Chi square formula in word document

Chi-square distribution

Find the median of the chi-square distribution with 22degrees of freedom. Round your answer to at least two decimal places

In the question above the resolution had the equation of P(Xsquare > M)=0.5. Where did the 0.5 come from? Is it coming from the previous question answer?

Chi-square distribution
Use the calculator provided to solve the following problems.
Suppose that follows a chi-square distribution with degrees of freedom. Compute . Round your answer to at least three decimal places.

Suppose again that follows a chi-square distribution with degrees of freedom. Find such that . Round your answer to at least two decimal places.

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

Additional Resources
Elementary Statistics (A Brief Version), 6th Ed.
Bluman
Chapter 7: Confidence Intervals and Sample Size
Section 7.4: Confidence Intervals for Variances and Standard Deviations

Supplementary Resources
We're asked to compute , where follows a chi-square distribution with degrees of freedom. Using the calculator, we can compute and use the complement rule to obtain :


.
(Note that, since the degrees of freedom in the above calculation must be specified for the ALEKS calculator, the expression appears as with in the calculator input.)

Figure 1

We are asked to find such that for a chi-square distribution with degrees of freedom. Note that such a is the value that cuts off an area of in the right tail of this distribution, that is, for the distribution. See Figure 1, which depicts the distribution and an area of shaded to the right of .
Using the calculator, we get

.
(Note that we have to use the input to specify the degrees of freedom.)

We're asked to find the median of the chi-square distribution with degrees of freedom. The median of a continuous distribution is the value that divides the distribution in half; in other words, the probability of obtaining a value greater than the median is and the probability of obtaining a value less than the median is .
In finding the median of the chi-square distribution with degrees of freedom, then, we are finding the value such that , where follows a chi-square distribution with degrees of freedom. In other words, we are finding . Using the calculator, we get

Comparing the mean and median of this distribution

.
The answer is:

Median

Median
Noun. (Statistics.)

The median of a list of

Example 1: The median of the numbers image1.png , image2.png , image3.png , image4.png , image5.png is image6.png. To see why this is so, arrange the numbers in order: image7.png , image8.png , image9.png , image10.png , image11.png. Then choose the middle number in the list, which is image12.png.

Example 2: The median of the numbers image13.png , image14.png , image15.png , image16.png , image17.png , image18.png is image19.png. To see why this is so, arrange the numbers in order: image20.png , image21.png , image22.png , image23.png , image24.png , image25.png. Then choose the "middle" number, which is defined to be the mean of the values in the middle of the list, that is, the mean of image26.png and image27.png.

Definition:

Given a list of image28.png numbers (not necessarily distinct), label the numbers image29.png so that image30.png.

1. If image31.png is odd, then the median of the list is the middle value in the ordered list, namely,

image32.png.

(This is to say that the median is the number in position image33.png in the list.)

2. If image34.png is even, then the median of the list is the mean of the two middle values in the ordered list, namely,

.

(This is to say that the median is the mean of the numbers in the positions just above and just below position image35.png in the list.)

Remarks:

· The median is the 50th percentile.

· Because the middle number of an ordered set of numbers (i.e., the median) will be the same whether the numbers are arranged in increasing or decreasing order, we may simply say "ordered" in the definition above without stating increasing or decreasing. However, in practice, it is more common to arrange the numbers in increasing order, as in calculating percentiles.

· The median is a measure of center of a set of numbers, like the mean and the mode. Unlike the mean, though, the median is resistant to outliers. This means that, in a sense, extreme values tend not to affect the value of median. For instance, the list of numbers image36.png , image37.png , image38.png , image39.png , image40.png is just like the list of numbers in Example 1 above, except that the extreme value image41.png in Example 1 has been replaced by the (much more extreme) value image42.png here. The median is the same in both cases, but the means are very different (image43.png in Example 1 and image44.png here.)