United States Military Academy

Department of Mathematical Sciences

Department of Mathematical Sciences

West Point, New York 10996 <mark.wroth@us.army.mil>

**Table of Contents**- 1. Introduction
- 2. Descriptive Statistics
- 2.1. Numerical Methods
- 2.2. Graphical Methods
- 2.2.1. Histograms
- 2.2.2. Box Plots

- 3. Basic Operations
- 3.1. Counting
- 3.2. Counting Techniques
- 3.2.1. Factorials
- 3.2.2. Permutations
- 3.2.3. Combinations

- 3.3. Random Variables and Probability Distributions
- 3.3.1. Discrete Random Variables
- 3.3.2. Continuous Random Variables

- 4. Estimation
- 4.1. Point Estimation
- 4.2. Interval Estimation

- 5. Hypothesis Testing
- A. Symbols and Abbreviations
- B. Program and Function Reference
- MA206() — Set up a custom menu allowing easy access to functions commonly used in the basic probability and statistics course.
- Hypergeometric Distribution — Compute probabilities related to a hypergeometric distribution, specifically the probability that a hypergeometric random variable lies between two constants a and b, inclusive.
`hygeopdf(x,n,succ,pop)`— Evaluate the PDF of a hypergeometric random variable.`hygeocdf`— Evaluate the CDF of a hypergeometric random variable.`TIStat.poissCdf`— Evaluate the probability that a Poisson random variable lies between`LOW`and`UP`inclusive.`TIStat.binomCdf`— Evaluate the probability that a Binomial random variable lies between`LOW`and`UP`inclusive.`unifCdf`— Evaluate the CDF of a uniformly distributed random variable.`expCdf`— Evaluate the CDF of an exponentially distributed random variable.`TIStat.normCDF`— Returns the probability that a normally distributed random variable with mean μ and standard deviation σ lies between`LOW`and`UP`.`TIStat.invNorm`—`TIStat.invNorm`returns the value of a normally distributed random variable such that a probability of`AREA`lies to the left of the value.`TIStat.inv_t`—`TIStat.inv_t`returns the value of a Student's T distributed random variable such that a probability of`AREA`lies to the left of the value.`TIStat.invChi2`— Compute the value of a χ^{2}distributed random variable such that a probability of`AREA`lies to the left of the value.`TIStat.zInt`— Compute a confidence interval for the mean of a random variable.`TIStat.tInt`— Compute a confidence interval for the mean of a normally distributed random variable.`chi2int`— The`chi2int()`program (and its companion`chi2gui()`, which provides a graphical user interface to the program) computes confidence intervals on the population variance or standard deviation of a normally distributed population.`binomInt`— The`binomint()`program computes confidence intervals on the population proportion for a binomially distributed random variable.

- C. Upgrading a TI-89 Calculator for MA206 Probability and Statistics
- Bibliography

**List of Figures**- 3-1. Hypergeometric PDF

Advanced calculators such as the
HP-48 and TI-89 present both an
opportunity and and a challenge to students (and teachers) of
probability and statistics. On one hand, the calculator makes
actually *performing* the sometimes tedious
calculations needed in P & S a matter of
punching a few buttons. Advanced calculators also larqely or
completely eliminate the need for cumbursum tables. But this
capability comes with a price. Not only does the student have to
master the concepts of the course—a challenge in
itself—but they must also learn what the capabilities of
the calculator are and how to invoke them.

This document is aimed at students (and teachers) who are trying to master the aspects of the advanced calculator (specifically the TI-89) that apply to the basic probability and statistics course. It supplements the course textbook and the calculator handbook and focuses on those uses of the calculator specifically needed for this course. It covers both the built in operations of the calculator, and programs written specifically to assist with the subject.

We assume that the student has been using the same calculator through the core math sequence, and is therefor familiar with basic calculator operations. In addition to basic arithmatic computation, this includes symbolic manipulation, basic calculus (particularly numerical quadrature) and graphing of functions.

One of the powerful features of the advanced calculators is programmability. In addition to briefly covering the built in functions of the calculator, this document discusses some programs written to assist with subjects covered in the basic P & S course.

The focus of many P & S courses—and many students—appears to be on mastering the basic computations of the subject. For example, a major goal during a block on the exponential random variable is being able to correctly compute probabilities involving such a random variable. Facility with this calculation is then assumed later in the course. With a properly set up calculator, the calculation itself is simple; the challenge is in knowing when to use the distribution, what value to use for the parameter, and how to interpret the result.

The calculator can also largely replace the use of tables, and hence of the need to standardize random variables for most purposes. The exception to this is that many statistical packages, including the TI-89, use and display standardized random variables in statistical tests, so some understanding of the process is needed.

The following sections are presented in an order generally conducive to a one semester course in probability and statistics, following the outline of MA206, the core course in the subject taught at USMA using [DEVORE] as the text. With some modification, it should be helpful in most basic probability and statistics courses.

