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 , we need to determine:
- Degrees of Freedom
- Confidence Level (which implies a significance level )
Given that we're working with a two-tailed confidence interval:
- is the chi-square value where the right tail has probability.
- is the chi-square value where the left tail has probability.
Since :
- , 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 :
- : This corresponds to the 99.5th percentile.
- : This corresponds to the 0.5th percentile.
After rounding to three decimal places, we get:
Final Answers
Would you like further details on chi-square distribution properties or help with similar problems?
Here are some follow-up questions:
- How are critical values used in hypothesis testing?
- What does it mean to have a 99% confidence level in this context?
- How would changing the sample size affect the critical values?
- What are chi-square distribution properties?
- 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)
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