Soft Smart Statistics and Probability Made Easy for Ti 89 Titanium and Voyage 200 Free

Chapter 1. Introduction

1.1. Why This Document?

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.

1.2. Using Advanced Calculators in Probabality and Statistics

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.

1.3. General Issues in Learning Probability and Statistics Armed with an Advanced Calculator

1.3.1. Interval Probabilities

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.

1.3.2. Standardized Random Variables

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.

Appendix A. Symbols and Abbreviations

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.

Appendix B. Program and Function Reference

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()

Name

MA206() -- Set up a custom menu allowing easy access to functions commonly used in the basic probability and statistics course.

Synopsis

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.

Description

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 the CATALOG key. Once in the Catalog window, pressing [F3] Flash Apps will bring up a list of the functions installed by any flash applications, and [F4] User-Defined will bring up a list of user defined functions.

Tip: When a function has been highlighted in either the [F3] Flash Apps or [F2] Built-in panes of the Catalog window, pressing [F1] Help will bring up a terse description of the inputs for the function.

Implementation

              1 :MA206()   2 :Prgm   3 :© Program to set up an MA206 custom menu   4 :© Rev 2.1 23 JUN 01   5 :© D/MathSci USMA (Mark Wroth)   6 :Custom   7 :Title  "Tools"   8 : Item  "CustmOff"   9 :Title "Calc"  10 : Item "∫"  11 : Item "SUM("                            12 : Item "√"  13 :Title  "Counting"  14 : Item  "nPr("  15 : Item  "nCr("  16 : Item  "!"  17 :Title  "Distr"  18 : Item  "TIStat.binomPdf("  19 : Item  "TIStat.binomCdf("  20 : Item  "TIStat.PoissPdf("  21 : Item  "TIStat.PoissCdf("  22 : Item  "hypergeo()"  23 : Item  "hygeoPdf("  24 : Item  "hygeoCdf("  25 : Item  "expCdf("  26 : Item  "unifCdf("  27 : Item  "TIStat.normCdf("  28 :Title "Intvl"  29 : Item "TIStat.zInt("  30 : Item "TIStat.tInt("  31 : Item "TIStat.zInt_1P("  32 : Item "BinomInt("  33 : Item "Chi2Int("  34 : Item "Chi2GUI()"  35 :EndCustm  36 :CustmOn  37 :EndPrgm  38

: The actual menu entry is the Greek letter Sigma, which the TI-89 uses as a summation operator. This symbol is not available in the HTML version of this document.

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).

Hypergeometric Distribution

Name

hypergeo()

-- Compute probabilities related to a hypergeometric distribution, specifically the probability that a hypergeometric random variable lies between two constants a and b, inclusive.

Synopsis

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.

Description

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.

TI-89 Implementation

              1 :hypergeo()   2 :Prgm   3 :©              Hypergeometric probabilities   4 :© Rev 2.0   5 :© Mark Wroth   6 :Local n,succ,pop,a,b,prob,usrmode   7 :getMode("ALL")→usrmode   8 :Dialog   9 :  Title "Hypergeometric Distn"  10 :  Request "Sample size:",n  11 :  Request "Pop size:",pop  12 :  Request "Successes:",succ  13 :  Request "lower limit:",a  14 :  Request "upper limit:",b  15 :EndDlog  16 :expr(n)→n              17 :expr(pop)→pop  18 :expr(succ)→succ  19 :expr(a)→a  20 :expr(b)→b  21 :© check inputs  22 :© compute  23 :If a<b Then  24 :  SUM(hygeopdf(x,n,succ,pop), x, a, b)→ prob              25 :Else  26 :  40 → main\err  27 :  PassErr  28 :EndIf  29 :© Disp prob  30 :Dialog  31 :  Title "Hypergeometric Distn ..."  32 :  Text "P("&string(a)&"≤X≤"&string(b)&")=&string(prob)  33 :  Text " "  34 :  Text "n = "&string;(n)&" N = "&string(pop)&" M = "&string(succ)              35 :EndDlog  36 :setMode(usrMode)  37 :EndPrgm  38

`hygeopdf(x,n,succ,pop)`

Name

hygeopdf(x,n,succ,pop)

-- Evaluate the PDF of a hypergeometric random variable.

Synopsis

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.

Description

hygeopdf computes the probability that a hypergeometric random variable with sample size n , possible number of successes succ , and population size pop assumes the value x .

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, Succ is the number of elements coded "success", n is the sample size, and max(0, n - Pop + Succ) ≤ x ≤ min(n, Succ).

Example

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.

TI-89 Implementation

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`

Name

hygeocdf

-- Evaluate the CDF of a hypergeometric random variable.

Synopsis

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.

Description

hygeocdf( x , n , succ , pop ) computes the probability that a hypergeometric random variable with sample size n , possible number of successes succ , and population size pop assumes a value less than or equal to x .

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, Succ is the number of elements coded "success", n is the sample size. Unlike the PDF, there are no limits (in principle) on x , although some care is needed to ensure that the function behaves properly at all values.

Example

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.

TI-89 Implementation

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)).

`TIStat.poissCdf`

Name

TIStat.poissCdf

-- Evaluate the probability that a Poisson random variable lies between

LOW

and

UP

inclusive.

Synopsis

TIStat.poissCdf(                LAMBDA              [,                  LOW                              ],                UP              )

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.

Description

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`

Name

TIStat.binomCdf

-- Evaluate the probability that a Binomial random variable lies between

LOW

and

UP

inclusive.

