Download Distance Formula Ti 84 Program Free

TI-83/84 PLUS BASIC MATH PROGRAMS (GEOMETRY) Archive Statistics Number of files. Volume, and circumferencen of 2D and 3D figures. If the shape you are looking for is not in this program, feel free to e-mail me and I will add it, or feel free to add it yourself. Distance Formula A program that calculates the distance between two points.

Safe reset of TI84

Important program to run the programs without getting run error problems. Further it resets your calculator without loosing programs and Applications

FAQ and problem solver for Ti84.

Math:Solvers

The Math-Solve command: examples how to use this command for physics or maths

Complex linear equation solver (4 variables): unique solver for 4 variables

Equation solvers with Polysmlt (Apps): explanation of the poly rooth finder

Math: Analytical

Calculating limits : Program which calculates limits

Control of derived derivate df/dx or integral fx(dx):Control your analytical derived derivates or integrals. Plot, add or substract (complex) vectors : graphical presentation of vectors

Vector Calculator 2 dimensional (inner product, cross product etc.)

Partial fraction decomposition : ( up to 4 fractions ) For integrals and Laplace Transformation

Math: Numerical

First order differential equation solver : (Euler or trapezoidal method )

Second order differential equations :(Euler or trapezoidal)

Signal builder for various programs : This program works as a function generator. The output can be used FOR EXAMPLE as input for the differential equation solvers and integrator.

Double integral: Calculate double integrals ( for example : calculate position from input parameter acceleration)

Complex help : assistent for quick calculations in complex network

Complex linear equation solver (4 variables) : Solve 4 complex equations for network theory

Network and Frequency analysis with the Y-editor : use the Y-editor for solving difficult problems (also serie and parallel resonance problems)

Filter design with Ti84 : Design methods for filters with Ti84

Power calculations one phase : Calculates real, blind, apperent power, and powerfactor

Delta wye, wye-delta transformation of a (complex ) network : Also for resistor network

Plot, add or substract (complex) vectors : graphic presentation of vectors

Calculates line currents and apperent, blind, real power and powerfactor of three phase voltage system for a symmetric load

Calculates line currents and apperent, blind, real power and powerfactor of three phase voltage system for assymetric load without and grounded (by Z0) load point,(including voltage loss in cables)

decimal->binary-hexadecimal coversions forward and reverse etc. )

Built your own periodic signal and use it for several programs like TRMS and AVERG calculator next

TRMS and AVERAGE value calculator for periodic signals : Input signal by function of with signal builder)

Triac light dimmer : Calculates current and voltage (TRMS) power, apperent power, powerfactor for every fire angle and source amplitude

Full bridge rectifier design : design your own rectifier (voltage, ripple, current etc.)

Fourier series with Ti84 : Calculate for all different periodic signals the spectrum

Electrical engineering : Transient Analysis

First order numerical / graphical differential equation solver : Transient analysis of RC or RL circuits

Second order differential equations : transients of RLC circuits

Control system programs

Nyquist diagram for Ti84 : controlling system stabilty by Nyquist (graphics'

Bode diagram with Ti84 : Bode diagram plot with Ti84

Eigenvalues : Calculating eigenvalues up to 4th order

Electromechanics : calculate shaft torque, power, rpm, omega

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BrownMath.com →TI-83/84/89 →Extra Statistics Utilities

Summary:This page presents a downloadable TI-83/84 program with easierversions of some calculator procedures plus new capabilities likecomputing skewness and kurtosis and making statistical inferencesabout standard deviation, correlation, and regression.See Using the Program below for afull list of features.

Your first course in statistics probably won’t use thesefeatures, but they’re offered here for advanced students andthose who are studying on their own.

If you’re in Stan Brown’s MATH200classes at TC3, this is the optional extra program; seeMATH200A Program — Basic Statistics Utilities for TI-83/84 for the required program.

Contents:

MATH200B Program Overview
1. Skewness and Kurtosis
5. Inferences about σ, the Standard Deviation of a Population
6. Inferences about Linear Correlation
7. Inferences about Linear Regression

See also:Troubles? See TI-83/84 Troubleshooting.

`MATH200B` Program Overview

Because this program helps you,
please click to donate!Because this program helps you,
please donate at
BrownMath.com/donate.

Getting the Program

The program is in two parts, MATH200B andMATH200Z. You need both on your calculator, even thoughyou won’t run MATH200Z directly. It works withall TI-83 Plus calculators and all TI-84 calculators, including thecolor models.

