Lesson 9a - Finding Critical Values from a Probability Distribution
docx
keyboard_arrow_up
School
Georgia State University *
*We aren’t endorsed by this school
Course
MATH2058
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
Pages
6
Uploaded by arissa2022
Lesson 9a - Finding Critical Values from a Probability Distribution
Decision making in statistical hypothesis is mostly based on probabilities associated with rare events. The values that form the boundaries between the rare value and the common or typical values are referred to as critical values
. They are essentially copercentiles.
The rare values or atypical values are found at the tails of the distribution. Therefore, we will be finding critical values at either left or right tail. Most of our discussion will center on the right tail. Our discussion can easily be modified to find critical values for the left tail.
Let FUN represent the name of a generic probability distribution for the random variable, X , and FUNcdf be the cumulative distribution function of X.
The right side critical value, v, for the probability of c (shaded area to the right tail is c, 0 < c< 1) will be obtained by solving the equation
P(X > v) = c for v.
In TI- lingo, we need to solve the critical value equation, FUNcdf(v, E99, parameter list of FUN) = c. This equation can written is slightly different form that suitable for all TI purposes as FUNcdf(v, E99, parameter list of FUN) – c = 0
Notes: FUN is only generic. The actual names of different distribution names are found by pressing the 2
nd
key, followed by the VARS key, then choose the appropriate distribution for the list. FUNcdf should be replaced by normalcdf(
for the normal distribution.
FUNcdf should be replaced by tcdf(
for the t distribution.
FUNcdf should be replaced by Fcdf(
for the F distribution.
FUNcdf should be replaced by X
2
cdf(
for the Chi-square distribution.
1
For example, if you are finding critical values for the normal distribution, you choose normalcdf( from the list. The parameters of the normal distribution are the mean and the standard deviation. Therefore, the equation to solve looks like this. Also replace the v by X to obtain
normalcdf(X, E99, µ, σ) -c = 0
How to solve equations on your TI
Example 1
Solve the equation 7x – 3 = 2x +17
The steps:
1.
Rearrange the equation to put all the terms on the left and only zero on the
right-hand side as follows. 7x – 3 -2x – 17 = 0
2.
Press he MATH key on the left column on your keyboard.
3.
Now under the MATH heading, go down to choose “Solver …”. This opens up the equation solving screen. There two types of such screens. Let us do the two-box screen.
Type the left-hand side 7x – 3 – 2x - 17 into the top box.
Type 0 into the box below
7x -3 – 2x -17
0
OK
4.
Now the computer needs you to supply it with a guess to help it do the correct calculations. Just give it any number between 2 and 10.
On the line that has “X = “, type your guess on this line, and DON’T GO BELOW THIS LINE.
5.
While on that line, press the “ALPHA” key below the 2
nd
Key, followed by the ENTER key and wait for your answer to show. The answer is 4.
Finding Critical Values for the Normal Distribution.
Example 2:
We are given a normal distribution with mean of 190 and a standard deviation of 12. Find the critical value with a tail probability of 8%.
Solution: we are asked to solve the critical value equation,
2
1.
We need to solve Rearrange the equation to put all the terms on the left and only zero on the right-hand side as follows. normalcdf(X, E99, 190, 12) - 0.08 = 0
2.
Press he MATH key on the left column on your keyboard.
3.
Now under the MATH heading, go down to choose “Solver …”. Fill in the boxes.
normalcdf(X, E99, 190, 12) - 0.08
0
OK
After you press OK, you will see
Eqn: 0 = normalcdf(X, E99, 190, 12) - 0.08
Note:
a.
In case your calculator layout gives you lower and upper, this is how to answer that.
Lower: type X
Upper: type E99 (for E, press 2
nd
followed by comma)
b.
To get normalcdf, press the 2
nd
Key, followed by VARS key, and option 2.
c.
To get the X, press the key just below the MODE key.
4.
Now the computer needs you to supply it with a guess to help it do the correct calculations. Just give it any number between 2 and 10.
On the line that has “X = “, type you guess on this line, and DON’T GO BELOW THIS LINE.
5.
While on that line, press the “ALPHA” key below the 2
nd
Key, followed by the ENTER key and wait
for your answer to show. The answer is 206.9
Notes for step 3 for Calculators with no Boxes
After step 2 in the above steps, some TI calculators will open up to the following screen
Eqn: 0 = Step 3 in the last example is simply to type the same expression in the upper box on the space after the equals sign to obtain the same result you had.
