Hello. This is Jim.
I believe a little bit of your question is lost in translation. I am assuming that the first 5 questions relate to a chi squared distribution, not an x2 distribution. The lowercase (and uppercase) Greek symbols for the letter chi look very much like an English x (or X). (PS: That’s how the term “Xmas” came about. The Greek word for Christ would look, with English letters, like Xristos and would be pronounced “Khris-tos). The first letter, chi, gives the KR sound.)
If you are doing any reports with your homework or answers, chi, (and any Greek letter) can be typed in any MS Office product (Word, Excel, PowerPoint) by merely typing the corresponding English letter and then changing the font of just that letter to the Symbol font. The letter chi used in a chi-squared distribution is the lowercase chi. Type a lowercase “c” and change the font of just that letter to show the Greek lowercase “chi” symbol. To show chi-squared, type “c2”. Then change the font of the “c” to Symbol and change the font of the “2” to superscript.
Your instructions said “Use the Table provided in the hand out to find the required x2 - values. Illustrate your work graphically.”
Presumably you have a chi-squared table as a handout. I also assume your table has a picture of a chi-squared distribution above the table. Since I do not have access to your handout, I’d direct you to http://www.medcalc.org/manual/chi-square-table.php so we can both look at the same table. The values in this table should match the values in your table; although your table may have more or fewer rows and/or columns.
Whenever using a table like this it is imperative to always look at the picture first. In the medcalc table I referenced on the web, the shaded area is to the right of the chi-squared value of concern. Notice the “P” pointing to the shaded area. That tells us that for whatever value of chi-squared we are concerned with, the probability values shown within the table will be the total of all probabilities to the RIGHT of that value, or the area under the curve to the right of your chi-squared value in question.
Now, let’s answer your questions.
1 - For a chi-squared curve with 19 degrees of freedom, find the x2 - value having area.
a. 0.025 to its right
If we want a probability of 0.025 (2.5%) to the right of our chi-squared value, look on the top row of numbers and find 0.025. Now look down the left column of numbers (degrees of freedom) and find 19. Find the cell where df=19 intersects with P=0.025. The number in the cell where these rows and columns intersect contain the number 32.852. This is the corresponding chi-squared value.
In plain English, for a chi-squared value of 32.852 and 19 degrees of freedom, we find that only 2.5% of the total area under the curve (or only 2.5% of the total probability) will be beyond this chi-squared value of 32.852. (Note: the table supplied as a handout may have lesser or greater decimal place accuracy. Use your table and answer to the level of accuracy given in your table.)
You are also asked to illustrate your work graphically. If the chi-squared axis on your handout picture contains a horizontal scale with numbers, these would be chi-squared values. Find the point where chi-squared is 32.852 and draw a vertical line up from there to intersect the curve. Shade in the area to the right of your line. Show 0.025 as the probability to the right of your line.
b. 0.95 to its right
Our medcalc table doesn’t show data for probabilities of 0.95 (95%) to the right. Please allow me to direct you to http://people.richland.edu/james/lecture/m170/tbl-chi.html. This table from Richland does have 0.95. This table does NOT have a picture, but we can note from the text just below the table title that says, “The areas given across the top are the areas to the right of the critical value.” We want 95% of the area under the curve to be to the right of our chi-squared value, so we find the intersection of the column with probability of 0.95 row with df=19. At this intersection we find that chi-squared = 10.117.
Remember to also show your work graphically.
2 - For an x2 - curve with df = 10, determine:
a. x2, 0.05
b. x, 20.0975
Your text in these two questions has lost a little in translation (or copy and paste).
I am assuming for (a) we want the chi-squared value where the probability to the right is 0.05 and df=10. The Richland table from the web tells me the chi-squared value is 18.307
For (b), a probability to the right of 0.0975 (9.75%) is not on any of the tables I can find. (This is an unusual value for a probability.) If I assume a typo and assume the actual probability desired is 0.975 (97.5%), the chi-squared value with this probability to the right when df = 10 is 3.247
Don’t forget to show your work graphically.
3. Consider an x2 - curve with df = 8. Obtain the x2 - value having area
a. 0.01 to its left.
b. 0.95 to its left
For (a), if we want 0.01 (1%) to the left, we just subtract 0.01 from the 100% total area under the curve (1.00) and get 0.99 probability to the right. For this probability and df=8, chi-squared = 1.646
For (b) we are looking for 0.05 to the right (1.00 total - 0.95 to the left = 0.05 to the right). For 0.05 to the right and df=8. Chi-squared = 15.507.
Remember to show your work graphically.
4. Determine the two x2 - values that divide the area under the curve into the middle 0.95 area and two outside 0.025 areas for a x2 - curve with
a. df = 5
b. df= 26
The first line will be at the point where 97.5% (0.975) of the probability (the area under the curve) is to the right.
The second line will be at the point where only 2.5% (0.025) of the probability is to the right.
The area in between these two lines should represent the middle 95% of the probabilities. The area on each side of this middle area, the area in each tail, will be 0.025 or 2.5% each. (Note: this is commonly used for 95% confidence levels.)
(a, line 1) P=0.975, df=5, chi-squared = 0.831
(a, line 2) P=0.025, df=5, chi-squared = 12.833
(b, line 1) P=.0975, df=26 Therefore chi-squared is 13.844.
(b, line 2) P = 0.025, df=26 Chi-squared is 41.923.
Remember to show your work graphically.
5. Determine the two x2 - values that divide the area under the curve into a middle 0.90 area and two outside 0.05 areas for a x2 - value with
a. df= 11
b. df = 28
We use the same method here only our lines are at P=0.95 and P=0.05. We’re defining our 90% confidence interval, with 90% of all probabilities within this interval, 5% to the left of it and 5% to the right of it.
(a) for df=11: P=0.95 at chi-squared value of 4.575 and P=0.05 is at chi-squared of 19.675
(b) for df=28: P=0.95 is at chi-squared value of 16.928. If P=0.05, chi-squared = 41.337
I will address questions 6 and 7 a little later. (It’s about 8 PM my time, CST.)
Hopefully I have not introduced any typographical errors in my answer. Please reply if you have any questions.