ࡱ> rtq` 0ibjbjss iXD$999PF::;^;`B@X@X@X@3A0cELF]]]]]]]$\`hb^iWG3A3AWGWG^X@X@|^UUUWGX@X@]UWG]UU:q[,[X@; K29N[ )\^0^[RcR.c[c[0WGWGUWGWGWGWGWG^^UWGWGWG^WGWGWGWG$$++ Statistics with the TI-84 Calculator Version 2.10 - 2004-01-09 - corrections & additions welcome - Dr. Wm J. Larson - william.larson@ecolint.ch Table of Contents  TOC \o "1-3" Creating a list  PAGEREF _Toc61428765 \h 1 Calculating the Mean, Median, Standard Deviation & Interquartile Range  PAGEREF _Toc61428766 \h 1 The Names of TI-84 Symbols & Alternative Symbols  PAGEREF _Toc61428767 \h 1 Range  PAGEREF _Toc61428768 \h 1 Interquartile Range  PAGEREF _Toc61428769 \h 1 Using a frequency list  PAGEREF _Toc61428770 \h 1 Redisplaying Data  PAGEREF _Toc61428771 \h 1 Be careful to clear the screen  PAGEREF _Toc61428772 \h 1 Making Histograms with a TI-84  PAGEREF _Toc61428773 \h 2 Enter your data  PAGEREF _Toc61428774 \h 2 Set Up Your Plot.  PAGEREF _Toc61428775 \h 2 Set Up Your Window  PAGEREF _Toc61428776 \h 2 Display Your Histogram  PAGEREF _Toc61428777 \h 2 Display the interval and frequency  PAGEREF _Toc61428778 \h 2 Calculating Probabilities for the Normal Distribution  PAGEREF _Toc61428779 \h 2 Using ShadeNorm  PAGEREF _Toc61428780 \h 2 A Program to Set the Window for ShadeNorm  PAGEREF _Toc61428781 \h Error! Bookmark not defined. Using normalcdf  PAGEREF _Toc61428782 \h 2 Significant Digits  PAGEREF _Toc61428783 \h 2 Calculating the Inverse Normal Distribution  PAGEREF _Toc61428784 \h 2 Using invNorm  PAGEREF _Toc61428785 \h 2 Calculating Probabilities for the t-Distribution  PAGEREF _Toc61428786 \h 3 Using tcdf  PAGEREF _Toc61428787 \h 3 Inverse t  PAGEREF _Toc61428788 \h 3 Calculating Probabilities for the Poisson Distribution (Higher Level only)  PAGEREF _Toc61428789 \h 3 Using poissonpdf and poissoncdf  PAGEREF _Toc61428790 \h 3 Calculating Probabilities for the Binomial Distribution (Higher Level core only)  PAGEREF _Toc61428791 \h 3 Using binompdf and binomcdf  PAGEREF _Toc61428792 \h 3 Confidence Intervals  PAGEREF _Toc61428793 \h 4 Calculating a Z interval  PAGEREF _Toc61428794 \h 4 Hypothesis Testing  PAGEREF _Toc61428795 \h 4 Conducting a Z-Test  PAGEREF _Toc61428796 \h 4 Conducting a t-Test (Higher Level only)  PAGEREF _Toc61428797 \h 4 Conducting a ( Test for Independence i.e. Contingency Tables  PAGEREF _Toc61428798 \h 4 Conducting a ( Test for Independence with the Yates Continuity Correction  PAGEREF _Toc61428799 \h 4 Conducting a ( Goodness of Fit Test (Higher Level only)  PAGEREF _Toc61428800 \h 5 Regression and Correlation Analysis  PAGEREF _Toc61428801 \h 5 Drawing a Scatter Diagram  PAGEREF _Toc61428802 \h 5 Fitting a line  PAGEREF _Toc61428803 \h 5 To get r & r to appear  PAGEREF _Toc61428804 \h 5 Covariance  PAGEREF _Toc61428805 \h 5 The equations you can fit:  PAGEREF _Toc61428806 \h 5  Creating a list Key STAT EDIT Edit and type your list in L1 or L2 etc. Calculating the Mean, Median, Standard Deviation & Interquartile Range First enter your data into a list as above. Key STAT, CALC, 1: 1-Var Stats, L1, Enter. Since L1 is the default for 1-Var Stats, if you entered your data into L1, you need not type L1 again. i.e. keying STAT, CALC, 1: 1-Var Stats, Enter would work. A set of statistics about L1 will appear. The mean,  EMBED Equation.3 , is at the top of the list. Scrolling down other statistics including Med (the median), Sx (sample standard deviation), (x (the population standard deviation), Q1 (the lower quartile) & Q3 (the upper quartile) will be displayed. The interquartile range = Q3Q1. The Names of TI-84 Symbols & Alternative Symbols TI-84 SymbolAlternative