If you have a “classic” TI-83, not a Plus or Silver,follow the directions below but put M20083B andM20083Z on your calculator, not MATH200B andMATH200Z. (M20083B and M20083Z aren’tbeing updated after version 4.2,which was released in August 2012, so you will see some differencesfrom the screen shots in this document.)

There are three methods to get the programsinto your calculator:

If you have a TI-84,download MATH200B.zip(110 KB,updated 18 July 2019), and unzip it.Use the USB cable that came with your calculator, and the freeTI Connect CE softwarefrom Texas Instruments, to transferthe MATH200B.8XP and MATH200Z.8XP programs to your calculator.
If a classmate has the programs on her calculator(any model TI-83/84),she can transfer them to yours, provided you both have a USB portor you both have a round I/O port. Connect the appropriate cable toboth calculators, inserting each end firmly.On your calculator, press [2ndx,T,θ,nmakesLINK][►] [ENTER]. Then on hers press[2ndx,T,θ,nmakesLINK] [3], selectMATH200B (or M20083B; see above),and finally press [►] [ENTER].If you get a prompt about a duplicate program, choose Overwrite.
Repeat forMATH200Z (or M20083Z; see above).
Or, as a last resort, key in the programs.See MATH200B.pdf, MATH200Z.pdf, and MATH200B_hints.htm in theMATH200B.zip file.

Using the Program

Press the [PRGM] key. If you can see MATH200Bin the menu, press its number; otherwise, scroll to it and press[ENTER]. When the program name appears on your home screen,press [ENTER] a second time to run it. Check the splash screen to make sure you have thelatest version (v4.4a), then press[ENTER].

The menu at right shows what the program can do:

Skew/kurtosis:compute skewness and kurtosis, which arenumerical measures of the shape of a distribution
Time series:plot time-series data
Critical t:find the t value that cuts thedistribution with a given probability in the right-hand tail
Critical χ²:find the χ² value that cuts thedistribution with a given probability in the right-hand tail
Infer about σ:hypothesis tests and confidence intervals forpopulation standard deviation and variance
Correlatn inf:hypothesis tests and confidence intervalsfor the linear correlation of a population; the hypothesis test forcorrelation doubles as a hypothesis test for slope of the regressionline
Regression inf:confidence intervals for slope of theregression line, y intercept, and ŷ for a particular x, plusprediction intervals for ŷ for a particular x

If you ever need to break out of the programbefore finishing the prompts, press [ON] [1].

If you run the program on a TI-84 with a higher-resolution screen,some displays will look slightly different, but all keystrokes will bethe same.

The program is protected so that you can’tedit it accidentally. If you want tolook at the program source code, see MATH200B.PDF andMATH200Z.PDF in the downloadableMATH200B.ZIP file.

Each procedure leaves its results in variables in case youwant to use them for further computations. For details, please see theseparate document MATH200B Program — Technical Notes.

1. Skewness and Kurtosis

Summary:

A histogram givesyou a general idea of the shape of a data set, but two numeric measures of shapeare also available. Skewness measures how far adistribution departs from symmetry, and in which direction.Kurtosis measures the height or shallowness of the centralpeak, using the normal distribution (bell curve) as a reference.

The 1:Skew/kurtosis part of the MATH200B program computes these statisticsfor a list of numbers or a grouped or ungrouped frequencydistribution. This section of the document explains how to use theprogram and how to interpret the numbers.

See also:For interpretation of skewness and kurtosis, andtechnical details of how they arecalculated, see Measures of Shape: Skewness and Kurtosis.

Using the Program

If you have a frequency or probability distribution, put thedata points or class midpoints (class marks) in one statistics listand the frequencies or probabilities in another. If you have a simplelist of numbers, put them in a statistics list.

Then press [PRGM], scroll if necessary and selectMATH200B, and in the program menu select 1:Skew/kurtosis. Specify yourdata arrangement, enter your data list, and if appropriate enter yourfrequency or probability list. The program will produce a great manystatistics.

Example: College Students’ Heights

Here are grouped data for heights of 100 randomlyselected male students:

Class boundaries	59.5–62.5	62.5–65.5	65.5–68.5	68.5–71.5	71.5–74.5
Class midpoints, x	61	64	67	70	73
Frequency, f	5	18	42	27	8
Data are adapted fromSpiegel 1999 [full citation in “References”, below], page 68.

A histogram, prepared with the MATH200Aprogram, shows the data are skewedleft, not symmetric.But how highly skewed are they? Andhow does the central peak compare to the normal distributionfor height and sharpness? To answer these questions, you have tocompute the skewness and kurtosis.