Computing the probability that a random variable lies in a stated interval is a common task in the probability and statistics course. Especially with the capabilities of the TI-89, there are several valid strategies students may use for computing such probabilities:

Manipulate the PDF (or PMF) directly. For example. integrate the PDF over the interval. This approach implies the need for a user-manipulable PDF (PMF) functions.

Subtract the endpoint CDF values. Most easily executed at the

**HOME**entry line, this approach implies the need for a user-manipulable CDF function for each distribution.Create (or find) a calculator program which computes the interval probability. Internally, such a program may use either computation approach.

There is no theoretical reason to choose between these techniques. Ideally, a student would master all of the different techniques and choose the technique appropriate to the particular problem.

Common practice in teaching the computation is to cover PDF-based approaches, but to emphasize CDF-based approaches. This fits well with the use of distribution tables, and may be easiest for some students because of this connection.

At the same time, the primary user interface for the probability computations in the TI-89 Statistics with List Editor is a program based GUI which allows the user to enter the distribution parameters and the ends of the interval. While entry line functions are also provided in the Statistics with List Editor, use of these functions is essentially undocumented. Consistency with the general approach of the TI-89 would appear to suggest GUI-based interfaces are desirable. This requires writing programs for the distributions which are not included in the Statistics with List Editor application. For MA206, this would include the Hypergeometric, Uniform, and Exponential distributions.

The TI-89 allows the student to approach the calculation of
interval probabilities any of the above ways—given the
availability of either existing programs or basic programming skills
for the third approach. To help gain understanding, it may be a good
idea for students to focus on one method and ensure
*it* is mastered. If GUI-based
programs are available for all of the distributions of interest,
focusing on this technique is likely to be the easiest.

The use of the calculator largely eliminates the need to use traditional probability tables. Since being able to use the standard normal probability tables is one of the main ways the use of a standardized random variable is presented, eliminating the need to use the tables at all also eliminates one of the major uses of standardized variables. It is tempting to simply ignore the topic completely if the student has adequate calculator skills.

However, there are several reasons to understand the basic manipulations surrounding standardized random variables, and the standard normal distribution in particular. Perhaps least important is the fact that traditional tables, while in some sense obsolescent as calculators with basic probability functions become more common, are still available when calculators are not, so some ability to use them is probably a good idea. More important from the perspective of the course material is that the manipulations to standardize the Normal random variable are the basis of the manipulations by which we derive the formulas for confidence intervals. So understanding how to standardize the normal random variable is a lead in to the material on confidence intervals. Finally, statistical packages—including the TI-89's advanced statistics functions— frequently state hypothesis test results in terms of standardized test statistics. Understanding the test results depends on understanding the normalized versions of the statistics.

The TI-89's one variable statistics application computes the sample mean, variance and standard deviation (using both the sample and population formulas), and the median and quartiles.

**Computing One-Variable Sample Statistics**

Enter the sample data into a list.

Select the 1: 1-Var Stats option from the

**[F4] Calc**menu (of the Statistics with List Editor application).Enter the name of the list containing the sample data, either by entering the variable name directly, or by selecting

**[2nd] VAR-LINK**and selecting the variable.Select

**ENTER**to confirm the selection, and again to compute the statistics.

A basic histogram is one of the standard plot types available. To create a histogram of data:

Enter the data into a list variable.

Select

**[F2] Plots**to bring up the Plots menu.Select 1:Plot Setup....

Highlight a plot line.

Select [F1] Define.

Select Plot Type; on the popup menu, select 4:Histogram

Enter the name of the list variable containing the data in the x box.

Enter an appropriate width for the histogram intervals in Hist. Bucket Width.

Ensure the NO option is selected in the Use Freq and Categories? popup.

Select

**ENTER**.Select

**[F5] ZoomData**.

A basic box plot is one of the standard plot types available.

Enter the data into a list variable.

Select

**[F2] Plots**to bring up the Plots menu.Select 1:Plot Setup....

Highlight a plot line.

Select [F1] Define.

Select Plot Type; on the popup menu, select 3:Box Plot

Enter the name of the list variable containing the data in the x box.

Ensure the NO option is selected in the Use Freq and Categories? popup.

Select

**ENTER**.Select

**[F5] ZoomData**.

The **[F3] Trace** function allows easy examination of the particular
values included in the plot.

The TI-89 computes several basic functions useful for counting problems.

For most of these operations, there are two or three different ways to access the same calculator function:

Select the function from a menu (usually the Math menu, accessed with the

**[2nd MATH**key).Type the name of the function in the Entry Line, using the alphabetic keys.

Add the function to the Entry Line using the

**CATALOG**. Functions defined from flash applications (such as Statistics with List Editor) and user-defined functions are also available through the**CATALOG**function.

The factorial function is accessed with the postfix operator
`!` which can be entered from the keyboard (using the
**[2nd] CHAR** function) or from the **[2nd]
MATH** 7: Probability menu.

The "permutations” function can be accessed with the
function `nPr(` function. This can be accessed via
the **[2nd] MATH** 7: Probability
nPr( menu pick.

