Question

Use the information Give the rejection region for a chi-square test of specified probabilities if the experiment involves \(k\) categories. $$ k=4, \alpha=.10 $$

Short answer

Answer: The rejection region for this test is when χ² > 6.251.

Step-by-step solution

Identify the given parameters

Here, we are given the number of categories (k) and the significance level (α), which are: $$ k = 4 \\ α = 0.10 $$

Determine the degrees of freedom

The degrees of freedom for the chi-square distribution are given by the formula: $$ \text{degrees of freedom} = k - 1 $$ In our case, $$ \text{degrees of freedom} = 4 - 1 = 3 $$

Find the chi-square critical value

Using the chi-square distribution table, we can find the critical value corresponding to the significance level (α) and the degrees of freedom. Since the significance level is 0.10 and the degrees of freedom are 3, the chi-square critical value is: $$ \chi^2_{critical} = 6.251 $$

Determine the rejection region

For a one-tailed chi-square test, the rejection region consists of chi-square values greater than the critical value obtained in the previous step. Therefore, the rejection region for this problem is: $$ \chi^2 > 6.251 $$

Key concepts

What is the Chi-Square Critical Value?

The chi-square critical value is a key threshold in hypothesis testing that determines whether the observed differences between expected and actual data are due to chance or are statistically significant. To find it, one must refer to a chi-square distribution table after calculating the degrees of freedom for the test. Upon establishing the significance level, which reflects the probability of rejecting a true null hypothesis, one locates the critical value. If the test statistic exceeds this value, we reject the null hypothesis, concluding there's a statistically significant effect. For instance, in an experiment with four categories and a 0.10 significance level, our chi-square critical value would be 6.251, which serves as the gateway to the rejection region in our hypothesis test.

Understanding the Chi-Square Test

Conducting a chi-square test involves comparing the counts observed in different categories to what we would expect if there were no effect or association. In this comparison, the chi-square statistic is calculated and then weighed against the critical value. If our chi-square statistic is bigger, this suggests that our data does not fit well with the null hypothesis of no association or effect.

Degrees of Freedom Explained

Degrees of freedom in a statistical test are a measure of the amount of independent information available to estimate a parameter. It often depends on the size of the sample data and the constraints imposed during the estimation process. Specifically, in the context of the chi-square test, degrees of freedom equals the number of categories minus one (\(k - 1\)). The concept is crucial because it influences the shape of the chi-square distribution we use to determine the critical value. For the provided exercise with four categories (\(k = 4\)), the degrees of freedom would be 3 (\(4 - 1\)).

Why Degrees of Freedom Matter

It's essential to accurately calculate degrees of freedom because this will directly impact the location of the critical value on the chi-square distribution. If the degrees of freedom are miscalculated, the critical value would be incorrect, potentially leading to erroneous conclusions about the data.

Understanding Significance Level

The significance level, denoted as \(\alpha\), is the threshold of probability for rejecting the null hypothesis when it is actually true. It is pre-selected by the researcher and commonly set at 0.05, 0.01, or 0.10. A lower significance level means that the test is more conservative, decreasing the chances of a Type I error (false positive). In the given exercise, a significance level of 0.10 implies that there is a 10% chance of rejecting the null hypothesis even if it is true, which can be considered a less conservative approach.

Choosing the Right Significance Level

Deciding on an appropriate significance level depends on the context of the study and the risk one is willing to accept for making a wrong decision. Higher stakes decisions often warrant a lower significance level to protect against incorrect findings. Thus, the chosen significance level directly influences how 'strict' our hypothesis test will be.

The Chi-Square Distribution Table

The chi-square distribution table is an essential tool for determining the chi-square critical value. It displays critical values of the chi-square distribution that correspond to different significance levels (\(\alpha\)) across various degrees of freedom. The table provides a quick reference to check against the chi-square test statistic calculated from the data. To use the table, one matches the desired significance level and degrees of freedom to find the critical value, as we did in our exercise with a 0.10 significance level and 3 degrees of freedom. If the calculated chi-square statistic exceeds the critical value from the table, we reject the null hypothesis.

Navigating the Distribution Table

When looking up critical values, it's crucial to understand that the table assumes a right-tailed test by default. The values correspond to cumulative probabilities from the right end of the distribution. Overlooking this convention can lead to misinterpreting the results of a chi-square test.