Enter the x’s in one statistics list and the f’s inanother. If you’re not sure how to create statistics lists,please see Sample Statistics on TI-83/84.

Then run the MATH200B program and select 1:Skew/kurtosis.Your data arrangement is 3:Grouped dist.When prompted, enter the list that contains thex’s and then the list that containsthe f’s. I’ve used L5 and L6, but you could use any lists.

The program gives its results on three screens of data.

The first screen shows some basic statistics: the sample size,the mean, the standard deviation, and the variance.As usual, you have to consider whether the data are a sample or thewhole population; the program gives you both σ and s,σ² and s².

The program stores keyresults in variables in case you want to do any further computationswith them. See MATH200B Program — Technical Notes for a complete listof variables computed by the program.

The second screen shows results for skewness. The third moment dividedby the 1.5 power of the variance is the skewness, which is about−0.11 for this data set. Again, you are given the values touse if this is the whole population and if it is a sample.

If this is the whole population, then you stop with the firstskewness figure and can state that the population is negatively skewed(skewed left).

But this is just a sample, so you use the “assample” figure for your skewness. (This is also the figure thatExcel reports.)The sample is negatively skewed (skewed left),but can you say anything about the skew of the population?To answer that question, use the standarderror of skewness, which is also shown on the screen.As a rule of thumb, if sampleskewness is more than about two standard errors either side of zero,you can say that the population is skewed in that direction.In this example, the standard error of skewness is 0.24, andthe statistic of −0.45 tells you thatthe skewness is only 0.45 standard errors below zero. This is not enough tolet you say anything about whether the population is skewed in eitherdirection or symmetric.

The last screen shows results for kurtosis. The fourth moment dividedby the square of the variance gives the kurtosis, which is 2.74.Some authors, and Microsoft Excel, prefer to subtract 3 and considerthe excess kurtosis: 2.74−3 is −0.26.

A bell curve (normal distribution) has kurtosis of 3 andexcess kurtosis of 0. If excess kurtosis is negative, as it is here,then the distribution has a lower peak and higher“shoulders” than a normal distribution, and it is calledplatykurtic.(An excess kurtosis greater than 0 would mean that the distributionwas leptokurtic, with a narrower and higher peak than a bellcurve.)

Since this is just a sample, and not the whole population, usethe “as sample” excess kurtosis of −0.21. (This isthe figure Excel reports.)Can you say anything about the kurtosis of the population fromwhich this sample was taken? Yes, just as you did for skewness.The rule of thumbis that an excess kurtosis of at least two standard errorsis significant. For this sample, the standard error of kurtosis is 0.48, and −0.21/0.48 =−0.44, so the excesskurtosis is only 0.44 standard errors below zero. (Or,the kurtosis is only 0.44 standard errors below 3.) Therefore youcan’t say whether the population is peaked like a normaldistribution, more than normal, or less than normal.

On high-resolution screens (the TI-84 Plus C and TI-84Plus CE), there’s enough room to show skewness and kurtosis onthe same screen, as shown at right.

Example: Throwing Dice

You can also use this part of the program to compute the shapeof a probability distribution. For instance, here’s the probabilitydistribution for the number of spots showing when you throw twodice:

Probability Distribution for Throwing Two Dice
Spots, x	2	3	4	5	6	7	8	9	10	11	12
Probability, P(x)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

The x’s go in one list and the P’s in another.(Enter the probabilities asfractions, not decimals, to ensure that theyadd to exactly 1. The calculator displays rounded decimals butkeeps full precision internally, and the program will tell you if yourprobabilities don’t add to 1.)Now run the MATH200B program and select 1:Skew/kurtosis. Your data arrangement is4:Discrete PD,and you’ll see the following results:

On the first screen, no sample size is shown becausea probability distribution is a population.

On the second screen, the skewness is essentially zero.This confirms what you can see in the histogram:the distribution is symmetric.Standard error and test statisticdon’t apply because you have a probability distribution(population) rather than a sample.

On the same screen, the kurtosis is 2.37 (not shown forreasons of space), and the excesskurtosis is −0.63; the dice make a platykurtic distribution.Compared to a normal distribution, this distribution of dice throwinghas a lower, less distinct peak and shorter tails.

On high-resolution screens, namely the TI-83 Plus C Silver Edition andTI-84 Plus CE, complete information about a probability distributionfits on one screen.

You may notice that, although the skewness is stillessentially zero, it’s a different very small number from thevery small number the older TI-84s gave us, on the screen shot above.I can’t account for this in detail, but I think it’slikely that the newer calculator’s chip processes floating pointwith very slightly different precision than the old one. Don’t obsess about it — for all practical purposes,both numbers are zero.