The "combinations” function
can be accessed through the `nCr(` function. This can
be accessed via the **[2nd] MATH** 7:
Probability nCr( menu
pick, or by typing the function name in the entry line..

The Binomial probability distribution is one of the pre-defined probability distribution in the Statistics with List Editor application. It is accessed via the [F5] Distr menu, using either the B: Binomial Pdf or C: Binomial Cdf menu items.

The hypergeometric probability distribution is not one of the pre-defined distributions in the Statistics with List Editor. Since it is not pre-defined for us, we can define the PDF and CDF as TI-89 functions.

The PDF of the hypergeometric distribution is shown in Figure 3-1.

**Figure 3-1. Hypergeometric
PDF**

(nCr(Succ, x) * nCr(Pop - Succ, n - x))/(nCr(Pop, n))

Where

Popis the number of elements in the population,Succis the number of elements coded “success”,nis the sample size, and max(0, n - Pop + Succ) ≤ x ≤ min(n, Succ)

The side conditions deal with the fact that the minimum number of successes in the sample is limited by the total number of failures in the population and the sample size (you can't have more failures in the sample than there are in the population), and the maximum number of successes in the sample is limited by the number of successes in the population.

The Poisson probability distribution is one of the pre-defined probability distribution in the Statistics with List Editor application. It is accessed via the [F5] Distr menu, using either the D: Poisson Pdf or E: Poisson Cdf menu items.

The TI-89‘s calculus
applications can significantly ease the manipulation of arbitrary
continuous probability distributions through their ability to find
both definite and indefinite integrals. You reach these functions
through the **HOME** screen, and should already be
familiar from earlier calculus courses.

The major caution in applying the basic calculus functions to the PDF is to ensure that the limits of integration are correctly applied. Like any computer, the TI-89 will do what you tell it to, which may not be what you intended, particularly for piecewise defined functions!

**Probabilities. **Finding the probability that an arbitrarily defined continuous
random variable lies in a given interval is, by definition, a matter
of integrating the PDF over the interval. For
simply defined functions (e.g. the Exponential distribution) this is
easily accomplished with the Integrate
function from the **HOME** **[F3] Calc**
menu.

**Expected Value. **Finding the expected value of an arbitrarily defined continuous
random variable can be accomplished by applying the definition of
expected value. For simply defined functions (e.g. the Exponential
distribution) this is easily accomplished with the
Integrate function from the
**HOME** **[F3] Calc** menu.

**Variance. **Finding the variance of an arbitrarily defined continuous random
variable can be accomplished by applying the definition of
variance. For simply defined functions (e.g. the Exponential
distribution) this is easily accomplished with the
Integrate function from the
**HOME** **[F3] Calc** menu. The
computational formula,
V(X)=E(X^{2})-[E(X)]^{2}
can be applied by integrating to find the expected value of
X^{2}; this may not be easier than applying
the definition directly.

Defining the PDF or the CDF as a TI-89 function allows it to be used in subsequent calculations. Examples of reasonable definitions are given for some of the probability distributions used in the basic probability and statistics course discussed below.

The Uniform distribution is not a separately defined probability distribution in the Statistics with List Editor application. Therefore all manipulations of random variables with this distribution depend on manipulating the PDF directly, user defined programs, or on the use of known formulas.

The cumulative distribution function can be defined as a
TI-89 function for convenience in
calculation. An example of such a definition is shown in in the `unifCdf` reference page.

The Exponential distribution is not a separately defined probability distribution in the Statistics with List Editor application. Therefore all manipulations of random variables with this distribution depend on manipulating the PDF directly, user-defined programs, or on the use of known formulas.

The `expcdf()` function (discussed in the `expCdf` reference page) can be used on the
TI-89 to compute probabilities related to
exponential random variables. It implements the piecewise definition
of the function, and thus can be used without careful reference to
the interval of definition.

The Normal, or Gaussian, probability distribution is one of the pre-defined distributions in the Statistics with List Editor application. Because of its promenance in statistical applications, there are a variety of built-in functions for accessing and manipulating this distribuution.

There are two main methods for computing probabilities involving
the Normal distribution; the Normal Cdf
function (accessed from the **[F5] Distr** menu of the
Statistics with List Editor), and with the
Shade function (also accessed from the
**[F5] Distr** menu of the Statistics with
List Editor). Both require the mean, standard deviation,
and limits of the interval; the Shade
function, in addition to computing the probability that the random
variable is in the interval, draws the PDF and
shades the area of interest.

The **[F2] Plots** menu includes the ability to create a normal
probability plot of data in one of the lists. To draw a normal
probability plot:

Start the Statistics with List Editor application.

Enter the data into a list variable.

Select the Plots menu by pressing

**[F2] Plots**.Select 2:Norm Prob Plot.

Fill out the resulting Norm Prob Plot... requestor:

Select an unused list variable at the Plot Number popup.

Enter the name of the list variable containing the data for which the probability plot is needed in the List: box.

Select values for the remaining entries on the requestor. The default values are probably acceptable.

