Level

In hypothesis testing, the significance level, denoted as α (alpha), is the probability of making a Type I error, which is rejecting a true null hypothesis. The significance level is a critical component in determining the threshold for statistical significance when interpreting test results.

The significance level is set by the researcher before conducting the test and commonly used values are 0.05, 0.01, or 0.10. These values correspond to a 5%, 1%, or 10% risk of incorrectly rejecting the null hypothesis. Lower significance levels mean that evidence must be stronger to reject the null hypothesis.

Linking the concept to the exercise, when you are asked to find the tabled values of the Chi-square distribution, you must know the significance level (α=0.05 in the exercise). This enables the identification of the correct column in the Chi-square table that aligns with the set α. The smaller the significance level, the further right (toward more extreme values) the critical value tends to be, indicating a stricter criterion for rejecting the null hypothesis. It's important to choose an appropriate significance level for your test as it affects the conclusions drawn from the statistical analysis.