SymbolsNameAlternative Name EMBED Equation.3 MeanAverageSxs, sn-1Sample standard deviationunbiased estimator of the population standard deviation (IB name)(x, snPopulation standard deviationSample standard deviation (IB name)minXLMinimum valuethe lowest valueQ1first quartileMedMMedianQ3third quartilemaxXHMaximum valuethe highest valueRange Range = MaxX - MinX Interquartile Range Interquartile range = Q3 Q1. Using a frequency list If you are given data points with frequencies for each data point, put the data points in L1 & the frequencies in L2. Then key STAT, CALC, 1: 1-Var Stats, L1, L2. L1 is the default for the data list, so if there is no frequency list & the data is in L1, you need not type L1. But there is no default for the frequency list. So if there is a frequency list in L2, you need to type 1-Var Stats L1, L2. Redisplaying Data If you cleared the screen (but did not run a new statistics calculation), you can redisplay your data. For example you can redisplay Q1 & Q3 by keying VARS 5:Statistics, PTS & then selecting 7:Q1 or 9:Q3. stdDev & variance Be careful stdDev( & variance( which are in LIST MATH and in the CATALOG return Sx (sn-1) and Sx (sn-1) respectively, not x (sn) and x (sn) as you might suppose. Be careful to clear the screen The TI-84 has a tendency to display information from a previous calculation, so when you are making a new calculation, always clear the screen first using CLEAR, CLEAR. Making Histograms with a TI-84 Enter your data If your data is just a set of numbers, enter your data into one list, say L1. If instead your data is a frequency distribution table, enter your data into two lists, say L1 for the values and L2 for the frequencies. If your data is grouped data, e.g. with class intervals, enter the midpoint of each class interval in L1 and the frequencies in L2. Set Up Your Plot. Now key 2nd STAT PLOT and set up your plot. Choose a plot, say Plot1, by putting the cursor on Plot1 and pressing ENTER. Turn Plot1 on by putting the cursor on On and pressing ENTER. Choose to plot a histogram by moving the cursor to the image of a histogram and pressing ENTER. If you have just a set of numbers in L1, key Xlist: L1 and Freq: 1. If instead you have the values in L1 and the frequencies in L2, key Xlist: L1 and Freq: L2. If you want to change Freq from L2 to 1, you must key ALPHA 1. Set Up Your Window Key WINDOW. Set Xmin a little less than your smallest value and Xmax a little more than your biggest value. Set Xscl to give the size of your class intervals. Xscl can be reset until you are satisfied that your interval size gives a good representation of the data. About 8 to 20 intervals usually give a good representation. Display Your Histogram Turn off any other plots and any graphs in Y=. Now key GRAPH and voila - the histogram! Display the interval and frequency To display the interval and frequency use TRACE Calculating Probabilities for the Normal Distribution Using ShadeNorm ShadeNorm will draw the graph and calculate the probability. Key 2nd DISTR DRAW1: ShadeNorm(lowerbound, upperbound [, (, (]) Example Find P(z < -0.5). (The default vales of ( = 0, (= 1 are desired, so they need not be entered.) Key DISTR DRAW 1: ShadeNorm(-100, -.5) The graph, the lower bound (-100, being 100 standard deviations from the mean, is effectively minus (), the upper bound and the P(z<-0.5), i.e. 0.3085 are displayed. If the graph is not visible, set the Window to: xmin = -3 Xmax = 3 Xscl = 1 Ymin = -.25 Ymax = .5 Yscl = .25 Xres = 1 Example If ( = 55, ( = 10, find P(40 < x < 65). Its tiresome to reset the window so