Problem 45 - An Introduction to Mathematical Statistics and Its Applications

Question

Find the value \(y\) that satisfies each of the following equations: (a) \(P\left(\chi_{9}^{2} \geq y\right)=0.99\) (b) \(P\left(\chi_{15}^{2} \leq y\right)=0.05\) (c) \(P\left(9.542 \leq x_{22}^{2} \leq y\right)=0.09\) (d) \(P\left(y \leq \chi_{31}^{2} \leq 48.232\right)=0.95\)

Step-by-Step Solution

Answer

The values of 'y' are obtained by consulting the chi-square distribution tables with specified degrees of freedom or by using a relevant statistical software. It's important to note that these values vary slightly based on different chi-square tables or algorithms used by software.

Step 1: Understand the Chi-Square Distribution

Chi-square distribution is used in the field of inferential statistics. It is applied when there is a need to test the goodness of fit, and to find out the independence of two datasets. The equation for Chi-square distribution is given as \(P(\chi_{n} ^{2} \leq y)\) or \(P(\chi_{n} ^{2} \geq y)\) where 'n' denotes degrees of freedom and 'y' is the variable whose value we will be finding using the statistical tables.

Step 2: Solve for (a) \(P(\chi_{9}^{2} \geq y)=0.99\)

For the chi-square distribution \(\chi^{2}_{9}\), we want to find the critical value of 'y' for which the tail probability is 0.99. That means, the cumulative probability from \(-\infty\) to 'y' is 1 - 0.99 = 0.01. Looking up this value in the chi-square distribution table for 9 degrees of freedom or by using statistical software will give the corresponding 'y' value.

Step 3: Solve for (b) \(P(\chi_{15}^{2} \leq y)=0.05\)

For the chi-square distribution \(\chi^{2}_{15}\), we want to find the critical value of 'y' for which the cumulative probability is 0.05. Consulting the chi-square table with 15 degrees of freedom or using a statistical software will give the respective 'y' value.

Step 4: Solve for (c) \(P(9.542 \leq \chi_{22}^{2} \leq y)=0.09\)

This question asks for the value of 'y' such that the probability of the chi-square distribution \(\chi^{2}_{22}\) lying between 9.542 and 'y' is 0.09. Find the cumulative probability for 9.542, let's say it is 'A'. Now we need to find 'y' such that the cumulative probability is 'A' + 0.09. Lookup 'A' + 0.09 in the chi-square distribution table for 22 degrees of freedom or use statistical software to get the corresponding 'y'.

Step 5: Solve for (d) \(P(y \leq \chi_{31}^{2} \leq 48.232)=0.95\)

In this problem, we need to find 'y' such that the probability of the chi-square distribution \(\chi^{2}_{31}\) lying between 'y' and 48.232 is 0.95. Find the cumulative probability for 48.232, let's say it is 'B'. Now we need to find 'y' such that 'B' - 'y' = 0.95. Lookup 'B' - 0.95 in the chi-square distribution table for 31 degrees of freedom or use statistical software to get the corresponding 'y'.

Key Concepts

Inferential Statistics

Inferential statistics allows us to draw conclusions about a population based on a sample taken from it. Unlike descriptive statistics, which simply describe the data we have, inferential statistics focuses on making predictions or inferences about a larger group.
The chi-square distribution is a key tool in inferential statistics, often used in hypothesis testing to determine if there is a significant association between categorical variables.
For example, in a chi-square goodness of fit test, we might use the chi-square distribution to assess if the observed distribution of data matches an expected distribution. Similarly, a chi-square test of independence helps evaluate if two categorical variables are related or not.

Degrees of Freedom

Degrees of freedom are an important concept in statistics that describe the number of values in a calculation that are free to vary. They are crucial in determining the shape of the chi-square distribution.
In a chi-square test, the degrees of freedom play a key role in interpreting the results. They are usually calculated based on the number of categories of data or the number of variables involved in the test.

• In a chi-square goodness of fit test, the degrees of freedom are typically the number of observed categories minus one.
• For a chi-square test of independence, it is calculated as \((r-1)(c-1)\), where \(r\) is the number of rows and \(c\) is the number of columns in a contingency table.

A proper understanding of degrees of freedom ensures accurate analysis of statistical data.

Cumulative Probability

The concept of cumulative probability is key in understanding probability distributions, including the chi-square distribution. Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value.
In the context of the chi-square distribution, cumulative probabilities are used to determine critical values, which are points in the distribution where the probability of the observed data is compared against a threshold.
For instance, if the cumulative probability is given as 0.05, it means there is a 5% chance of the random variable being less than or equal to the critical value you are observing. This helps in identifying how extreme a specific outcome is within the context of your data set.

Chi-Square Table

A chi-square table is an essential tool used to find critical values for chi-square hypothesis tests. It lists the critical values of the chi-square distribution for different degrees of freedom at various significant levels.
To use a chi-square table, one must know:

• The degrees of freedom for the data set.
• The desired level of significance, such as 0.05 or 0.01.

With this information, one can look up the critical value corresponding to these parameters in the table. These critical values can then be used to determine the significance of the test results.
Remember, these tables help in making decisions about the outcomes of chi-square tests, such as whether to reject or accept a null hypothesis.