Problem 47 - An Introduction to Mathematical Statistics and Its Applications

Question

For df values beyond the range of Appendix Table A.3, chi square cutoffs can be approximated by using a formula based on cutoffs from the standard normal pdf, \(f_{Z}(z) .\) Define \(\chi_{p, n}^{2}\) and \(z_{p}^{*}\) so that \(P\left(\chi_{n}^{2} \leq \chi_{p, n}^{2}\right)=p\) and \(P\left(Z \leq z_{p}^{*}\right)=p\), respectively. Then $$ \chi_{p, n}^{2} \doteq n\left(1-\frac{2}{9 n}+z_{p}^{*} \sqrt{\frac{2}{9 n}}\right)^{3} $$ Approximate the 95 th percentile of the chi square distribution with 200 df. That is, find the value of \(y\) for which $$ P\left(\chi_{200}^{2} \leq y\right) \doteq 0.95 $$

Step-by-Step Solution

Answer

The value of \(y\), the 95th percentile of chi-square distribution with 200 degrees of freedom is approximately 219.4.

Step 1: Find the z-score for the 95th percentile

For a standard normal distribution, the \(z_{p}^{*}\) value corresponding to the 95th percentile (p=0.95) can be found from standard normal distribution tables or through calculation. Its approximately 1.645.

Step 2: Substitution

Now that we have \(z_{p}^{*}\), the number of degrees of freedom, n, is given as 200. We can substitute these values into the given chi-square approximation formula \( \chi_{p, n}^{2} \doteq n\left(1-\frac{2}{9 n}+z_{p}^{*} \sqrt{\frac{2}{9 n}}\right)^{3} \).

Step 3: Compute y

After substituting \(n=200\) and \(z_{p}^{*} = 1.645\), perform the arithmetics to compute the value for y.

Key Concepts

Degrees of freedom

The concept of degrees of freedom is essential in statistics and particularly in the context of the chi-square distribution. Degrees of freedom ( "df" ) refers to the number of independent values or quantities that are free to vary when calculating a statistic. In the case of the chi-square distribution, the degrees of freedom often denote the sample size minus the number of parameters used as estimates. **Why are Degrees of Freedom Important?**

• Degrees of freedom allows for more accurate interpretations and estimation of statistical results.
• They play a crucial role in determining the shape of the chi-square distribution: as the degrees of freedom increase, the chi-square distribution approaches a normal distribution.
• They help in providing the cutoff values from chi-square tables, needed for hypothesis testing or constructing confidence intervals.

In the given exercise, we use 200 degrees of freedom in the formula provided to approximate the chi-square cutoff value. This high df value suggests that the chi-square distribution closely resembles the bell shape of the standard normal distribution.

Standard normal distribution

The standard normal distribution is a special type of normal distribution where the mean is 0 and the standard deviation is 1. This bell-shaped curve is symmetrical around the mean.Understanding the standard normal distribution is a cornerstone of statistical knowledge since many statistical methods and tests rely on it.**Key Features:**

• The mean, median, and mode of the distribution are all equal, at 0.
• About 68% of the data falls within one standard deviation from the mean.
• It is used to transform data to a common scale: a process called standardization.

In this exercise, the value that corresponds to the 95th percentile of the standard normal distribution is used. This is denoted as \(z_{p}^{*}\) and typically found in z-tables or through calculation, giving us approximately 1.645 for the 95th percentile. This value is pivotal in approximating percentile cutoffs for other distributions, such as in the given chi-square approximation formula.

95th percentile

The 95th percentile is a statistical measure indicating the value below which 95% of the observations in a dataset lie. This metric is frequently used in statistics to evaluate or measure key outcomes above which only the top 5% of occurrences are found.For continuous probability distributions like the normal and chi-square distributions, this concept is crucial in determining threshold values:**Significance:**

• In hypothesis testing, the 95th percentile is often used to identify critical values that help in deciding if you should reject the null hypothesis.
• This percentile is associated with a 5% Type I error probability when conducting two-tailed tests.

In the context of this exercise, we're approximating the value, \(y\), for which 95% of the distribution's values are below it, using the formula \(\chi_{p, n}^{2}\). By calculating this, we understand where the data tends to cluster and the point that separates the vast majority of data from the highest values.