2. Time Series Plots

Summary: To plot a time series or trend line,put the numbers in a statistics list and use the 2:Time series part ofthe MATH200B program.

Example:Let’s plot the closing prices of Cisco Systems stockover a two-year period. The following table is adapted fromSullivan 2008 [full citation in “References”, below],page 82, which credits NASDAQ as the source.

Month	3/03	4/03	5/03	6/03	7/03	8/03	9/03	10/03
Closing	12.98	15.00	16.41	16.79	19.49	19.14	19.59	20.93
Month	11/03	12/03	1/04	2/04	3/04	4/04	5/04	6/04
Closing	22.70	24.23	25.71	23.16	23.57	20.91	22.37	23.70
Month	7/04	8/04	9/04	10/04	11/04	12/04	1/05	2/05
Closing	20.92	18.76	18.10	19.21	18.75	19.32	18.04	17.42

Enter the closing prices in a statistics list such as L1,ignoring the dates.

Now run the MATH200B program and select 2:Time series. The program prompts youfor the data list. (Caution: The program assumes thetime intervals are all equal. If they aren’t, the horizontalscale will not be uniform and the graph will not be correct.)

It’s usually good practice to start the vertical scale atzero, or in other words to show the x axis at its proper level onthe graph. But the program gives you the choice. If you have goodreason, you can let the program scale the data to take up the entire screen.This exaggerates the amount of change from one time period to thenext.(If the data include any negative or zero values, thex axis will naturally appear in the graph, and program skips theyes/no prompt.)

Below you see the effect of a “yes” at left and theeffect of a “no” at right.

As you can see, the graph that doesn’t include the zerolooks a lot more dramatic, with bigger changes. But that can bedeceptive. A more accurate picture is shown in the first graph,the one that does include the x axis.

optional extra: Tracing the Graph

If you wish, you can press the [TRACE] key and displaythe closing prices, scrolling back and forth with the[◄] and [►] keys. If you wantto jump to a particular month, say June 2004, the 16th month, type16 and then press [ENTER].

3. Critical t or Inverse t

The TI-83 doesn’t have an invT function asthe TI-84 does, but if you need to find critical t or inverse t oneither calculator you can use this part of the MATH200B program.

Caution: our notation of t(df,rtail) matches most booksin specifying the area of the right-hand tail for critical t. But theTI calculator’s built-in menus specify the area of the left-hand tail. Make sureyou know whether you expect a positive or negative t value.

Some textbooks interchange the arguments: t(rtail,df).Since degrees of freedom must always be a whole number and the tailarea must always be less than 1, you’ll always know whichargument is which.

Example: find t(27,0.025), the t statistic with 27 degrees offreedom (sample size 28) for a one-tailed significance test withα = 0.025, a two-tailed test withα = 0.05, or a confidence interval with1−α = 95%.

Solution: run the MATH200B program and select 3:Critical t. Whenprompted, enter 27 for degrees of freedom and 0.025 for the area ofthe right-hand tail, as shown in the first screen. After a shortpause, the calculator gives you the answer: t(27,0.025) =2.05.

Interpretation: with a sample of 28 items (df=27), a t scoreof 2.05 cuts the t distribution with 97.5% of the area to the left and2.5% to the right.

4. Critical χ² or Inverse χ²

χ²(df,rtail) is the critical value for the χ²distribution with df degrees of freedom and probabilityrtail. (In the context of a hypothesis test, rtail isα, the significance level of the test.)

In the illustration, rtail is the area of theright-hand tail, and the asterisk * marks the critical valueχ²(df,rtail). The critical value or inverse χ² isthe χ² value such that a higher value of χ² has only anrtail probability of occurring by chance.

You can compute critical χ² only for the right-handtail, because the χ² distribution has no left-hand tail.

Caution: Some textbooks write the function theother way, χ²(rtail,df). Since df is a whole numberand rtail is a decimal between 0 and 1, you will beable to adapt.

Example:What is the critical χ² for a 0.05 significance testwith 13 degrees of freedom?

Run the MATH200B program and select4:Critical χ². Enter the number of degrees offreedom and the area of the right-hand tail. Be patient: thecomputation is slow. But the program gives you the critical χ²value of 22.36, as shown in the second screen.

Interpretation: For a χ² distribution with 13 degreesof freedom, the value χ² = 22.36 divides thedistribution such that the area of the right-hand tail is 0.05.

