Question

Use the calculator provided to solve the following problems.

Suppose that ?2 follows a chi-square distribution with 28 degrees of freedom. Compute P??216. Round your answer to at least three decimal places.

Suppose again that ?2 follows a chi-square distribution with 28 degrees of freedom. Find k such that =P>?2k0.05. Round your answer to at least two decimal places.

Find the median of the chi-square distribution with 28 degrees of freedom. Round your answer to at least two decimal places.

Answer 1

Suppose we conduct the following statistical experiment. We select a random sample of size n from a normal population, having a standard deviation equal to ?. We find that the standard deviation in our sample is equal to s. Given these data, we can define a statistic, called chi-square, using the following equation:

?2 = [ ( n - 1 ) * s2 ] / ?2

The distribution of the chi-square statistic is called the chi-square distribution. The chi-square distribution is defined by the following probability density function:

Y = Y0 * ( ?2 ) ( v/2 - 1 ) * e-?2 / 2

where Y0 is a constant that depends on the number of degrees of freedom, ?2 is the chi-square statistic, v = n - 1 is the number of degrees of freedom, and e is a constant equal to the base of the natural logarithm system (approximately 2.71828). Y0 is defined, so that the area under the chi-square curve is equal to one.

In the figure below, the red curve shows the distribution of chi-square values computed from all possible samples of size 3, where degrees of freedom is n - 1 = 3 - 1 = 2. Similarly, the green curve shows the distribution for samples of size 5 (degrees of freedom equal to 4); and the blue curve, for samples of size 11