Select

**ENTER**to close the requestor.

Select the Plots menu by pressing

**[F2] Plots**.Select 1:Plot Setup.

Select the plot variable containing the normal scores (the name of this variable was chosen in the Plot Number popup of the Norm Prob Plot...) by highlighting it using the cursor keys and pressing

**[F4]**.Display the plot by pressing

**[F5] ZoomData**.

The primary mechanism for computing point estimates is the 1: 1-Var Stats menu item of the F4 Calc menu of the Stats/List Editor Flash App. This application returns a variety of point estimates based on the sample data contained in one of the data lists.

A basic familiarity with the List
Editor is very useful in computing point estimates
from sample data. This subject is covered in
*Statistics with List Editor Application for
TI-89/TI-92* manual.

Tip:A particularly useful technique with the 1-Variable Statistics application is to use theVAR-LINKmenu to select the specific list for which to calculate statistics.

The main TI-89 function supporting estimation (other than the functions used for calculating sample statistics) is the [F7] Ints menu, which includes functions for Z and T-based confidence intervals on the mean (among others). These functions allow the interval to be calculated directly from sample data, or from previously computed sample statistics.

The 5: 1-PropZInt menu can be
used to calculate confidence intervals on the population
proportion of a binomial distribution. However, this function
appears to use the approximate formula defined by [DEVORE] in Equation 7.11 (which is the standard form
used by most texts rather than the more exact form defined in
Equation 7.11. The binomial interval programs `binomInt` defined in this document supplement the
`1-proportion Z Interval` program by using
the more exact formulation ([DEVORE] Equation
7.11).

The `chi2int()` and its companion
GUI `chi2gui()` are
user-defined programs that compute confidence intervals on the
variance or standard deviation of a normal population. They are
discussed in `chi2int`.

The [F5] Distr menu’s 2: Inverse submenu includes functions for computing the critical values of the Normal, Student’s T, and Chi-squared distributions (among others not covered in MA206).

Tip:The various inverse functions ask for theAREA(probability) at which the inverse is to be calculated. This area is the probability that the random variable isless thanthe returned inverse value. This is consistent with the general definition of a CDF. However, the critical values of a distribution are defined in terms of the probability that the random variable isgreater thanthe critical value. The translation between the two is, of course, that the area above the critical value is 1 minus the area the inverse function is expecting.This difference can be ignored by taking the absolute value of the resulting critical value—if the distribution is symmetric around zero. Becasue this relationship does

nothold true for distribution not symmetric about zero (i.e. the Chi-squared distribution or the general normal distribution), relying on this property can lead the student into mistakes. It is more correct and more generally applicable to always remember that1-α=AREA, and make the entry into the invers function accordingly.

The [F6] Tests menu includes applications for (among others), Z and T-based hypothesis tests. These tests allow the test statistic to be provided, or to be computed from data entered in one of the lists. They also can display the distribution with the acceptance region shaded or simply provide the numerical results. In both cases, the calculator provides the p-value relevant to the test, rather than drawing a conclusion.

This appendix lists various symbols and abbreviations used in the text. In particular, it lists non-ASCII symbols; depending on the medium in which this document appears, these symbols may be differently rendered.

- ©
Represents the TI-89 comment symbol. Usually rendered with the closest available glyph, which is frequently the copyright symbol.

- ∫
Integral sign.

- ∩
Set intersection.

- √
Square root (surd or radical).

- α
Lower case Greek letter alpha.

- LAMBDA
Lower case Greek letter lambda; usually the parameter of a Poisson or exponential distribution.

- SUM
Summation operator, usually the upper case Greek Sigma.

- →
The TI-89 “

**STORE**” command as represented in the editor window.- ∪
Set union.

The TIStat applications
(belonging to the Statistics with List
Editor flash application) need their own
reference pages; the application manual ([TI-89STATSLE]) does not describe how to use the
applications from the Home screen, although they are available
from the **CATALOG** screen as well as from the
`MA206()` program. Some of these have been
added to this appendix, but this incomplete.

ma206() |

**Inputs**

- none
This program has no inputs.

**Outputs**

- none
This program returns no direct outputs; it sets up a custom menu accessible by the user.

This program sets up a TI-89
*custom menu*, which allows function
names to be easily inserted into the Entry
Line

Tip:Function and program names can also be easily pasted into the Entry Line by using theCATALOGkey. Once in the Catalog window, pressing[F3] Flash Appswill bring up a list of the functions installed by any flash applications, and[F4] User-Definedwill bring up a list of user defined functions.

Tip:When a function has been highlighted in either the[F3] Flash Appsor[F2] Built-inpanes of the Catalog window, pressing[F1] Helpwill bring up a terse description of the inputs for the function.

It does not appear to be possible to insert a function prototype (i.e. to give variable names for arguments to a function to be pasted into the entry line).

hypergeo() |

**Inputs**

- Sample size
The size of the sample drawn.

- Pop size
The total size of the population from which the sample is drawn.

- Successes
The number of successes in the population.

- lower limit
The lower limit of the interval for which the probability is desired.