Eqn: 0 = normalcdf(X, E99, 190, 12) - 0.08
3
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
SHORT CUT for critical values for the normal distribution
The invnorm( ) command helps find the critical values for the normal distribution with ease. Press the 2
nd
key followed by VARS and select option 3, invnorm
Invnorm(1 – 0.08, 190, 12)
Finding Critical Values for the t Distribution.
There is only one parameter for the t distribution; its number of degrees of freedom. Example 3:
We are given a t distribution with 23 degrees of freedom. Find the critical value with a tail probability of 8%.
Solution: we are asked to solve the critical value equation,
tcdf(X, E99, 23) = 0.08
1.
Rearrange the equation to put all the terms on the left and only zero on the
right-hand side as follows. tcdf(X, E99, 23) - 0.08 = 0
2.
Press he MATH key on the left column on your keyboard.
3.
Now under the MATH heading, go down to choose “Solver …”. Fill in the boxes.
tcdf(X, E99, 23) - 0.08
0
OK
After you press OK, you will see
Eqn: 0 = tcdf(X, E99, 23) - 0.08
Note:
d.
In case your calculator layout gives you lower and upper, this how to answer that.
Lower: type X
Upper: type some E99.
e.
To get tcdf(, press the 2
nd
Key, followed by VARS key, and option 6.
f.
To get the X, press the key just below the MODE key.
4.
Now the computer needs you to supply it with a guess to help it do the correct calculations. Just give it any number between 2 and 10.
On the line that has “X = “, type you guess on this line, and DON’T GO BELOW THIS LINE.
4
5.
While on that line, press the “ALPHA” key below the 2
nd
Key, followed by the ENTER key and wait for your answer to show. The answer is 1.45.
SHORT CUT for critical values for the t distribution
The invT( ) command helps us find the critical values for the t distribution with ease. Press the 2
nd
key followed by VARS and select option 4, invT(
invT(1 – 0.08, 23)
Finding Critical Values for the F Distribution.
There F distribution has two parameter - numerator degrees of freedom and denominator degrees of freedom. Example 4:
We are given an F distribution with degrees of freedom; 5 for numerator and 42 for denominator. Find the critical value with a tail probability of
8%.
Solution: we are asked to solve the critical value equation,
Fcdf(X, E99, 5, 42) = 0.08
1.
Rearrange the equation to put all the terms on the left and only zero on the
right-hand side as follows. Fcdf(X, E99, 5) , 42 - 0.08 = 0
2.
Press he MATH key on the left column on your keyboard.
3.
Now under the MATH heading, go down to choose “Solver …”. Fill in the boxes.
Fcdf(X, E99, 5, 42) - 0.08
0
OK
After you press OK, you will see
Eqn: 0 = Fcdf(X, E99, 5, 42) - 0.08
Note:
4.
In case your calculator layout gives you lower and upper, this how to answer that.
Lower: type X
Upper: type E99
5.
Now the computer needs you supply it with a guess to help it do the correct calculations. Just give it any number between 2 and 10.
5
On the line that has “X = “, type you guess on this line, and DON’T GO BELOW THIS LINE.
6.
While on that line, press the “ALPHA” key below the 2
nd
Key, followed by the ENTER key and wait for your answer to show. The answer is 2.13.
Finding Critical Values for the Chi Square Distribution.
There is only one parameter for the Chi-square distribution; its number of degrees
of freedom. Example 5: We are given a chi square distribution with 12 degrees of freedom. Find the critical value with a tail probability of 8%.
Solution: we are asked to solve the critical value equation,
X
2
cdf(X, E99, 12) = 0.08
1.
Rearrange the equation to put all the terms on the left and only zero on the
right-hand side as follows. X
2
cdf (X, E99, 23) - 0.08 = 0
2.
Press he MATH key on the left column on your keyboard.
3.
Now under the MATH heading, go down to choose “Solver …”. Fill in the boxes.
X
2
cdf (X, E99, 23) - 0.08
0
OK
After you press OK, you will see
Eqn: 0 = X
2
cdf (X, E99, 23) - 0.08
Note:
4.
In case your calculator layout gives you lower and upper, this how to answer that.
Lower: type X
Upper: type E99.
5.
Now the computer needs you to supply it with a guess to help it do the correct calculations. Just give it any number between 2 and 10.
On the line that has “X = “, type your guess on this line, and DON’T GO BELOW THIS LINE.
6.
While on that line, press the “ALPHA” key below the 2
nd
Key, followed by the ENTER key and wait for your answer to show. The answer is 19.37. 6
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help