5. Inferences about σ, the Standard Deviation of a Population

Summary:This part performs hypothesis testsand computes confidence intervals for the standard deviation of apopulation. Since variance is the square of standard deviation, it canalso do those calculations for the variance of a population.

Cautions:

The tests on standard deviation or variance of a population require thatthe underlying population must be normal.They are not robust, meaning that even moderate departures fromnormality can invalidate your analysis.See MATH200A Program part 4for procedures to test whether apopulation is normal by testing the sample.

Outliers are also unacceptable and must be ruled out.See MATH200A Program part 2for an easy way to test for outliers.

See also:Inferences about One Population Standard Deviation gives the statisticalconcepts with examples of calculation “by hand” andin an Excel workbook.

You already know how to test the mean of apopulation with a t test, or estimate a population mean using at interval. Why would you want to do that for the standarddeviation of a population?

The standard deviation measures variability. In manysituations not just the average is important, butalso the variability. Another way to look at it is thatconsistency is important: the variability must not be toogreat.

For example, suppose you are thinking aboutinvesting in one of two mutual funds. Both show an average annualgrowth of 3.8% in the past 20 years, but one has a standard deviationof 8.6% and the other has a standard deviation of 1.2%. Obviously youprefer the second one, because with the first one there’squite a good chance that you’d have to take a loss if you needmoney suddenly.

Industrial processes, too, are monitored not only for averageoutput but for variability within a specified tolerance. If thediameter of ball bearings produced varies too much, many of themwon’t fit in their intended application. On the other hand, itcosts more money to reduce variability, so you may want to make surethat the variability is not too low either.

To use the program, first check the requirements for yoursample; see Cautions above.Then run the MATH200B program and select 5:Infer about σ.When prompted, enter the standard deviation and size of the sample,pressing [ENTER] after each one. If you know the variance ofthe sample rather than the standard deviation, use the square rootoperation since s is the square root of the variance s² (seeexample below).

The program then presents you with a five-item menu:confidence interval for the population standard deviation σ,confidence interval for the population variance σ², andthree hypothesis tests for σ or σ² less than,different from, or greater than a number. Make your selection bypressing the appropriate number.

Confidence Intervals

If you select one of the confidence intervals, the program willprompt you for the confidence level and then compute the interval.Because this involves a process of successive approximations, it cantake some time, so please be patient.

The program displays the endpoints of the interval on screenand also leaves them in variables L and H incase you want to use them in further calculations. You can includethem in any formula by pressing [ALPHA)makesL] and [ALPHA^makesH].

By the way, confidence intervals about a populationstandard deviation are not symmetric around the sample standarddeviation. That’s different from the simpler cases of means andproportions. In this example, the 95% interval for σ extends2.7 units below the sample standard deviation, but 4.3 units aboveit.

Hypothesis Tests

If you select one of the hypothesis tests, the program willprompt you for σ, the population standard deviation in thenull hypothesis. If your H₀ is about population varianceσ² rather than σ, use the square root symbol toconvert the hypothetical variance to standard deviation.

The program then displays the χ² test statistic, thedegrees of freedom, and the p-value. These are also left in variablesX, D, and P in case you wish to usethem in further calculations. You can include them in any formula with[x,T,θ,n], [ALPHAx^-1makesD], and [ALPHA8makesP].

Examples

Example 1: A machine packs cereal intoboxes, and you don’t want too much variation from box to box. You decide that a standard deviation of no more than fivegrams (about 1/6 ounce) is acceptable. To determine whether the machine isoperating within specification, you randomly select 45 boxes. Here arethe weights of the boxes, in grams:

386	388	381	395	392	383	389	383	370
379	382	388	390	386	393	374	381	386
391	384	390	374	386	393	384	381	386
386	374	393	385	388	384	385	388	392
400	377	378	392	380	380	395	393	387

Solution: First, use 1-VarStats to find the sample standard deviation,which is 6.42 g. Obviously this is greater than the targetstandard deviation of 5 g, but is it enough greater that you cansay the machine is not operating correctly, or could it have come froma population with standard deviation no more than 5 g?Your hypotheses are

H₀: σ = 5, the machine is within spec(some books would say H₀: σ ≤ 5)

H₁: σ > 5, the machine is not working right

No α was specified, but for an industrial process with nopossibility of human injury α = 0.05 seemsappropriate.

Next, check the requirements: is the samplenormally distributed and free of outliers?Use MATH200A part 2 to make a box-whisker plot to rule out outliers, andMATH200A part 4 to check normality. The outputs are shown at right. You can seethat the sample has no outliers andthat it is extremely close to normal,so requirements are met and you can proceed with thehypothesis test.