- upper limit
The upper limit of the interval for which the probability is desired.

**Outputs**

- probability
The primary output of the program is the probability that the random variable lies in the closed interval [a,b]. The program also echoes the parameters entered into the program as a check on data entry error.

`hypergeo` is a program which prompts
the user for the parameters of a hypergeometric distribution
and the endpoints of an interval, and then computes the
probability that the random variable lies in that
interval.

The hypergeometric distribution models a situation where
a sample is taken from a finite population consisting of a
fixed number of successes and failures
*without* replacement. The random variable
is the number of successes drawn in the sample.

The format of the program is intended to be similar to the format used in the Statistics with List Editor application.

- The © symbol is used to indicate the TI-89 comment symbol.
- The
`Request`function returns a string value; since we need a numeric value, we have to convert each stored value. - The calculation here should probably either include
checks on the validity of the input parameters, or call
`hygeopdf`to allow that function to do the error checking. - The choice of variable names in the output summary matches the convention used in [DEVORE] rather than mirroring the descriptive strings used in the input dialog.

hygeopdf( |

**Inputs**

*x*The value at which the PDF is to be evaluated.

*n*The sample size.

*succ*The total number of successes in the population.

*pop*The total number of elements (successes and failures) in the population.

**Outputs**

- probability
The PDF value.

`hygeopdf` computes the
probability that a hypergeometric random variable with sample
size ` n`, possible number of successes

The hypergeometric
PDF is defined as (nCr(Succ, x) * nCr(Pop -
Succ, n - x))/(nCr(Pop, n)), where ` Pop`
is the number of elements in the population,

To find the probabilitity that a random variable from a hypergeometric distribution with a population size of 50 with 15 successes and a sample size of 10 has exactly 5 successes:

Enter

`hygeopdf(5.,10,15,50)`in the entry line of the**HOME**window.Press

**Enter**The expression you entered and the answer,

`.094903`, will be displayed in the History Area.**Note:**If you enter all of the parameters using exact forms, the calculator will display the exact answer (in this case`904332/9529015`). Entering any parameter using a decimal form (the`5.`in the example) cause the calculator to provide the approximate answer.

Because of the very simple definition of
`hygeocdf()`, it is important that we
define `hygeopdf()` to return zero for
invalid values of x. It is also appropriate to test for
invalid parameter inputs; an invalid input here can
propogate up to the CDF.

hygeocdf( |

**Inputs**

*x*The value at which the CDF is to be evaluated.

*n*The sample size.

*succ*The total number of successes in the population.

*pop*The total number of elements (successes and failures) in the population.

**Outputs**

- probability
The CDF value.

`hygeocdf(x,
n, succ,
pop)` computes the
probability that a hypergeometric random variable with sample
size

The hypergeometric
CDF is defined as
SUM^{x}_{i=0}(nCr
(Succ, i) * nCr (Pop - Succ, n - i))/(nCr (Pop, n)), where
` Pop` is the number of elements in the
population,

To find the probability that a random variable from a hypergeometric distribution with a population size of 50 with 15 successes and a sample size of 10 has 5 or fewer successes:

Enter

`hygeocdf(5.,10,15,50)`in the entry line of the**Home**window.Press

**Enter**The expression you entered and the answer,

`.969998`, will be displayed in the History Area.**Note:**If you enter all of the parameters using exact forms, the calculator will display the exact answer (in this case`2813126/2900135`). Entering any parameter using a decimal form (the`5.`in the example) cause the calculator to provide the approximate answer.

The CDF for the Hypergeometric can
be implemented easily given the existence of a
PDF function which correctly returns
zero for values of x which violate the side conditions (see
`hygeopdf(x,n,succ,pop)`).

1 :hygeocdf(x, n, succ, pop) 2 :Func 3 : SUM(hygeopdf(i, m, succ, pop),i,0,x) 4 :© CDF for a 5 :© hypergeometric RV 6 :© Rev 1.0 7 :© Mark Wroth 8 :EndFunc |

TIStat.poissCdf( |

**Inputs**

`LAMBDA`(required)The parameter of the Poisson distribution.

`LOW`(optional)The lower bound of the interval. Defaults to negative infinity.

`UP`(required)The upper bound of the interval.

**Outputs**

- Probability
The probability of that the random variable lies within the given interval.

This program computes various probabilities connected
with Poisson random variables. The use of the optional
` LOW` argument allows the program to be
used to compute the PDF,
CDF, or the probability that the random
variable lies in a specific interval.

It is important to understand that the interval over
which the probability is computed is a
*closed* interval; in other words, the
endpoints are included in the interval.

TIStat.binomCdf( |

**Inputs**

`n`(required)The number of trials.

`p`(required)The probability of success.

`LOW`(optional)The lower bound of the interval. Defaults to negative infinity.

`UP`(required)The upper bound of the interval.

**Outputs**

- Probability
The probability of that the random variable lies within the given interval.

This program computes various probabilities connected
with binomial random variables. The use of the optional
` LOW` argument allows the program to be
used to compute the PDF,
CDF, or the probability that the random
variable lies in a specific interval.

