Question

Find the tabled values of \(q_{\alpha}(k, d f)\) using the information given. $$ \alpha=.05, k=5, d f=10 $$

Short answer

Answer: [Using the instructions given, the student should look up an F-distribution table and find the critical F-value themselves at the intersection of row 5 and column 10.]

Step-by-step solution

Locate the correct F-distribution table

Search for "F-distribution table 0.05" on the internet or in a statistics textbook. The table should be labeled with the given significance level of 0.05.

Identify the numerator degrees of freedom (k) and the denominator degrees of freedom (df)

Using the information given in the exercise, we have: Numerator degrees of freedom (\(k\)) = 5 Denominator degrees of freedom (\(df\)) = 10

Find the tabled value

Locate the row of the F-distribution table corresponding to the numerator degrees of freedom (\(k\)) and the column corresponding to the denominator degrees of freedom (\(df\)). Then, find the value at the intersection of the row and column you identified. In this case, find the value at the intersection of row 5 (numerator \(k\)) and column 10 (denominator \(df\)).

Report the tabled value

The tabled value found at the intersection of row 5 and column 10 in the F-distribution table is the required critical F-value for the given significance level, numerator degrees of freedom, and denominator degrees of freedom. Report this value as \(q_{\alpha}(k, df)\). Note that the values in F-distribution tables might slightly vary depending on the source; however, the value should be approximately the same.

Key concepts

Critical F-value

The critical F-value plays a crucial role in hypothesis testing, particularly when dealing with variances or analyzing variances across multiple groups. It is the cut-off point beyond which we reject the null hypothesis. To find the critical F-value, as demonstrated in the problem, we refer to an F-distribution table which requires two types of degrees of freedom: the numerator (representing the number of groups minus one) and the denominator (representing the total number of observations minus the number of groups).The critical value corresponds to a specific level of significance, typically represented by \(\alpha\), which denotes the probability of rejecting the null hypothesis when it is actually true. In the given problem, with \(\alpha=.05\), \(k=5\), and \(df=10\), you use these numbers to navigate the F-distribution table. The resulting critical F-value is the threshold above which the actual F-statistic calculated from the data would suggest that the variances are significantly different, and thus, you would reject the null hypothesis in an ANOVA test, for example.

Degrees of freedom

Degrees of freedom (df) are a fundamental concept in statistics, indicating the number of independent values that can vary in an analysis without breaking any constraints. It is crucial for determining the appropriate distribution to reference when assessing statistical significance.

In the context of the F-distribution, we differentiate between two types of degrees of freedom: the numerator degrees of freedom \(k\), which are associated with the number of groups or treatment levels in your test (minus one), and the denominator degrees of freedom \(df\), which are linked to the total sample size across all groups (minus the number of groups). These 'degrees of freedom' help shape the F-distribution curve, which is different for each pair of \(k\) and \(df\). The larger the degrees of freedom, the closer the F-distribution gets to a normal distribution shape.

Statistical significance

Statistical significance is the measure of whether the results obtained from a study or experiment are likely not due to chance. It's a way to quantify the reliability of an effect or difference found in the data. In hypothesis testing, statistical significance is denoted by \(\alpha\), the significance level, which is the threshold of probability below which we reject the null hypothesis, indicating that the findings are noteworthy. Common \(\alpha\) levels include 0.05, 0.01, and 0.001.When analyzing data, if the observed test statistic is more extreme than the critical value from the relevant distribution table (be it F-distribution, t-distribution, etc.), and falls into the tail ends representing a combined area of \(\alpha\), we declare the results statistically significant. This would suggest that the observed effect or difference is pronounced enough to be unlikely due to random chance and merits further consideration or action.