Now, run the MATH200B program and select5:Infer about σ. Enter s:6.42 andn:45, and select 5:Test σ>const. Enter 5for σ in H₀.

The results are shown at far right. The test statistic isχ² = 72.54 with 44 degrees of freedom, and thep-value is 0.0043.

Since p<α, youreject H₀ and accept H₁.At the 0.05 level of significance, the population standard deviationσ is greater than 5, and the machine is not operating withinspecificaton.

Example 2:You have a random sample of size 20, with a standard deviation of 125. Youhave good reason to believe that the underlying population is normal,and you’ve checked the sample and found no outliers.Is the population standard deviation different from 100, at the 0.05significance level?

Solution:n = 20, s = 125, σ_o = 100,α = 0.05. Your hypotheses are

H₀: σ = 100

H₁: σ ≠ 100

This time in the INFER ABOUT σ menu you select4:Test σ≠const.

Results are shown at right. χ² = 29.69 with19 degrees of freedom, and the p-value is 0.1118.

p>α; fail to reject H₀. At the 0.05 significance level,you can’t say whether the population standard deviationσ is different from 100 or not.

Example 3: Of several thousand studentswho took the same exam, 40 papers were selected randomly andstatistics were computed. The standard deviation of the sample was 17points. Estimate the standard deviation of the population, with 95%confidence. (Recall that test scores are normally distributed.)

Solution:Check the data and make sure there are no outliers.Run MATH200B and select [2] in the firstmenu. Enter s and n, and in the second menu select1:σ interval with a C-level of 95 or .95.The results screen is shown at right.

Conclusion: You’re 95% confident that the standarddeviation of test scores for all students is between 13.9 and21.8.

Remark: The center of the confidenceinterval is about17.9, which is different from the point estimate s=17. This is afeature of confidence intervals for σ or σ²:they are asymmetric because the χ² distribution used tocompute them is asymmetric.

Example 4:Heights of US males aged 18–25 are normally distributed. Youtake a random sample of 100 from that population and find a mean of65.3 in and a variance of 7.3 in². (Remember thatthe units of variance are the square of the units of the originalmeasurement.)

Estimate the mean and variance of the height of US malesaged 18–25, with 95% confidence.

Solution for mean:Computing a confidence interval for the mean is a straightforwardTInterval. Just remember that for Sx thecalculator wants the sample standard deviation, but you have thesample variance, which is s². Therefore you take the square rootof sample variance to get sample standard deviation, as shown in theinput screen at near right.

The output screen at far right shows the confidence interval.You’re 95% confident that the mean height of US males aged18–25 is between 64.8 and 65.8 in.

Solution for variance:Run the MATH200B program and select5:Infer about σ. Enter s:√7.3 and n:100.Select 2:σ² interval and enter C-Level:.95 (or95). The program computes the confidence interval for populationvariance as 5.6 ≤ σ² ≤ 9.9.Notice that the output screen shows the point estimate for variance, s²,and that as expected the confidence interval is not symmetric.

You’re 95% confident that the variance in heights of US malesaged 18–25 is between 5.6 and 9.9 in².

Complete answer:You’re 95% confident that the heights of US males aged18–25 have mean 64.8 to 65.9 in and variance5.6 to 9.9 in².

6. Inferences about Linear Correlation

Summary:With linear correlation, you computea sample correlation coefficient r. But what can you say about thecorrelation in the population, ρ? The MATH200B program computes aconfidence interval about ρ or performs a hypothesis test totell whether there is correlation in the population.

See also:Inferences about Linear Correlation gives the statisticalconcepts with examples of calculation “by hand” andin an Excel workbook.

To perform inferences about linear regression, first load yourx’s and y’s in any two statistics lists. Then runthe MATH200B program and select 7:Regression inf.

Example: Thefollowing sample of commuting distances and times for fifteen randomlyselected co-workers is adapted fromJohnson & Kuby 2004 [full citation at https://BrownMath.com/swt/sources.htm#so_Johnson2004], page 623.

Commuting Distances and Times
Person	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Miles, x	3	5	7	8	10	11	12	12	13	15	15	16	18	19	20
Minutes, y	7	20	20	15	25	17	20	35	26	25	35	32	44	37	45

The TI’s LinReg(ax+b) command can tell youthat the correlation of the sample is 0.88. But what can you inferabout ρ, the correlation of the population? You can get a confidenceinterval estimate for ρ, or you can perform a hypothesis testfor ρ≠0.

