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# HP 48g Series Calculators - Inverses of the Upper-Tail Probability Function

Description
The following routines can be used on the HP 48g Series calculator. The HP 48g provides four upper-tail probability commands: UTPC, UTPN, UTPF, and UTPT, and a normal distribution command (NDIST), which can be used to determine the statistical significance of test statistics.
The following table provides descriptions of the test statistic commands for the 48g series calculator:
 Keys Programmable Command Description UTPC UTPC Upper tail chi-square distribution: Prefix. Takes the degrees of freedom from level 2 and a real number (x) from level 1 and returns the probability that a x 2random variable is greater than x. UTPF UTPF Upper-tail f distribution: Prefix. Takes the numerator degrees of freedom from level 3, the denominator degrees of freedom from level 2, and a real number (x) from level 1, and returns the probability that a Snedecor's F random variable is greater than x. UTPN UTPN Upper-tail normal distribution: Prefix. Takes the mean from level 3, the variance from level 2, and a real number (x) from level 1, and returns the probability that a normal random variable is greater than x for a normal distribution. UTPT UTPT Upper-tail t distribution: Prefix. Takes the degrees of freedom from level 2 and a real number (x) from level 1, and returns the probability that the Student's t random variable is greater than x. NDIST NDIST Normal distribution: Prefix. Takes the mean from level 3, the variance from level 2, and a real number (x) from level 1, and returns the density function at x for a normal distribution.
##### note:
When used as an argument for these commands, the number of degrees of freedom must be between 0 and 499. Also, in the calculations, the degrees of freedom are rounded to the nearest integer.
The inverses of the upper-tail distribution functions are useful for constructing confidence intervals. The probability arguments of an inverse upper-tail probability function is a number between 0 and 1, inclusive (0, less than or equal to x, less than or equal to 1).
The built-in HP Solve application, (Solve equation), can be used to obtain the four upper-tail distribution functions (Chi-Square, Normal, Snedecor's F, and Student's t) by using the programs listed below.
 Function HP Solve Program Chi-Square p, n, x, UTPC, then [ - ] Normal p, m, v, x, UTPN, then [ - ] Snedecor's F p, n1, n2, x, UTPF, then [ - ] Student's t p, n, x, UTPT, then [ - ]
p = Upper-tail probability
n = number of degrees of freedom
m = mean of the distribution variable
v = variance variable
x = normal random variable
n1 = numerator of degrees of freedom
n2 = denominator of degrees of freedom
Calculator symbol key
The procedures in this document use the following text to represent symbol keys:
 Key Description Text Representation Right shift key RS Left-shift key LS
Example of using the normal distribution
Determine what is the value of x when the upper-tail probability for the normal distribution is .05, the mean of the distribution is 0, and the variance is 1.
Instructions for example of using normal distribution
1. Begin the Solve application by pressing RS, then SOLVE and highlight "Solve equation...", then press OK.
Figure : Beginning the Solve equation
2. Move the highlight to the EQ: field (if necessary) and enter the program for Normal distribution: p, m, v, x, UTPN, then [ - ]. Press OK
(To get the lower case characters, press ALPHA, then LS)
Figure : Entering the program
3. Move the highlight to each variable name field and key in the value.
Figure : Keying in values
4. Move the highlight to the variable x and press the menu key SOLVE
Figure : Solving
The result for x is: 1.64485362695.

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