It is important to understand that the interval over
which the probability is computed is a
*closed* interval; in other words, the
endpoints are included in the interval.

unifCdf( |

**Inputs**

*x*The value at which the CDF is to be evaluated.

*a*The lower limit of the region for which the PDF is non-zero.

*b*The upper limit of the region for which the PDF is non-zero.

**Outputs**

- Cumulative probability
The probability that a uniformly distributed random variable with the specified parameters is less than or equal to

.*x*

This function evaluates the CDF of a
uniformly distributed random variable. It will return zero
for values less than the lower limit,
` a`, one for values above the upper
limit,

To find the probabilitity that a random variable uniformly distributed between 1 and 10 is less than 5:

Enter

`unifcdf(5.,1,10)`in the entry line of the**Home**window.Press

**Enter**The expression you entered and the answer,

`.444444`, will be displayed in the History Area.**Note:**If you enter all of the parameters using exact forms, the calculator will display the exact answer (in this case`4/9`). Entering any parameter using a decimal form (the`5.`in the example) cause the calculator to provide the approximate answer.

expCdf( |

**Inputs**

*x*The value of the random variable at which the CDF is to be evaluated.

*LAMBDA*The parameter of the distribution. LAMBDA is one over the mean of the distribution.

**Outputs**

- Cumulative probability
The probability that the random variable is less than or equal to the supplied

.*x*If an invalid parameter LAMBDA is supplied, an error string is returned, rather than a numeric result.

This function implements the CDF for
an exponentially distributed random variable with parameter
` LAMBDA`. Such a random variable has
PDF f(x) = LAMBDA e

To compute the probability that an exponentially distributed random variable with mean 5 is less than or equal to 3:

Enter

`expcdf(3.,1/5)`in the entry line of the**Home**window.Press

**Enter**The expression you entered and the answer,

`.451188`, will be displayed in the History Area.**Note:**If you enter both parameters using exact forms, the calculator will display the exact answer (in this case`1-e`). Entering either parameter using a decimal form (the^{-3/5}`3.`in the example) cause the calculator to provide the approximate answer.

```
1 :expcdf(x,LAMBDA)
2 :Func
3 : when(LAMBDA<0, when(x≥0,1-e^(-LAMBDA*x),0),“LAMBDA must be > 0”)
4 : © CDF of an exponential The “©” symbol is used here to represent the TI-89 comment symbol
5 : © RV with parameter LAMBDA
6 : © Rev 1.0 JUN 00
7 : © D/MathSci USMA (Mark Wroth)
8 :EndFunc
``` |

The `expCdf` function wraps a simple call
to the usual mathematical definition inside two tests. The first of
these tests checks that the required parameter LAMBDA is greater than
zero, as required by the definition of the function. The second test
checks whether the input value x is greater than or less than zero,
branching to the two piecewise definitions of the
CDF depending on the result. Both tests use the
`where()` function, which is in essence a simple
branching structure.

TIStat.normCDF( |

**Inputs**

`LOW`(required)The lower bound of the interval over which the probability is desired.

`UP`(required)The upper bound of the interval over which the probability is desired.

`μ`(optional)The mean of the normally distributed random variable. If the mean is not supplied, it defaults to 0.

`σ`(optional)The standard deviation of the normally distributed random variable. If the standard deviation is not supplied, it defaults to 1.

**Outputs**

- Probability
The probability of that the random variable lies in the interval [LOW, UP].

This function is used from the command line of the **HOME**
screen, and may be entered either by typing the name of the
function or selecting it from the **CATALOG**
screen, where it is found under F3: Flash
Apps.

This function may also be accessed from the menu system, under F5 Distributions, 4: Normal CDF.

TIStat.invNorm( |

**Inputs**

`AREA`(required)The cumulative probability that the random variable is less than the returned value.

`μ`(optional)The mean of the random variable.

`σ`(optional)The standard deviation of the random variable.

**Outputs**

- Value
The value of the random variable below which the input probability falls.

This function is used from the command line of the **HOME**
screen, and may be entered either by typing the name of the
function or selecting it from the **CATALOG**
screen, where it is found under F3: Flash
Apps.

This function may also be accessed from the menu system, under F5 Distributions, 2: Inverse, 1: Inverse Normal.

TIStat.inv_t( |

**Inputs**

`AREA`(required)The cumulative probability that the random variable is less than the returned value.

`DF`(required)The number of degrees of freedom.

**Outputs**

- Value
The value of the random variable below which the input probability falls.

This function is used from command line of the **HOME**
screen, and may be entered either by typing the name of the
function or selecting it from the **CATALOG**
screen, where it is found under F3: Flash
Apps.

This function may also be accessed from the menu system, under F5 Distributions, 2: Inverse, 2: Inverse t.

TIStat.invChi2( |

**Inputs**

`AREA`(required)The cumulative probability that the random variable is less than the returned value.

`DF`(required)The number of degrees of freedom.