Synopsis

TIStat.binomCdf(                n              ,                p              [,                  LOW                              ],                UP              )

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.

Description

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`

Name

unifCdf

-- Evaluate the CDF of a uniformly distributed random variable.

Synopsis

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 .

Description

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, b , and (x-a)/(b-a) between those two values.

Example

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.

TI-89 Implementation

`expCdf`

Name

expCdf

-- Evaluate the CDF of an exponentially distributed random variable.

Synopsis

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.

Description

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

Example

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^-3/5 ). Entering either parameter using a decimal form (the 3. in the example) cause the calculator to provide the approximate answer.

TI-89 Implementation

              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`

Name

TIStat.normCDF

-- Returns the probability that a normally distributed random variable with mean μ and standard deviation σ lies between

LOW

and

UP

Synopsis

TIStat.normCDF(                LOW              ,                UP              [,                                  μ                ,                  σ                              ])

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].

Usage

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, .

`TIStat.invNorm`

Name

TIStat.invNorm

returns the value of a normally distributed random variable such that a probability of

AREA

lies to the left of the value.

Synopsis

TIStat.invNorm(                AREA              [,                                  μ                ,                                  σ                              ])

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.

Usage

This function may also be accessed from the menu system, under F5 Distributions, , .

`TIStat.inv_t`

Name

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.

Synopsis

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.

Usage

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, , .

`TIStat.invChi2`

Name

TIStat.invChi2

-- Compute the value of a χ² distributed random variable such that a probability of

AREA

lies to the left of the value.

Synopsis

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.

Usage

This function may also be accessed from the menu system, under F5 Distributions, , .

`TIStat.zInt`

Name

TIStat.zInt

-- Compute a confidence interval for the mean of a random variable.

Synopsis

zInt(σ,List[,FRQ,CLEV] | σ,XBAR,N[,CLEV])

Usage

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.

`TIStat.tInt`

Name

TIStat.tInt

-- Compute a confidence interval for the mean of a normally distributed random variable.

Synopsis

TIStat.tInt(LIST[,FRQ,CLEV] |              XBAR,SX,N[,CLEV])

Usage

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`

Name

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.

Synopsis

chi2int(                n              ,                s2              ,                clevel              ,                type              )

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

Outputs

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.

In addition, the

chi2gui

will store the following user inputs to the indicated variables (and will use the values in those variables as the default choices when it opens).

statvars\n: The sample size
statvars\ssdevx: The sample standard deviation (square root of the entered sample variance.
statvars\clevel: The confidence level.

Usage

This function can be called in either of two ways: from the Home command line, as chi2int( n , s2 , clevel , type ) or by calling chi2gui() . If the chi2int for is used, the input arguments are:

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.

Example

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 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 entering chi2gui() at the Home screen.
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.

TI-89 Implementation

The `chi2int` Program

                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 σ²"→tstr   9 :If type=2 Then  10 : √(l)→l  11 : √(u)→u  12 : "CI on σ"→tstr  13 :EndIf  14 :Dialog  15 : Title tstr  16 : Text "Cint = ( "&string(l)&" , "&string(u)&" )"  17 : Text "n    = "&string(n)  18 : Text "s²                = "&string(s2)  19 :EndDlog  20 :l→statvars\lower  21 :u→statvars\upper  22 :EndPrgm

The `chi2gui` Program

                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 "s²                ",s2  13  Request "C level",clevel  14  DropDown "CI on ",{"variance","std dev"},type  15 EndDlog  16 If ok=1 Then  17 expr(n)→n  18 expr(s2)→s2  19 expr(clevel)→clevel  20 chi2int(n,s2,clevel,type)  21 n→statvars\n  22 √(s2)→statvars\ssdevx  23 clevel→statvars\clevel  24 EndIf  25 EndPrgm

`binomInt`

Name

binomInt

-- The

binomint()

program computes confidence intervals on the population proportion for a binomially distributed random variable.

Synopsis

binomint(                success              ,                n              ,                clevel              )

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).

Usage

This function is called by entering binomint( success , n , clevel ) at the HOME screen.

Example

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.

TI-89 Implementation

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

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

C.1. Overview

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.

C.2. Installing the Advanced Mathematics Software using another TI-89

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 menu
1. Select [2nd ] VAR-LINK
2. Select F-3 LINK.
On the receiving calculator, select
1. Cursor down until option is highlighted
2. Press ENTER
3. A warning message will display. Press ENTER to continue (or ESC to abort).
On the sending calculator, select
1. Cursor down until option is highlighted
2. Press Enter
3. 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.

C.3. Installing the Statistics with List Editor Flash Application Using Another TI-89

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 flash application
1. Select F-7 Flashapp to display the list of flash applications
2. Highlight the application (it may already be highlighted, for example if its the only one there).
3. Press F-4 to check mark the application. A small check mark should appear next to the application name.
On the receiving calculator, select the menu by selecting [2nd] VAR-LINK. Both calculators should now be in the VAR-LINK screen.
On the receiving calculator, select the receive option
1. Select F3 LINK
2. Move the highlight to option
3. Press Enter. The messages VAR-LINK WAITING TO RECEIVE and BUSY should appear on the status line.
On the sending calculator, select the option
1. Select F3 LINK
2. Move the highlight to option
3. 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 Guidebook, Texas Instruments.

This is the users manual for the TI-89 itself, as purchased.

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Source: https://www.west-point.org/users/usma1978/36200/Calculators/TI89/old/ti89ma206.html