input this program. It prompts for (, (, the lower and upper bounds, sets the widow size and then runs ShadeNorm. PROGRAM: NORMWIND :Input "MEAN: ",M :Input "ST DEV: ",S :-.3/(S* EMBED Equation.3 ) STO% Ymin :-3.6*Ymin STO% Ymax :M-3.5*S STO% Xmin :M+3.5*S STO% Xmax :Input "LOWER BND:",L :Input "UPPER BND:",U :ShadeNorm(L,U,M,S) :Stop Using normalcdf Drawing the cumulative probability distribution graph is very illustrative, but a little time consuming. Its faster just to run normalcdf. Key 2nd DISTR DISTR (the default) 2: normalcdf(lowerbound, upperbound [, (, (]) Example If ( = 55, (= 10, find P(x < 65). Keying DISTR DISTR normalcdf (-E99, 65, 55, 10) will display 0.841. We usually want the cumulative distribution function (cdf) for the normal distribution. The probability distribution function (pdf) would be useful to graph the normal curve in Y=, but ShadeNorm already does that. Significant Digits Notice that more significant digits are available with the TI-84 than with a normal distribution table in a textbook. However in the real world ( & ( are usually not known with enough accuracy to make this meaningful. Calculating the Inverse Normal Distribution Using invNorm For ((a) ( P(Z < a) if ( is known but a is not known, invNorm will calculate a. Key 2nd DISTR DISTR (the default) 3: invNorm(area, [, (, (]) Example If P(Z < a) = .6, find a. (The default vales of ( = 0, (= 1 are desired, so they need not be entered.) Keying DISTR 3: invNorm(.6) will give 0.253347 Example If x ~ N(100, 5) & P(x < a) = .20, find a. Keying DISTR 3: invNorm(.2, 100, 5) will give 95.8. Calculating Probabilities for the t-Distribution Using tcdf tcdf will calculate the probability, i.e. the t cumulative probability distribution function. Key 2nd DISTR DISTR (the default) 5: tcdf(lowerbound, upperbound, df) Example If df = 20, find P(t < 2). Keying DISTR DISTR tcdf (-E99, 2, 20) will display 0.9704. We usually want the cumulative distribution function (cdf) for the normal distribution. The probability distribution function (pdf) would be useful to graph the normal curve in Y=, but Shade_t already does that. Inverse t There is no function corresponding to invNorm for the t-distribution, but you can use the TI-84 equation solver. MATH 0: Solver. Then using tcdf( from 2nd DISTR, key in eqn: 0 = tcdf(-10000, x, 27) - 0.95. Then key ALPHA SOLVE, giving 1.703. We needed a large negative number, so we used -10000. Except for v < 5, -10 would have been large enough. Or get t from a graph. This method takes longer, but it is more illustrative. Graph 2nd DISTR 5: tcdf(-10000, x, df), where 10000 is the lower bound, df is the degrees of freedom and x is the variable to be graphed. Set Xmin = -4, Xmax = 4, (because the tails beyond t = ( 4 are almost zero, Ymin = -.1, Ymax = 1.1 (because tcdf is the probability that t < x, which must be zero for x = -( and one for x = + (), Xres = 8 (because the TI-84 calculates t and thats very slow). Now graph the desired probability (e.g. for P(t < t*) = 0.90, graph Y2 = 0.90) and find the intersection of the curves. Example Find t for n = 10 and p = 0.95. df = n - 1 = 9. Key Y1 = tcdf(-10000, x, 9) Y2 = .95. Use 2nd CALC intersect to find t (x on the screen) = 1.833 at p (y on the screen) = .95, of course. Calculating Probabilities for the Poisson Distribution (Higher Level only) Using poissonpdf and poissoncdf Since the Poisson distribution is discrete, either the cumulative distribution function (cdf) or the probability distribution function (pdf) would be useful. Use the pdf to find the probability that one value is observed (X = Xo) & the