Question

Find the tabled values of \(q_{\alpha}(k, d f)\) using the information given. $$ \alpha=.01, k=4, d f=8 $$

Short answer

Answer: 20.09

Step-by-step solution

Identify the Chi-Square Distribution Parameters

First, we need to identify the parameters of the chi-square distribution which are the significance level \(\alpha = 0.01\), the number of groups \(k = 4\), and the degrees of freedom \(df=8\).

Look up the Tabled Values for \(\chi^2_{\alpha}(k, d f)\)

Using a chi-square table or a chi-square calculator, we can look up the tabled values for \(\chi^2_{0.01}(4, 8)\).

Find the Tabled Values for \(q_{\alpha}(k, d f)\)

After examining the chi-square table or using a chi-square calculator, we find that the tabled value for \(\chi^2_{0.01}(4, 8)\) is 20.09.

Report the Result

The tabled value for \(q_{0.01}(4, 8)\) is 20.09. This means that, if the test statistic exceeds 20.09, we reject the null hypothesis at a 0.01 significance level when comparing four groups with 8 degrees of freedom.

Key concepts

Chi-Square Table

When working with chi-square distribution, the chi-square table is an essential tool. It presents the critical values for the chi-square test statistic for various degrees of freedom and significance levels. Understanding this table is key for anyone conducting a hypothesis test.

Here's how to decipher it: For a given significance level and degrees of freedom, the table provides the threshold value your test statistic must exceed to consider the results statistically significant. Essentially, it tells you how extreme the observed data must be relative to the null hypothesis to warrant a second look.

In the textbook exercise provided, you would consult the chi-square table to determine the critical value at a 0.01 significance level with 8 degrees of freedom. Intuitively, this table is a quick reference that saves time and effort instead of calculating the values from scratch every time you conduct a test. It's a staple in statistics, especially for analyses in fields like psychology, biology, and political science, where categorical data is often analyzed.

Degrees of Freedom

Degrees of freedom, in statistics, is a concept that describes the number of independent values in a study that are free to vary. When conducting a chi-square test, the degrees of freedom are calculated as the number of categories minus the number of parameters estimated.For instance, when comparing observed and expected frequencies in a chi-square goodness of fit test, the degrees of freedom would be the number of categories minus one. Meanwhile, for a chi-square test of independence, you calculate the degrees of freedom by multiplying the number of rows minus one by the number of columns minus one.Why is this important? The degrees of freedom factor into determining the critical values from the chi-square table, as seen in our exercise where the degrees of freedom were 8. It serves as a correction factor so that the size of the test statistic is adjusted for the sample size and the constraints imposed by the data, ensuring the validity of the test results.

Significance Level

The significance level, usually denoted by \( \alpha \), is a threshold set by the researcher to determine whether to reject the null hypothesis. It represents the probability of making a Type I error – that is, falsely rejecting the null hypothesis when it is true.

A typical significance level used in many scientific studies is 0.05, but more stringent criteria, such as 0.01 or 0.001, can also be used depending on the research context and the need for precision. The smaller the \( \alpha \) value, the stronger the evidence must be to reject the null hypothesis.

In the context of our exercise, an \( \alpha \) of 0.01 indicates a 1% risk of concluding there is an effect when there is not one, reflecting a high level of confidence in the results of the test. Consequently, the corresponding critical value from the chi-square table will be higher to reflect this stringent requirement.

Null Hypothesis

The null hypothesis is a default statement that there is no effect or no difference, and it's the hypothesis that a statistical test aims to challenge or verify. In the context of chi-square tests, the null hypothesis often posits that there is no association between the categorical variables in question.Accepting or rejecting the null hypothesis is not a statement of absolute certainty but rather a decision made with respect to the level of evidence stipulated by the significance level. If our test statistic is higher than the critical value from the chi-square table, we have sufficient evidence to reject the null hypothesis at the chosen significance level.For our exercise, using a chi-square test with a significance level of 0.01 and 8 degrees of freedom, the null hypothesis would be rejected if the test statistic exceeded 20.09, which is a high benchmark, indicating strong evidence against the null hypothesis.