**Outputs**

- Value
The value of the random variable below which the input probability falls.

This function is used from command line of the **HOME**
screen, and may be entered either by typing the name of the
function or selecting it from the **CATALOG**
screen, where it is found under F3: Flash
Apps.

This function may also be accessed from the menu system, under F5 Distributions, 2: Inverse, 3: Inverse Chi-square.

This program takes two forms depending on whether the sample statistics are to be computed from data contained in a list or entered directly by the user.

This program can also be accessed from the menu system.

This program takes two forms depending on whether the sample statistics are to be computed from data contained in a list or entered directly by the user.

This program can also be accessed from the menu system.

chi2int( |

**Inputs**

- n
The number of samples in the sample.

- s2
The sample variance.

- clevel
The desired confidence level for the confidence interval.

- type
The type of interval desired, where

indicates a confidence interval on the variance, and**1**a confidence interval on the standard deviation.**2**

The `chi2int` program provides its
outputs in two forms: a graphical requester that provides
the requested confidence interval and echoes the user
inputs, and by storing the user inputs and the desired
confidence interval endpoints in the
`statvars` directory.

The set of stored variables are different for the
`chi2int` and the
`chi2gui` programs. The
`chi2int` stores:

- statvars\lower
The lower end of the desired confidence interval.

- statvars\upper
The upper end of the desired confidence interval.

- statvars\n
The sample size

- statvars\ssdevx
The sample standard deviation (square root of the entered sample variance.

- statvars\clevel
The confidence level.

This function can be called in either of two ways: from
the Home command line, as
` chi2int(n,
s2,
clevel,
type)` or by calling

If the ` chi2gui()` form is used,
there are no command line inputs; the program will raise a
requester to allow the user to supply the needed values.

Given a sample of size n = 17, and a sample variance of 137,324.3, compute a 95% confidence interval on the population variance.

Begin at the Home screen.

Enter the command

.**chi2int(17,137324.3,.95,1)****Enter****Tip:**As a shortcut to entering the command name, use the Catalog function and select the F4 User-Defined tab. Then select the desired function from the list.Read the confidence interval (76171.3, 318080) on the resulting requester.[1]

Alternatively, using the `chi2gui`
to solve the same problem:

Start the

`chi2gui`by enteringat the Home screen.**chi2gui()**Enter the values for n, the sample variance, and the confidence level in the open requesters.

Select the desired confidence interval type from the drop down menu.

Press

**Enter**.Read the confidence interval (76171.3, 318080) on the resulting requester.

1 :chi2int(n,s2,clevel,type) 2 :Prgm 3 :© D/MathSci USMA (Mark Wroth) 4 :© Revision 1.1 21 JUN 01 5 :Local l,u,tstr 6 :(n-1)*s2/(tistat.invchi2(1-(1-clevel)/2,n-1))→l 7 :(n-1)*s2/(tistat.invchi2((1-clevel)/2,n-1))→u 8 :"CI on σ |

1 chi2gui() 2 Prgm 3 © D/MathSci USMA 4 © Version 1.1 21 JUN 01 5 Local n,s2,clevel,type 6 string(statvars\n)→n 7 string(statvars\ssdevx^2)→s2 8 string(statvars\clevel)→clevel 9 Dialog 10 Title "Chi Squared CI" 11 Request "n ",n 12 Request "s |

binomint( |

**Inputs**

- success
The number of successes in the sample.

- n
The size of the sample.

- clevel
The desired confidence level (1-α).

**Outputs**

- statvars\lower
The lower endpoint of the resulting confidence interval.

- statvars\upper
The upper endpoint of the resulting confidence interval.

- statvars\n
The size of the sample.

- statvars\clevel
The desired confidence level (1-α).

To compute a 90% confidence interval on the binomial parameter p based on a sample of 100 observations, 50 of which were successes:

At the

**HOME**screen, enter.**binomint(50,100,.9)****ENTER**Read the resulting interval (.418848, .581152) in the output requester. This requester also echoes the input values.

The ends of the confidence interval are also stored in
`statvars\lower` and
`statvars\upper`.

This function implements Equation 7.10 from [DEVORE]. This expression includes several terms
which are usually neglected (as in the
`TIStat.zInt_1P` or Devore's
Equation 7.11. The expanded function implemented here is
considered acceptably accurate even if np or n(1-p) are not
sufficiently large.

1 binomint(success,n,clevel) 2 Prgm 3 © Find a "7.10" CI for the binomial parameter "p" 4 © D/MathSci USMA (Mark Wroth) 5 © Revision 1.1 21 JUN 01 6 Local l,u,phat,t1,t2,z 7 success/n→phat 8 tistat.invnorm(1-(1-clevel)/2,0,1)→z 9 z*√(phat*(1-phat)/n+z^2/(4*n^2))→t1 10 1+z^2/n→t2 11 (phat+z^2/(2*n)-t1)/t2→l 12 (phat+z^2/(2*n)+t1)/t2→u 13 Dialog 14 Title "CI on p" 15 Text "Cint = ( "&string(l)&" , "&string(u)&" )" 16 Text "trials = "&string(n) 17 Text "successes = "&string(success) 18 Text "C Level = "&string(clevel) 19 Text "p-hat = "&string(phat) 20 EndDlog 21 l→statvars\lower 22 u→statvars\upper 23 EndPrgm |

To load the TI-89 Statistics with List Editor flash application, you need to load the Advanced Mathematics Software Operating System/base code, and then install the Statistics with List Editor application. The TI manual indicates that it is about four times faster to do this from calculator to calculator than from desktop to calculator.

