Finding Critical Values of (chi)^2 For large numbers of degrees of freedom, we can approximate critical values of as follows:
(chi)^2 = (1/2)(z + sqrt(2k-1))

Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of (chi)^2 in the right-tailed hypothesis test with α = 0.01 and a sample size of 56, we let k =55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of for Exercise 12. How close is it to the critical value of (chi)^2 = 82.292 obtained by using Statdisk and Minitab?

Verified step by step guidance

Step 1: Understand the formula for approximating the critical value of chi-squared: \( \chi^2 = \frac{1}{2}(z + \sqrt{2k - 1}) \). Here, \( k \) is the degrees of freedom, and \( z \) is the critical value obtained from technology or a table.

Step 2: Identify the given values from the problem. The degrees of freedom \( k \) is 55, and the critical value \( z \) is provided as 2.33 (or the more accurate value 2.326348).

Step 3: Substitute \( k = 55 \) into the formula \( \sqrt{2k - 1} \) to calculate \( \sqrt{2 \times 55 - 1} \). This simplifies to \( \sqrt{109} \).

Step 4: Substitute \( z = 2.33 \) (or \( z = 2.326348 \)) and \( \sqrt{109} \) into the formula \( \chi^2 = \frac{1}{2}(z + \sqrt{2k - 1}) \). This becomes \( \chi^2 = \frac{1}{2}(2.33 + \sqrt{109}) \) or \( \chi^2 = \frac{1}{2}(2.326348 + \sqrt{109}) \).

Step 5: Compare the approximated critical value of \( \chi^2 \) obtained using the formula to the exact critical value of \( \chi^2 = 82.292 \) provided in the problem. This step involves evaluating how close the approximation is to the exact value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chi-Squared Distribution

The Chi-Squared distribution is a statistical distribution commonly used in hypothesis testing, particularly in tests of independence and goodness-of-fit. It is defined by its degrees of freedom, which typically correspond to the number of categories or groups being analyzed. As the degrees of freedom increase, the distribution approaches a normal distribution, allowing for approximations in critical value calculations.

Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the relevant statistical distribution. In the context of the Chi-Squared distribution, the critical value indicates the point beyond which the observed statistic would lead to the rejection of the null hypothesis, thus providing a basis for decision-making.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the Chi-Squared test, degrees of freedom are typically calculated as the number of categories minus one. They play a crucial role in determining the shape of the Chi-Squared distribution and influence the critical values used in hypothesis testing.

Watch next

Master Step 1: Write Hypotheses with a bite sized video explanation from Patrick

Start learning