cdf to find the probability that one of a range of values is observed (X ( Xo). Key 2nd DISTR DISTR (the default), B: poissonpdf((, x) or Key 2nd DISTR DISTR (the default), C: poissoncdf((, x) Example If ( = 3.75, find P(x = 6). Keying DISTR DISTR poissonpdf (3.75,6) will display 0.0908. Example If ( = 1.4, find P(x ( 2) = 1 - P(x ( 1) Keying DISTR DISTR poissoncdf (1.4,1) will display 0.408. P(x ( 2) = 1 - 0.408 = 0.592. Graphing the Poisson distribution Use poissonpdf to graph the Poisson distribution. For example use Y1 = poissonpdf (6, x). Since the Poisson distribution is discrete, there will only be output for integer values of x. Therefore the grapher has to be set so that integer values of x fall on the pixel elements. In WINDOW set xmin = 0 and xmax to a multiple of 4.7 equal to about 3(. Trace can be used to read out the values. Since only integer values of x will be traceable, key in 0, 1, 2, ... Unfortunately the values for x = 0 and for p small will be hidden by the axes. If necessary turn off the axes with 2nd FORMAT AxesOff. Calculating Probabilities for the Binomial Distribution (Higher Level core only) Using binompdf and binomcdf Since the Binomial distribution is discrete, either the cumulative distribution function (cdf) or the probability distribution function (pdf) would be useful for calculating probabilities. Use the pdf to find the probability that one value is observed (X = Xo) & the cdf to find the probability that one of a range of values is observed (X ( Xo). Key 2nd DISTR DISTR (the default) 0: binompdf(n, p[, x]) or Key 2nd DISTR DISTR (the default) A: binomcdf(n, p[, x]), where n is the number of trials, p is the probability of a success in one trial and x the desired number of success. If no x is specified, then a list of probabilities will be generated for x equals zero to n. Example If n = 6, p = .75, find P(x = 6). Keying DISTR DISTR pdf (6, .75, 6) will display 0.178. Example If n = 6, p = .75, find P(x > 2) = 1 - P(x ( 3). Keying DISTR DISTR binomcdf (6, .75, 3) will display 0.169. P(x > 2) = 1 - 0.169 = 0.831. Graphing the Binomial distribution Use binompdf to graph the Binomial distribution. For example use Y1 = binomcdf (6, .75, x). Since the Binomial distribution is discrete, there will only be output for integer values of x. Therefore the grapher has to be set so that integer values of x fall on the pixel elements. In WINDOW set xmin = 0 and xmax to the smallest multiple of 4.7 bigger than n. Trace can be used to read out the values. Since only integer values of x will be traceable, key in 0, 1, 2, ... Unfortunately the values for x = 0 and for p small will be hidden by the axes. If necessary turn off the axes with 2nd FORMAT AxesOff. Confidence Intervals Calculating a Z interval Zinterval can be used to calculate a Confidence Interval. You can enter your entire sample & have the TI-84 calculate ( or you can enter ( directly. Key STAT TESTS 7: Zinterval. Then if you are entering ( directly select Stats & key ENTER. Then enter (, (, n & the desired confidence level (as a decimal, not as a % - its called the C-Level), select Calculate & key ENTER. If you are given the actual sample numbers, i.e. not (, enter them into a list and then you can either calculate ( as described above (key STAT, CALC, 1: 1-Var Stats, L1) & then use Zinterval Stats. Or you can use Zinterval Data. In Data you must enter (, n & the desired confidence level as before, but instead of ( you enter the name of the list containing your data, e.g. L1, select Calculate & key ENTER. Hypothesis Testing Conducting a Z-Test Z-Test is