If X is χ2(23), find the following:
(a) P(14.85 < X < 32.01).
(b) Constants a and b such that P(a < X < b) = 0.95 and P(X < a) = 0.025.
(c) The mean and variance of X.
Step 1 of 5:
Given that X has a distribution with 23 degrees of freedom.That is n=23.
Therefore mean of X is 23 and variance of X is 2*23=46. That is =23 and =46.
Standard deviation of X becomes =6.7823.That is =6.7823.
Step 2 of 5:
Here we have to find the value of P(14.85<X<32.01).
To find this value we make use of standard normal approximation.
It is gives by,
We have to use the table of standard normal distribution table representing the area to the left of the Z score statistic to get the probability values.
Step 3 of 5:
Here we have to find the values of the constants ‘a’ and ‘b’ using the fact that P(a<X<b)=0.95 and P(X<a)=0.025).
We know that P(-1.96<Z<1.95)=0.95. That is 95% of the area under the normal curve lies between -1.96 and +1.96. Using this we can find the values of a and b.
Using standard normal approximation, we get
It is given that P(<Z<)=0.95.
=-1.96 and =1.96
Thus the value of constant a is 9.70662 and value of constant b is 36.293308.