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Chapter 3: Problem 2

If \(X\) is \(\chi^{2}(5)\), determine the constants \(c\) and \(d\) so that \(P(c

Short Answer

Expert verified
The constants \(c\) and \(d\) for which \(P(c < X < d) = 0.95\) and \(P(X < c) = 0.025\) in a \(\chi^{2}(5)\) distribution are approximately \(c 鈮� 0.5543\) and \(d 鈮� 11.0705\).

Step by step solution

01

Determine the percentile for the lower bound

First, the percentile corresponding to \(c\) should be determined. This is the value below which a given percentage falls. Here, the given condition \(P(X < c) = 0.025\) equates to the 2.5th percentile of the chi-square distribution. The use of a chi-square distribution table or a statistical software function, like qchisq in R, provide the value. Using this, it can be found that \(c 鈮� 0.5543\).
02

Determine the percentile for the upper bound

Next, find \(d\) which is the 97.5th percentile of the chi-square distribution because 2.5% of the data lies below \(c\) and 95% lies between \(c\) and \(d\). Hence, 2.5% of data lies above \(d\) making \(d\) the 97.5th percentile. Using the same chi-square distribution table or software function as before, it can be found that \(d 鈮� 11.0705\).

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