used to test a hypothesis. You can enter your entire sample & have the TI-84 calculate ( or you can enter ( directly. Key STAT, select TEST 1: Z-Test. Then if you are entering  EMBED Equation.3  directly, select Stats & key ENTER. Then enter (o, (,  EMBED Equation.3 , n & the alternative hypothesis. Select Calculate & key ENTER. If you are using the actual sample numbers, i.e. not  EMBED Equation.3 , enter the data into a list and then use Z-Test Data. In Data you must enter (o, (, n & the alternative hypothesis as before, but instead of  EMBED Equation.3  you enter the name of the list containing your data, e.g. L1, select Calculate & key ENTER. Conducting a t-Test (Higher Level only) t-Test is used to test a hypothesis. It is more realistic than the z test in that s, the standard deviation calculated from the sample, is used, but it requires that the sample be approximately normal or large. For large samples the z & t tests give the same answer. You can enter your entire sample & have the TI-84 calculate  EMBED Equation.3  & s or you can enter ( & s directly. Key STAT, select TEST 2: t-Test. Then if you are entering  EMBED Equation.3  & s directly, select Stats & key ENTER. Then enter (o,  EMBED Equation.3 , Sx (i.e. s) n & the alternative hypothesis. Select Calculate & key ENTER. If you are using the actual sample numbers, i.e. not  EMBED Equation.3  & s, enter the data into a list and then use Z-Test Data. In Data you must enter (o, (, n & the alternative hypothesis as before, but instead of  EMBED Equation.3  you enter the name of the list containing your data, e.g. L1, select Calculate & key ENTER. Conducting a ( Test for Independence i.e. Contingency Tables (-Test is used to test a hypothesis of independence with a 2-way contingency table. First enter your data in a matrix. Key MATRIX, select EDIT, select a matrix to fill or edit, key ENTER, change the r ( c (number of rows & columns), if necessary & enter your data. Now key STAT, TESTS, C: (-Test. Key in the name of the matrix containing your data (Observed) and the name of the matrix where you want the expected values placed by keying MATRIX NAMES, selecting the desired matrix name and keying ENTER. Otherwise use matrices A & B which will appear by default as the observed & expected matrices. Then choose how to display your results: Draw or Calculate. Draw will draw the ( distribution, and report ( (the value of () & P (the probability of the observed values, if the null hypothesis of independence were true). Calculate will report (, P & df (the number of degrees of freedom). To view the expected value matrix, key MATRIX, EDIT 2:B (assuming you used B, the default). Note that for a ( test df = (r - 1)(c- 1). Conducting a ( Test for Independence with the Yates Continuity Correction When the df = 1, i.e. when the observed is a 2 ( 2 table, the IB requires that Yates Continuity Correction be applied. ( (corrected) ( EMBED Equation.3 . Enter your observed data in a matrix, say [A]. Make sure matrix B is set to be 2 ( 2, using Matrix EDIT and keying in 2 ( 2. Key STAT, TESTS, C: (-Test [A] [B] ENTER. Unfortunately I have not been able to find a way to get the TI-84 to do the Yates Continuity Correction, so now you have to do it by hand. Copy out the 4 expected values from [B] & do the math. Example suppose the Observed is A =  EMBED Equation.3 . The TI-84 will give the expected B =  EMBED Equation.3 . So now by hand do (|18 - 14.56| - 0.5)/14.56 + (|10 - 13.44| - 0.5)/13.44 + (|8 - 11.44 - 0.5)/11.44 + (|14 - 10.56| - 0.5)/10.56. Luckily it turns out that the numerator of these 4 terms is