Confidence Interval for Chi-Squared - read the quantile

In summary, a confidence interval for Chi-Squared is a range of values that estimates the precision and accuracy of the Chi-Squared statistic with a certain level of confidence. It is calculated using the Chi-Squared distribution and the quantile function, and the sample size has a direct impact on its width. The quantile represents the critical values used in the calculation and the confidence level represents the probability of the true population value falling within the interval. A higher confidence level results in a wider interval.
  • #1
nickyveronica
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0

Homework Statement



1000 samples each of size n=121, where observations Xi ~ iid N ( m, sigma^2), for each sample we calculate Confidence Interval: I = [ 0.75 s^2, s^2 + s^2/4 ] where s^2=(1/(n-1))[tex]\Sigma[/tex](Xi-X bar)^2

how many samples include the true value of sigma^2, ie 1000 * prob.

Homework Equations



we know that (n-1) s^2/sigma^2 ~ Chi-sqrd with (n-1) degrees of freedom

The Attempt at a Solution



so i started:

P (0.75s^2 < sigma^2 < s^2 + s^2/4) = P (0.75 < sigma^2/s^2 < 5/4) =
=P (0.75/(n-1) < sigma^2/(s^2(n-1)) < 1.25/(n-1)) =
=P (1.25/(n-1) < (n-1)s^2/sigma^2 < 0.75/(n-1)) =
=P (1.25/120 < Chi-sqrd with (n-1) df < 0.75/120) =
=[tex]\chi[/tex] n-1 df (0.75/120) - [tex]\chi[/tex] n-1 df (1.25/120) =
=[tex]\chi[/tex] n-1 df (0.00625) - [tex]\chi[/tex] n-1 df (0.0104167) ... =
=1 -[tex]\alpha[/tex]

and now I am stuck, since i am not sure if i am allowed to deduct them, and whether the numbers represent the quantiles of chi-squared, and if yes, i have a complications to find them in the table since was trying to get each separaetely and then get the p value for v=120 and then deduct but don't like it...

how do i finish it now to get the probability...

thank you, if any

nicky
 
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  • #2


Dear nicky,

Thank you for your post. Let me help you complete your solution.

First, you are correct in using the fact that (n-1) s^2/sigma^2 ~ Chi-sqrd with (n-1) degrees of freedom. This means that the confidence interval you have calculated for each sample is based on a chi-squared distribution with (n-1) degrees of freedom.

Next, you are on the right track in using the chi-squared distribution to calculate the probability of including the true value of sigma^2. The probability that the true value of sigma^2 falls within the confidence interval for each sample is equal to the area under the chi-squared distribution curve between the two values of 0.75/(n-1) and 1.25/(n-1).

To find this probability, you can use a chi-squared distribution table or a statistical software program. For example, if you use a chi-squared distribution table, you can look up the values for 0.75/(n-1) and 1.25/(n-1) in the chi-squared distribution table for (n-1) degrees of freedom. You can then find the corresponding probabilities for these values, and subtract the smaller probability from the larger probability to find the probability of including the true value of sigma^2 in the confidence interval for each sample.

Alternatively, you can use a statistical software program to find this probability. For example, if you use R, you can use the pchisq function to calculate the probability of including the true value of sigma^2 in the confidence interval for each sample. The syntax for this function is pchisq(q, df, lower.tail=TRUE), where q is the upper limit of the confidence interval, df is the degrees of freedom, and lower.tail=TRUE specifies that you want to calculate the probability of the true value falling below the upper limit of the confidence interval. You can then subtract this probability from 1 to find the probability of including the true value of sigma^2 in the confidence interval for each sample.

I hope this helps you complete your solution. If you have any further questions or need clarification, please don't hesitate to ask. Best of luck with your research!
 

1. What is a confidence interval for Chi-Squared?

A confidence interval for Chi-Squared is a range of values that is likely to include the true population value of the Chi-Squared statistic with a certain level of confidence. It is used to estimate the precision and accuracy of a statistical test.

2. How is confidence interval for Chi-Squared calculated?

The confidence interval for Chi-Squared is calculated using the Chi-Squared distribution and the quantile function, which determines the critical values for a given level of confidence. The formula for calculating the confidence interval is: CI = (X̄ ± Zα/2 * √(X̄/n)), where X̄ is the sample mean, n is the sample size, and Zα/2 is the critical value from the Chi-Squared distribution table.

3. What does the quantile represent in the confidence interval for Chi-Squared?

The quantile in the confidence interval for Chi-Squared represents the critical values used to calculate the interval. It is a specific value on the Chi-Squared distribution that corresponds to a given level of confidence. For example, a quantile of 0.95 represents a 95% confidence level.

4. How does the sample size affect the confidence interval for Chi-Squared?

The sample size has a direct impact on the width of the confidence interval for Chi-Squared. As the sample size increases, the width of the interval decreases, indicating a more precise estimate of the true population value. This is because a larger sample size provides more information and reduces the margin of error in the calculation.

5. What is the significance of the confidence level in the confidence interval for Chi-Squared?

The confidence level in the confidence interval for Chi-Squared represents the probability that the true population value falls within the calculated interval. For example, a confidence level of 95% means that there is a 95% chance that the true population value falls within the calculated interval. The higher the confidence level, the wider the interval will be.

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