Question

Use Table 5 in Appendix I to bound the \(p\) -value for a chi-square test: a. \(X^{2}=4.29, d f=5\) b. \(X^{2}=20.62, d f=6\)

Short answer

Answer: a) The p-value is bounded between 0.1 and 0.2. b) The p-value is bounded between 0.001 and 0.01.

Step-by-step solution

Locate the Appropriate Row in Appendix I Table 5

First, you need to find the row that corresponds to the degrees of freedom in the given problem. In this case, the degrees of freedom (df) are 5. Locate the row labeled "5" in the "df" column of Table 5 in Appendix I.

Find the Column(s) Where the Chi-Square Statistic Lies

Next, you need to identify the columns where the given \(X^{2}\) statistic (4.29) lies between. Look for the values in the row corresponding to the degrees of freedom that are closest to, but greater than and less than 4.29. In this case, you will find that the \(X^{2}\) statistic lies between the values in the columns labeled "0.20" (4.351) and "0.10" (6.626) of the table.

Identify the Bounds of the P-Value

Since the given \(X^{2}\) statistic lies between the values in the columns labeled "0.20" and "0.10," we can conclude that the p-value is bounded between 0.1 and 0.2. Case b: \(X^{2}=20.62, d f=6\)

Locate the Appropriate Row in Appendix I Table 5

First, you need to find the row that corresponds to the degrees of freedom in the given problem. In this case, the degrees of freedom (df) are 6. Locate the row labeled "6" in the "df" column of Table 5 in Appendix I.

Find the Column(s) Where the Chi-Square Statistic Lies

Next, you need to identify the columns where the given \(X^{2}\) statistic (20.62) lies between. Look for the values in the row corresponding to the degrees of freedom that are closest to, but greater than and less than 20.62. In this case, you will find that the \(X^{2}\) statistic lies between the values in the columns labeled "0.01" (16.812) and "0.001" (22.457) of the table.

Identify the Bounds of the P-Value

Since the given \(X^{2}\) statistic lies between the values in the columns labeled "0.01" and "0.001," we can conclude that the p-value is bounded between 0.001 and 0.01.

Key concepts

Understanding the P-Value in a Chi-Square Test

In a chi-square test, the p-value is a crucial component. It indicates the probability of observing a test statistic as extreme as, or more extreme than, the value calculated from the sample data. This is under the assumption that the null hypothesis is true. A smaller p-value suggests that there is stronger evidence against the null hypothesis.

In practical terms, calculating the p-value involves looking at a chi-square distribution table and identifying the bounds between which the observed chi-square statistic falls:

• Locate the degrees of freedom row that corresponds to your data.
• Identify the chi-square column nearest to your calculated statistic.
• The p-value will be between the corresponding probability values.

For example, if the statistic is less than the chi-square value for the 0.05 column and more than the 0.01 column, the p-value is between 0.01 and 0.05. Thus, understanding and interpreting p-values accurately can determine the direction and confidence of your hypothesis testing.

Degrees of Freedom in Statistics

The concept of degrees of freedom (df) is central in statistics, particularly when conducting a chi-square test. It describes the number of values in a calculation that are free to vary.

Degrees of freedom are important because they are used to determine the appropriate chi-square distribution to reference when evaluating a test statistic. In essence, it adjusts the distribution to better fit the data size and complexity you're working with.

• Degrees of freedom in a chi-square test is calculated as the number of categories minus one.
• They help to shape the distribution curve used in calculating p-values.
• The more the degrees of freedom, the closer the chi-square distribution will resemble a normal distribution.

Understanding degrees of freedom helps to ensure that the analysis reflects the true variability present in the data.

Chi-Square Statistic Explained

The chi-square statistic is the test statistic used in a chi-square test of independence or goodness of fit. It quantifies the difference between observed and expected frequencies in categorical data.

A higher chi-square statistic suggests a greater degree of deviation from what was expected in the null hypothesis. To compute it:

• Calculate the expected frequencies for each category, which are often based on the proportions expected if the null hypothesis were true.
• Use the formula: \[ X^{2} = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency.
• Compare this statistic to the critical value from a chi-square distribution table.

Proper interpretation of the chi-square statistic within the context of the degrees of freedom is essential to making informed decisions regarding the null hypothesis.