always the same for a 2 ( 2 table, in our example 8.6436. So you only need to calculate 8.6436 ( (1/14.56 + 1/13.44 + 1/11.44 + 1/10.56) = 2.81. Now go to the ( table & find that for df = 1 the critical 5% value is 3.841. Since 2.81 < 3.841, we fail to reject (accept) the assumption of independence. Conducting a ( Goodness of Fit Test (Higher Level only) A Goodness of Fit Test tests whether the population fits a model, e.g. binomial, Poisson, uniform, normal, etc. The normal, binomial, Poisson, & geometric probability distributions are in 2nd DISTR. There is no ( Goodness of Fit function in the TI-84, but it is easy to calculate. Put the Observed Values in L1 and the Expected Values (the values that you would get if the model you are testing is correct) in L2. In L3, enter the formula (L1 - L2) /L2. (To enter a formula scroll up to L3, key ENTER & type it in.) To find the ( test statistic, enter sum(L3). To find p, enter (cdf(sum(L3),E99,df). (cdf is in 2nd DISTR. E99 is a very good approximation to (. Note that for a best fit model df = k - m - 1, where k = the number of data categories and m = the number of parameter values estimated on the basis of the sample data. Regression and Correlation Analysis Drawing a Scatter Diagram First enter your data into lists. See above. Then Key 2nd STAT PLOT, choose a Plot, ENTER, Select ON, Type: scatter (the squiggle of dots in the upper left), the names of your x & y lists (E.g. L1 & L2 - note that these are 2nd 1 & 2nd 2). Then Key GRAPH and ZOOM 9: ZoomStat. Fitting a line Eleven kinds of regressions for fitting data to a particular type of equation are available. Only 8: LinReg(a+bx) is needed for the IB. Each of them except D accept the following optional parameters Xlistname, Ylistname, freqlist, regeq. regeq is where the fitted regression equation will be stored. The defaults are L1, L2, 1, RegEQ. If you type the independent variable into L1 & the dependent variable into L2, you can use the defaults, i.e. avoid keying in the list names. It is useful to have the regression equation, so that you can plot it on top of the scatterplot to see if the fit looks good. You can paste regeq to Y1 by going to Y1 in Y= and then keying VARS 5: Statistics EQ 1: RegEq. Or if you want the Equation saved to Y1 instead of RegEQ, key LinReg(a+bx) L1, L2, Y1. (Or whatever lists & equation you are using.) Note if you are using the defaults (L1, L2, freqlist =1) they are not needed. The commas between L1, L2 & Y1 are required. Y1 must be keyed as VARS Y-VARS 1: Function Y1. For example key LinReg(ax+b) Y1 (Y1 is in the VARS, Y-VARS, 1:Function menu.) To get r & r to appear To get r & r to appear in the screen, set the diagnostics on by keying 2nd CATALOG, (x-1 - to get to d faster), DiagnosticOn, ENTER, ENTER. Key STAT, CALC, 8: LinReg(a+bx), ENTER. a, b, r & r are displayed. Covariance Covariance = (xy -  EMBED Equation.3  EMBED Equation.3 . Covariance can be calculated from the data displayed by STAT CALC 2: 2-VAR STATS L1, L2. Scrolling down will display (xy,  EMBED Equation.3  &  EMBED Equation.3 . The equations you can fit: 3: MedMed (a sophisticated linear regression which is less sensitive to outliers than LinReg) 4: LinReg(ax+b) (the standard linear regression) 5: QuadReg (a quadratic regression {y = ax + bx + c} ) 6: CubicReg (a cubic regression{y = ax + bx + cx + d}) 7: QuartReg (a quartic regression {y = ax4 + bx + cx + dx + e}) 8: LinReg(a+bx) (a duplication of 4, but useful because some textbooks use one definition of a linear equation, some the other. 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