Math Problem Statement

Find the critical values chi squared Subscript Upper Lχ2L and chi squared Subscript Upper Rχ2R for the given confidence level c and sample size n. cequals=0.80.8​, nequals=2525 Question content area bottom Part 1 chi squared Subscript Upper Lχ2Lequals=enter your response here ​(Round to three decimal places as​ needed.)

Solution

To find the critical values χL2\chi^2_L and χR2\chi^2_R for a given confidence level cc and sample size nn, we need to determine the critical points from the chi-square distribution.

Given:

  • Confidence level c=0.8c = 0.8
  • Sample size n=25n = 25

The degrees of freedom (dfdf) is calculated as n1n - 1: df=n1=251=24df = n - 1 = 25 - 1 = 24

The confidence level of 0.8 corresponds to the middle 80% of the chi-square distribution. This means that 10% of the distribution is in each tail (since 100% - 80% = 20%, and this is divided equally between the two tails).

We need to find the critical values for the chi-square distribution at the lower 10% and upper 90% tails:

  • χL2\chi^2_L is the 10th percentile (0.1 quantile).
  • χR2\chi^2_R is the 90th percentile (0.9 quantile).

Using a chi-square distribution table or calculator for df=24df = 24:

  1. For the lower tail critical value (χL2\chi^2_L): χL2=χ0.1,242\chi^2_L = \chi^2_{0.1, 24}

  2. For the upper tail critical value (χR2\chi^2_R): χR2=χ0.9,242\chi^2_R = \chi^2_{0.9, 24}

Using a chi-square calculator or table for df=24df = 24:

  • χ0.1,24215.659\chi^2_{0.1, 24} \approx 15.659 (rounded to three decimal places)
  • χ0.9,24235.172\chi^2_{0.9, 24} \approx 35.172 (rounded to three decimal places)

Therefore: χL215.659\chi^2_L \approx 15.659 χR235.172\chi^2_R \approx 35.172

Would you like more details or have any questions? Here are some related questions:

  1. How do you find the degrees of freedom for a chi-square test?
  2. What is the chi-square distribution used for?
  3. How do you interpret the critical values in a chi-square test?
  4. What are the properties of the chi-square distribution?
  5. How do you find the chi-square critical value using a statistical table?
  6. What is the relationship between confidence level and critical values in a chi-square test?
  7. How do changes in sample size affect the critical values in a chi-square test?
  8. What are the applications of the chi-square test in real-world scenarios?

Tip: Always double-check your calculations and use accurate tables or calculators to ensure precision in your statistical analysis.

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Math Problem Analysis

Mathematical Concepts

Chi-square Distribution
Statistical Inference

Formulas

Chi-square critical values calculation

Theorems

Chi-square distribution properties

Suitable Grade Level

Advanced