Question

Suppose that x² follows a chi-square distribution with 30 degrees of freedom. Compute P(17≤x²≤ 46). Round your answer to at least three decimal places.

Suppose again that x² follows a chi-square distribution with 30 degrees of freedom. Find k such that P(x²≥ k) = 0.01.   Round your answer to at least two decimal places.

Find the median of the chi-square distribution with 30 degrees of freedom. Round your answer to at least two decimal places.

Answer 1

Solution:

We are given that: x² follows a chi-square distribution with 30 degrees of freedom.

Part a) Compute P(17 ≤ x² ≤ 46) =.............?

We need to use Excel command to get exact probability:

=CHISQ.DIST(x , df , cumulative)

Thus to find P(17 ≤ x² ≤ 46),

we use:

=CHISQ.DIST(46,30,TRUE) - CHISQ.DIST(17,30,TRUE)

=0.9415

=0.942

Thus P(17 ≤ x² ≤ 46) = 0.942

Part b) Find k such that P(x²≥ k) = 0.01

Use following excel command:

=CHISQ.INV.RT( probability , df )

Thus

=CHISQ.INV.RT( 0.01 , 30)

=50.892

=50.89

Thus

P(x²≥ k) = 0.01

P(x²≥ 50.89 ) = 0.01

Part c) Find the median of the chi-square distribution with 30 degrees of freedom.

Median is the middle most value in the ordered data.

That is: 50% of the data is below median and 50% of the data is above median.

Thus find Median value such that:

P( x² ≤ Med ) =50%

P( x² ≤ Med ) = 0.50

Thus use following excel command:

=CHISQ.INV( probability , df)

=CHISQ.INV( 0.50 , 30)

=29.336

=29.34

Thus Median = 29.34