Requirements

Before you can make any inference (hypothesis test orconfidence interval) about correlation or regression in thepopulation, check these requirements:

The data are a simple random sample.
The plot of residuals versus x is featureless —no bending, no thickening or thinning trend from left to right, and nooutliers.
The residuals are normally distributed. You can checkthis with a normal probability plot, available in most statisticspackages and in MATH200A part 4. Since the test statistic is a t, and thet test is robust, moderate departures from normality are okay.

To make a scatterplot of residuals, perform a regressionwith LinReg(ax_b) L1,L2 (or whichever lists containyour data). This computes the residuals automatically. You can thenplot them by following the procedure in Display the Residuals, part ofLinked Variables. As you see from the graph at right, theresiduals don’t show any problem features.

To check normality of the residuals, run MATH200A part 4 and whenprompted for the data list press [2ndSTATmakesLIST][▲], scroll to RESID if necessary, andpress [ENTER] [ENTER]. The graph at right shows that theresiduals are approximately normally distributed.

It can be hard to tell whether a normal probabilityplot is close enough to a straight line. But MATH200A part 4 shows the rand critical values from theRyan-Joinertest. When r > the critical value, the points are nearenough to a normal distribution. Here r=0.9772 > crit=0.9383,so the residuals are close enough to normal.

Confidence Interval about ρ

Enter your x’s and y’s in two statistics lists,such as L1 and L2. Run the MATH200B program and select 6:Correlatn inf. Whenprompted, enter your x list and y list, select1:Conf interval, and enter your desired confidencelevel, such as .95 or 95 for 95%.

The output screen is shown at right. For this sample ofn = 15 points, the sample correlation coefficient isr = 0.88. For the correlation of the population (distancesand times for all commuters at this company),you’re 95% confident that0.67 ≤ ρ ≤ 0.96.

(Just like confidence intervals about σ, confidenceintervals about ρ extend different amounts above and below thesample statistic.)

Hypothesis Test about ρ

You can also do a hypothesis test to see whether there is anycorrelation in the population. The null hypothesis H₀ is thatthere is no correlation in the population, ρ = 0; thealternative H₁ is that there is correlation in the population,ρ ≠ 0.

Select your α; 0.05 is a common choice.Run the MATH200B program and select 6:Correlatn inf. Enter your x and ylists and select 2:Test ρ≠0.

The output screen is shown at right. Sample size n = 15,and sample correlation is r = 0.88. The t statistic for thishypothesis test is 6.64, and with 13 (n−2) degrees of freedomthat yields a p-value of <0.0001.

p<α; reject H₀ and accept H₁. At the 0.05level of significance, ρ ≠ 0: there is somecorrelation in the population. Furthermore, the populationcorrelation is positive. (See p < α in Two-Tailed Test: What Does It Tell You? for interpreting theresult of a two-tailed test in a one-tailed manner like this.)

Remark: When p is greater than α, youfail to reject H₀. In that case, you conclude thatit is impossible to say, at the 0.05 level of significance,whether there is correlation in the population or not.

7. Inferences about Linear Regression

Summary:A linear regression fits anequation of the formŷ = b₁x + b₀ to thesample data, but the slope b₁ and the y interceptb₀ are just sample statistics. If you took a differentsample you would likely get a different regression line.

The MATH200B program findsconfidence intervals for the slope β₁ and intercept β₀of the line that best fits the entire population of points, not just aparticular sample. It can also find aconfidence interval about the mean ŷ for a particular xand aprediction interval about all ŷ’s for a particular x.

The program doesn’t do any hypothesis tests on theregression line. The standard test is totest whether the regression line has a nonzero slope, β₁ ≠ 0.But thattest is identical to the test for a nonzero correlation coefficient,ρ ≠ 0, which the MATH200B program performs as part ofthe 6:Correlatn inf menu selection.

See also:Inferences about Linear Regression explains the principles andcalculations behind inferences about linear regression; there’seven an Excel workbook.

The Example

Let’s use the same data on commutingdistances and times from Inferences about Linear Correlation.The TI-83/84 command LinReg(ax+b)will show the bestfitting regression line for this particular sample, but what can you say aboutthe regression for all commuters at that company?

Download Distance Formula Ti 84 Program Free

Requirements

The requirements for inference about regression are the sameas the requireemnts for inference about correlation, listed above.

Regression Coefficients for the Population

Solution: Enter the x’s and y’s in any twostatistics lists, such as L1 and L2. Run the MATH200B program and select7:Regression inf. Specify the two lists and your desired confdence level,such as .95 or 95 for 95%.

