Math Problem Statement

Find the critical values 

𝜒21 − 𝛼/2

 and 

𝜒2𝛼/2

 for the given confidence level and sample size. (Round your answers to three decimal places.)

confidence level 99%, 

n = 90

𝜒21 − 𝛼/2

𝜒2𝛼/2

Solution

To find the critical values for a chi-square distribution for a 99% confidence level with a sample size of n=90n = 90, we need to determine:

  1. Degrees of Freedom df=n1=901=89df = n - 1 = 90 - 1 = 89
  2. Confidence Level 99%99\% (which implies a significance level α=10.99=0.01\alpha = 1 - 0.99 = 0.01)

Given that we're working with a two-tailed confidence interval:

  1. χ1α/22\chi^2_{1 - \alpha/2} is the chi-square value where the right tail has α/2\alpha/2 probability.
  2. χα/22\chi^2_{\alpha/2} is the chi-square value where the left tail has α/2\alpha/2 probability.

Since α=0.01\alpha = 0.01:

  • α/2=0.005\alpha/2 = 0.005, meaning we need the 0.995 and 0.005 quantiles for the chi-square distribution with 89 degrees of freedom.

Calculation

Using chi-square tables or a statistical calculator for a chi-square distribution with df=89df = 89:

  1. χ1α/22=χ0.9952\chi^2_{1 - \alpha/2} = \chi^2_{0.995}: This corresponds to the 99.5th percentile.
  2. χα/22=χ0.0052\chi^2_{\alpha/2} = \chi^2_{0.005}: This corresponds to the 0.5th percentile.

After rounding to three decimal places, we get:

χ1α/22=χ0.9952124.116\chi^2_{1 - \alpha/2} = \chi^2_{0.995} \approx 124.116 χα/22=χ0.005259.342\chi^2_{\alpha/2} = \chi^2_{0.005} \approx 59.342

Final Answers

χ1α/22=124.116\chi^2_{1 - \alpha/2} = 124.116 χα/22=59.342\chi^2_{\alpha/2} = 59.342

Would you like further details on chi-square distribution properties or help with similar problems?


Here are some follow-up questions:

  1. How are critical values used in hypothesis testing?
  2. What does it mean to have a 99% confidence level in this context?
  3. How would changing the sample size affect the critical values?
  4. What are chi-square distribution properties?
  5. How do we interpret the two-tailed critical region?

Tip: Chi-square distributions are skewed to the right, especially with lower degrees of freedom. Higher degrees reduce this skew.

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Math Problem Analysis

Mathematical Concepts

Statistics
Chi-square distribution
Confidence interval

Formulas

Degrees of freedom: df = n - 1
Significance level: α = 1 - confidence level
Critical values for chi-square: 𝜒²(1 - 𝛼/2) and 𝜒²(𝛼/2)

Theorems

Chi-square distribution properties

Suitable Grade Level

Undergraduate (Statistics or Advanced Placement)