This discussion assume you have two TI-89 calculators, one with the Advanced Mathematics Software and Statistics with Lists application installed, and one which you are upgrading to that configuration, and that you have a calculator to calculator link cable.

Ensure both calculators have fresh batteries.

**Warning**A power loss (or any other interruption) during this operation will mean the receiving unit has to be reloaded using a computer.

Ensure any data which is to be retained on the receiving calculator is backed up to another calculator or computer.

**Warning**This procedure will delete all user variables and reset the receiving calculator to its factory state. This may include deleting flash applications.

Link the two TI-89s using the calculator to calculator cable (as described on page 366 of the

*TI-89 and TI-92 Plus Guidebook*.On both calculators, select the LINK menu

Select

**[2nd ] VAR-LINK**Select

**F-3 LINK**.

On the receiving calculator, select Receive Product Code

Cursor down until option 5: Receive Product Code is highlighted

Press

**ENTER**A warning message will display. Press

**ENTER**to continue (or ESC to abort).

On the sending calculator, select Send Product Software

Cursor down until option 4:Send Product SW is highlighted

Press

**Enter**A warning message will display. Press

**ENTER**to continue (or**ESC**to abort).

After a short pause (about five seconds), the receiving calculator will display a status message and progress indicator. Wait until the display clears (about six minutes). When the display clears, the transfer is complete.

**Warning**Interrupting the transmission will result in the receiving calculator becoming inoperable until it is reloaded from a computer.

Reload any backed-up data to be retained on the receiving calculator

For more information on installing base code updates, see Upgrading Product Software
(Base Code), beginning on page 373 of the *TI-89 and TI-92 Plus Guidebook*.

Link the two TI-89s using the calculator to calculator cable (as described on page 366 of the

*TI-89 and TI-92 Plus Guidebook*.On the sending calculator, select the LINK menu by selecting [2nd ] VAR-LINK

On the sending calculator, select the Stats/List Edi flash application

Select

**F-7 Flashapp**to display the list of flash applicationsHighlight the Stats/List Edi application (it may already be highlighted, for example if its the only one there).

Press

**F-4**to check mark the Stats/List Edi application. A small check mark should appear next to the application name.

On the receiving calculator, select the LINK menu by selecting

**[2nd] VAR-LINK**. Both calculators should now be in the VAR-LINK screen.On the receiving calculator, select the receive option

Select

**F3 LINK**Move the highlight to option 2:Receive

Press Enter. The messages

`VAR-LINK WAITING TO RECEIVE`and`BUSY`should appear on the status line.

On the sending calculator, select the Send to TI-89/92 Plus option

Select

**F3 LINK**Move the highlight to option 3: Send to TI-89/92 Plus

Press

**ENTER**.

The message

`SENDING TISTATLE`, a progress bar, and the`BUSY`indicator should be displayed on the receiving calculator.Wait until the screen clears on the receiving calculator (about 75 seconds). When the receiving calculators VAR-LINK screen returns, the transmission is complete.

For more information on installing flash applications, see
Transmitting Variables, Flash Applications, and Folders, beginning on
page 367 of the *TI-89 and TI-92 Plus
Guidebook*

*Probability and Statistics for Engineering and the Sciences*, Jay L. Devore, Duxbury Press, Belmont, 1995, Fourth.

*TI-89 and TI-92 Plus Guidebook for
Advanced Mathematics Software Version 2.0*, Texas
Instruments.

http://www.ti.com/calc/docs/8992pguide.htm. This is the updated manual for the Advanced Mathematics Software; it replaces the package.

*Downloading Applications to the TI-89*, Texas
Instruments.

http://www.ti.com/calc/pdf/gb/eng/8992p/22link.pdf (Chapter 22 of the*TI-89 and TI-92 Plus Guidebook*)

*Advanced Mathematics Software*, Texas Instruments.

http://www.ti.com/calc/flash/89ams.htm. This is an overview page for the AMS upgrade.

*Statistics with List Editor*, Texas Instruments.

http://www.ti.com/calc/flash/89stats.htm. This is an overview page for the Statistics with List Editor package

*Statistics with List Editor*, Texas Instruments.

http://www.ti.com/calc/flash/pdf/statsle.pdf. This is the users guide for the Statistics with List Editor application

[1] | This is Example 7.15 from Devore's 5th edition. However, the use of the full accuracy of the Chi-squared inverse function rather than the five significant figures available from a set of tables results in a slightly different answer than Devore obtains. |