Results: Always look first at the sample size (bottomof the screen) to make sure you haven’t left out any points.The slope of the sample regression line is1.89, meaning that on average each extra mile of commute takes 1.89minutes (a speed of about 32 mph). But the 95% confidenceinterval for the slope is 1.28 to 2.51: you’re 95% confidentthat the slope of commuting time per distance, for all commuters atthis company, is between 1.28 and 2.51 minutes per mile.

The second section of the screen shows that they intercept of the sample is 3.6: this represents the “fixedcost” of the commute, as opposed to the “variable cost”per mile represented by the slope. But the 95% confidence interval is−4.5 to +11.8 minutes.

Interpretation: the line that best fits the sample data is

ŷ = 1.89x + 3.6

and the regression line for the whole population is

ŷ = β₁x + β₀

where you’re 95% confident that

1.28 ≤ β₁ ≤ 2.51 and −4.5 ≤ β₀ ≤ +11.8

Let’s think a bit more about that intercept, with a 95%confidence interval of −4.5 to +11.8 minutes. This is a goodillustration thatit’s a mistake to use a regression line too far outside your actual data.Here, the x’s runfrom 3 to 20. The y intercept corresponds to x = 0,and a commute of zero miles is not a commute at all. (Yes, there arepeople who work from home, but they don’t get in their cars anddrive to work.) While the y intercept can be discussed as amathematical concept, it really has no relevance to this particularproblem.

Inferences about a Particular x Value

The first output screen was about the line as a whole; now theprogram turns to predictions for a specific x value.First it asks for the x value you’re interested in. This time,let’s make predictions about a commute of 10 miles.

Caution: You should only use x values that arewithin the domain of x values in your data, or close to it. Nomatter how good the straight-line relationship of your data, youdon’t really know whether that relationship continues for loweror higher x values.

The program arbitrarily limits you to the domain plus orminus 15% of the domain width, but even that may be too much in someproblems. In this problem, commuting distances range from 3 mi to20 mi, a width of 17 mi. The program will let you makepredictions about any x value from3−.15*12 = 0.45 mi to15+.15*12 = 22.55 mi, but you have to decide how faryou’re justified in extrapolating.

The input and output screens are shown at right.ŷ (y-hat) is simply the y value on the regressionline for the given x value, found byŷ = (slope)×10+(intercept) = 22.6. That is aprediction for μ_y|x=10, the average time for many10-mile commutes. The screen shows a 95% confidence interval for thatmean: you’re95% confident that the average commute time for all 10-mile commutes(not just in the sample) is between 19.3 and 25.9minutes.

But that is an estimate of the mean. Can we say anything aboutindividual commutes? Yes, that is the prediction interval at thebottom of thescreen. It says that 95% of all 10-mile commutes take between10.4 and 34.7 minutes.

References

Spiegel, Murray R., and Larry J. Stephens. 1999.: Theory and Problems of Statistics. 3d ed. McGraw-Hill.
Sullivan, Michael. 2008.: Fundamentals of Statistics. 2d ed. Pearson Prentice Hall.

What’s New

18 July 2019, program version 4.4a:With some very unlikely data sets,the program could falsely tell you that class widths were unequal orthat discrete probabilities didn’t add to 1, and refuse tocompute any results. I changed the floating-point tests involved,eliminating that possibility. My thanks to Ernest Brock for drawingthis potential problem to my attention.

Don’t worry that you might have been getting incorrectcomputations! If MATH200B computed results for you, they were correct.The only problem could have been in the program saying your data setwas invalid when it was actually valid — and Idon’t know of any actual case where even that has happened.

15 May 2016: Move citations to the newReferences section.

Ti 84 Free Calculator Download

27 Dec 2015:

Program version 4.4:
- MATH200B now senses whether it’s being run on ahi-resolution color screen or low-resolution b&w screen andadjusts itself acordingly, so there’s no more need for aseparate MATH200C. I learned the technique in the forums atCemetech.
- On high-resolution TI-84s, skewness and kurtosis fiton a single screen.
- Round correation coefficients to four decimalplaces.
- On screens for regression confidence intervals, show theconfidence leval.
Mention the newer TI-84 models’ different floating-point processing ascompared to the older models.
Add a paragraph explaining the normalitytest for residuals.
Show the results screen for a confidenceinterval about the standard deviation of a population, and note thatthe interval is not symmetric.
Lose the comparisons to Sullivan’s book.

17 Dec 2015: