Casio FX 2.0 PLUS manual

ALGEBRA FX 2.0 PLUS FX 1.0 PLUS User’s Guide 2 ( Additional Functions ) E http://world.casio.com/edu_e/[...]

CASIO ELECTRONICS CO ., L TD . Unit 6, 1000 North Circular Road, London NW2 7JD, U.K. Important! Please k eep your manual and all inf ormation handy for future ref erence.[...]

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20010101 Contents Chapter 1 Ad vanced Statistics Application 1-1 Adv anced Statistics (ST A T) .............................................................. 1-1-1 1-2 T ests (TEST) .................................................................................... 1-2-1 1-3 Confidence Interval (INTR) ..............................................[...]

20010101 Adv anced Statistics Application 1-1 Ad vanced Statistics (ST A T) 1-2 T ests (TEST) 1-3 Confidence Interval (INTR) 1-4 Distribution (DIST) 1 Chapter[...]

20010101 1-1 Adv anced Statistics (ST A T) u u u u u Function Menu The follo wing shows the function menus for the ST A T Mode list input screen. Pressing a function key that corresponds to the added item displays a menu that lets you select one of the functions listed below . • 3 (TEST) ... T est (page1-2-1) • 4 (INTR) ... Confidence interval [...]

20010101 • Logar ithmic Reg ression ... MSE = Σ 1 n – 2 i =1 n ( y i – ( a + b ln x i )) 2 •E xponential Repression ... MSE = Σ 1 n – 2 i =1 n ( ln y i – ( ln a + bx i )) 2 •P ow er Regression ... MSE = Σ 1 n – 2 i =1 n ( ln y i – ( ln a + b ln x i )) 2 •S in Reg ression ... MSE = Σ 1 n – 2 i =1 n ( y i – ( a sin ( bx i [...]

20010101 4. After you are finished, press i to clear the coordinate values and the pointer from the displa y . · The pointer does not appear if the calculated coordinates are not within the display range. ·T he coordinates do not appear if [Off] is specified for the [Coord] item of the [SETUP] screen. · The Y -CAL function can also be used with [...]

20010101 u u u u u Common Functions • The symbol “ ■ ” appears in the upper right cor ner of the screen while e xecution of a calculation is being performed and while a graph is being drawn. Pressing A during this time terminates the ongoing calculation or draw operation (AC Break). • Pressing i or w while a calculation result or graph is[...]

20010101 1-2 T ests (TEST) The Z T est pro vides a var iety of diff erent standardization-based tests. The y mak e it possib le to test whether or not a sample accurately represents the population when the standard deviation of a population (such as the entire population of a country) is known from previous tests. Z testing is used f or market rese[...]

20010101 The following pages e xplain various statistical calculation methods based on the principles descr ibed abov e . Details concer ning statistical principles and terminology can be found in any standard statistics textbook. On the initial ST A T Mode screen, press 3 (TEST) to display the test men u, which contains the following items. • 3 [...]

20010101 Pe rf or m the f ollowing key operations from the statistical data list. 3 (TEST) b (Z) b (1-Smpl) The following shows the meaning of each item in the case of list data specification. Data ............................ data type µ .................................. population mean v alue test conditions (“ G µ 0 ” specifies two-tail t[...]

20010101 Calculation Result Output Example µ G 11.4 ........................ direction of test z .................................. z score p .................................. p-value o .................................. mean of sample x σ n -1 ............................. sample standard deviation (Displayed only f or Data: List setting.) n ..[...]

20010101 u u u u u 2-Sample Z T est This test is used when the standard deviations f or tw o populations are known to test the h ypothesis . The 2-Sample Z T est is applied to the normal distr ib ution. Z = o 1 – o 2 σ n 1 1 2 σ n 2 2 2 + o 1 : mean of sample 1 o 2 : mean of sample 2 σ 1 : population standard deviation of sample 1 σ 2 : popul[...]

20010101 o 1 ................................. mean of sample 1 n 1 ................................. siz e (positive integer) of sample 1 o 2 ................................. mean of sample 2 n 2 ................................. siz e (positive integer) of sample 2 After setting all the parameters, align the cursor with [Execute] and then press [...]

20010101 u u u u u 1-Prop Z T est This test is used to test for an unknown proportion of successes. The 1-Prop Z T est is applied to the normal distr ibution. Z = n x n p 0 (1– p 0 ) – p 0 p 0 : e xpected sample proportion n : s i z e of sample Pe rf or m the f ollowing key operations from the statistical data list. 3 (TEST) b (Z) d (1-Prop) Pr[...]

20010101 u u u u u 2-Prop Z T est This test is used to compare the propor tion of successes. The 2-Prop Z T est is applied to the nor mal distribution. Z = n 1 x 1 n 2 x 2 – p (1 – p ) n 1 1 n 2 1 + x 1 : data value of sample 1 x 2 : data value of sample 2 n 1 : s i z e of sample 1 n 2 : s i z e of sample 2 ˆ p : estimated sample propor tion P[...]

20010101 p 1 > p 2 ............................ direction of test z .................................. z score p .................................. p-value ˆ p 1 ................................. estimated propor tion of sample 1 ˆ p 2 ................................. estimated propor tion of sample 2 ˆ p .................................. e[...]

20010101 k k k k k t T ests u u u u u t T est Common Functions Y ou can use the f ollowing graph analysis functions after dr a wing a g r aph. • 1 (T) ... Displa ys t score . Pressing 1 (T) di spla ys the t score at the bottom of the display , and displa ys the pointer at the corresponding location in the graph (unless the location is off the gra[...]

20010101 u u u u u 1-Sample t T est This test uses the hypothesis test for a single unkno wn population mean when the population standard deviation is unkno wn. The 1-Sample t T est is applied to t -distr ib ution. t = o – 0 µ σ x n –1 n o : mean of sample µ 0 : assumed population mean x σ n -1 : sample standard deviation n : s i z e of sam[...]

20010101 Calculation Result Output Example µ G 11.3 ...................... direction of test t ................................... t score p .................................. p-value o .................................. mean of sample x σ n -1 ............................. sample standard deviation n .................................. size of sa[...]

20010101 u u u u u 2-Sample t T est 2-Sample t T est compares the population means when the population standard deviations are unknown. The 2-Sample t T est is applied to t -distribution. The following applies when pooling is in eff ect. t = o 1 – o 2 n 1 1 + n 2 1 x p n –1 2 σ x p n –1 = σ n 1 + n 2 – 2 ( n 1 –1) x 1 n –1 2 +( n 2 ?[...]

20010101 The following shows the meaning of each item in the case of list data specification. Data ............................ data type µ 1 ................................. sample mean v alue test conditions (“ G µ 2 ” specifies two-tail test, “< µ 2 ” specifies one-tail test where sample 1 is smaller than sample 2, “> µ 2 ?[...]

20010101 Calculation Result Output Example µ 1 G µ 2 ........................... direction of test t ................................... t score p .................................. p-value df ................................. degrees of freedom o 1 ................................. mean of sample 1 o 2 ................................. mean of s[...]

20010101 u u u u u LinearReg t T est LinearReg t T est treats paired-v ar iab le data sets as ( x , y ) pairs , and uses the method of least squares to deter mine the most appropriate a , b coefficients of the data for the regression f or mula y = a + bx . It also determines the correlation coefficient and t value , and calculates the e xtent of th[...]

20010101 Calculation Result Output Example β G 0 & ρ G 0 .............. direction of test t ................................... t score p .................................. p-value df ................................. degrees of freedom a .................................. constant ter m b .................................. coefficient s ....[...]

20010101 k k k k k χ 2 T est χ 2 T est sets up a n umber of independent groups and tests h ypothesis related to the propor tion of the sample included in each group . The χ 2 T est is applied to dichotomous variab les (var iab le with tw o possible values , such as yes/no). Expected counts F ij = Σ x ij i =1 k × Σ x ij j =1 k ΣΣ i =1 j =1 x[...]

20010101 After setting all the parameters, align the cursor with [Execute] and then press one of the function k e ys shown below to perf orm the calculation or dr a w the g r aph. • 1 (CALC) ... P erforms the calculation. • 6 (DRA W) ... Draws the g r aph. Calculation Result Output Example χ 2 ................................. χ 2 val ue p ..[...]

20010101 k k k k k 2-Sample F T est 2-Sample F T est tests the h ypothesis for the ratio of sample variances . The F T est is applied to the F distr ibution. F = x 1 n –1 2 σ x 2 n –1 2 σ Pe rf or m the f ollowing key operations from the statistical data list. 3 (TEST) e (F) The following is the meaning of each item in the case of list data s[...]

20010101 After setting all the parameters, align the cursor with [Execute] and then press one of the function k e ys shown below to perf orm the calculation or dr a w the g r aph. • 1 (CALC) ... P erf or ms the calculation. • 6 (DRA W) ... Draws the g r aph. Calculation Result Output Example σ 1 G σ 2 .......................... direction of t[...]

20010101 k k k k k ANO V A ANO V A tests the hypothesis that the population means of the samples are equal when there are multiple samples. One-W ay ANO V A is used when there is one independent v ar iable and one dependent variab le . Two - Wa y ANOV A is used when there are tw o independent variab les and one dependent variab le . Pe rf or m the [...]

20010101 Calculation Result Output Example One-W ay ANO V A Line 1 (A) .................... Factor A df valu e, SS val ue , MS valu e, F value , p-value Line 2 (ERR) ............... Error df va lu e, SS val ue, MS va lue Tw o - W a y ANO V A Line 1 (A) .................... Factor A df valu e, SS val ue , MS valu e, F value , p-value Line 2 (B) ....[...]

20010101 k k k k k ANO V A (T w o-W a y) u u u u u Description The nearby tab le shows measurement results f or a metal product produced by a heat treatment process based on two treatment levels: time (A) and temper ature (B). The e xperiments were repeated twice each under identical conditions . Pe rf or m analysis of v ar iance on the f ollo wing[...]

20010101 u u u u u Input Example u u u u u Results 1-2-25 T ests (TEST)[...]

20010101 1-3 Confidence Interval (INTR) A confidence inter v al is a r ange (interv al) that includes a statistical value, usually the population mean. A confidence inter v al that is too broad makes it difficult to get an idea of where the population value (true value) is located. A narro w confidence inter val, on the other hand, limits the popul[...]

20010101 u u u u u General Confidence Interval Precautions Inputting a value in the range of 0 < C-Level < 1 for the C-Le vel setting sets you value y ou input. Inputting a v alue in the r ange of 1 < C-Lev el < 100 sets a value equiv alent to your input divided by 100. # Inputting a value of 100 or greater , or a negative value causes [...]

20010101 k k k k k Z Interval u u u u u 1-Sample Z Interval 1-Sample Z Interval calculates the confidence inter val f or an unknown population mean when the population standard deviation is kno wn. The following is the confidence interval. Left = o – Z α 2 σ n Right = o + Z α 2 σ n Ho w e ver , α is the le vel of significance. The value 100 [...]

20010101 After setting all the parameters, align the cursor with [Execute] and then press the function key shown belo w to perform the calculation. • 1 (CALC) ... P erforms the calculation. Calculation Result Output Example Left .............................. inter v al lower limit (left edge) Right ............................ inter v al upper l[...]

20010101 The following shows the meaning of each item in the case of list data specification. Data ............................ data type C-Le vel ........................ confidence level (0 < C-Lev el < 1) σ 1 ................................. population standard de viation of sample 1 ( σ 1 > 0) σ 2 ................................. [...]

20010101 u u u u u 1-Prop Z Interv al 1-Prop Z Interval uses the n umber of data to calculate the confidence inter val f or an unknown propor tion of successes. The following is the confidence interval. The v alue 100 (1 – α ) % is the confidence le vel. Left = – Z α 2 Right = + Z x n n 1 n x n x 1 – x n α 2 n 1 n x n x 1 – n :s i z e of[...]

20010101 u u u u u 2-Prop Z Interval 2-Prop Z Interval uses the n umber of data items to calculate the confidence interval for the defference between the proportion of successes in two populations. The following is the confidence interval. The v alue 100 (1 – α ) % is the confidence le vel. Left = – – Z α 2 x 1 n 1 x 2 n 2 n 1 n 1 x 1 1– [...]

20010101 Left .............................. inter v al lower limit (left edge) Right ............................ inter v al upper limit (r ight edge) ˆ p 1 ................................. estimated sample propotion for sample 1 ˆ p 2 ................................. estimated sample propotion for sample 2 n 1 ................................[...]

20010101 o .................................. mean of sample x σ n -1 ............................. sample standard deviation ( x σ n -1 > 0) n .................................. size of sample (positive integer) After setting all the parameters, align the cursor with [Execute] and then press the function key shown belo w to perform the calcul[...]

20010101 The following confidence interv al applies when pooling is not in effect. The v alue 100 (1 – α ) % is the confidence le vel. Left = ( o 1 – o 2 )– t df α 2 Right = ( o 1 – o 2 )+ t df α 2 + n 1 x 1 n –1 2 σ n 2 x 2 n –1 2 σ + n 1 x 1 n –1 2 σ n 2 x 2 n –1 2 σ C = df = 1 C 2 n 1 –1 + (1 – C ) 2 n 2 –1 + n 1 x 1[...]

20010101 o 1 ................................. mean of sample 1 x 1 σ n -1 ............................ standard deviation ( x 1 σ n -1 > 0) of sample 1 n 1 ................................. size (positive integer) of sample 1 o 2 ................................. mean of sample 2 x 2 σ n -1 ............................ standard deviation ( x[...]

20010101 1-4 Distrib ution (DIST) There is a variety of diff erent types of distr ibution, b ut the most well-known is “normal distr ib ution, ” which is essential f or perfor ming statistical calculations. Nor mal distribution is a symmetr ical distribution centered on the g reatest occurrences of mean data (highest frequency), with the freque[...]

20010101 u u u u u Common Distribution Functions After drawing a graph, you can use the P-CAL function to calculate an estimated p-value for a par ticular x va lu e. The following is the general procedure f or using the P-CAL function. 1. After dr awing a graph, press 1 (P-CAL) to display the x value input dialog bo x. 2. Input the v alue you want [...]

20010101 k k k k k Normal Distribution u u u u u Normal Probability Density Nor mal probability density calculates the probability density of nomal distribution from a specified x value. Nor mal probability density is applied to standard nor mal distribution. πσ 2 f ( x ) = 1 e – 2 2 σ ( x – µ ) 2 µ ( σ > 0) Pe rf or m the f ollowing k[...]

20010101 u u u u u Normal Distribution Pr obability Nor mal distrib ution probability calculates the probability of nor mal distribution data f alling between two specific values. πσ 2 p = 1 e – dx 2 2 σ ( x – µ ) 2 µ a b ∫ a : lo w er boundar y b : upper boundar y Pe rf or m the f ollowing key operations from the statistical data list. [...]

20010101 Calculation Result Output Example p .................................. nor mal distribution probability z:Low ........................... z:Low value (con ver ted to standardize z score for lower value) z:Up ............................. z:Up value (conv er ted to standardize z score for upper v alue) u u u u u In verse Cumulative Normal D[...]

20010101 After setting all the parameters, align the cursor with [Execute] and then press the function key shown belo w to perform the calculation. • 1 (CALC) ... P erforms the calculation. Calculation Result Output Examples x ....................................... inverse cum ulativ e nor mal distr ibution (T ail:Left upper boundar y of integ r[...]

20010101 k k k k k Student- t Distribution u u u u u Student- t Pr obability Density Student- t probability density calculates t probability density from a specified x va lu e. f ( x ) = Γ Γ df π – df + 1 2 2 df 2 df + 1 df x 2 1+ Pe rf or m the f ollowing key operations from the statistical data list. 5 (DIST) c (T) b (P .D) Data is specified[...]

20010101 u u u u u Student- t Distribution Pr obability Student- t distrib ution probability calculates the probability of t distr ib ution data f alling between two specific values. p = Γ Γ df π 2 df 2 df + 1 – df +1 2 df x 2 1+ dx a b ∫ a :l ow er boundar y b : upper boundary Pe rf or m the f ollowing key operations from the statistical da[...]

20010101 Calculation Result Output Example p .................................. Student- t distrib ution probability t:Lo w ........................... t:Low v alue (input lower v alue) t:Up ............................. t:Up v alue (input upper v alue) k k k k k χ 2 Distribution u u u u u χ 2 Pr obability Density χ 2 probability density calcula[...]

20010101 Calculation Result Output Example p .................................. χ 2 probability density # Current V -Window settings are used f or graph drawing when the SET UP screen's [Stat Wind] setting is [Manual]. The V - Window settings below are set automatically when the [Stat Wind] setting is [A uto]. Xmin = 0, Xmax = 11.5, Xscale = [...]

20010101 u u u u u χ 2 Distrib ution Probability χ 2 distribution probability calculates the probability of χ 2 distribution data falling betw een two specific values. p = Γ 1 2 df df 2 x e dx 2 1 df 2 –1 x 2 – a b ∫ a :l ow er boundar y b : upper boundary Pe rf or m the f ollowing key operations from the statistical data list. 5 (DIST) d[...]

20010101 Calculation Result Output Example p .................................. χ 2 distribution probability k k k k k F Distrib ution u u u u u F Probability Density F probability density calculates the probability density function f or the F distr ib ution at a specified x va lu e. Γ n 2 x d n n 2 – 1 2 n Γ 2 n + d Γ 2 d d nx 1 + n + d 2 f [...]

20010101 Calculation Result Output Example p .................................. F probability density # V-Windo w settings f or graph dr awing are set automatically when the SET UP screen's [Stat Wind] setting is [A uto]. Current V - Window settings are used for graph drawing when the [Stat Wind] setting is [Manual]. 1-4-13 Distribution (DIST)[...]

20010101 u u u u u F Distribution Pr obability F distribution probability calculates the probability of F distr ib ution data falling between two specific values. p = Γ n 2 dx x d n n 2 –1 2 n Γ 2 n + d Γ 2 d d nx 1 + n + d 2 – a b ∫ a : lower boundary b : upper boundar y Pe rf or m the f ollowing key operations from the statistical data l[...]

20010101 Calculation Result Output Example p .................................. F distribution probability 1-4-15 Distribution (DIST)[...]

20010101 k k k k k Binomial Distribution u u u u u Binomial Probability Binomial probability calculates a probability at a specified value for the discrete binomial distr ib ution with the specified number of tr ials and probability of success on each trial. f ( x ) = n C x p x (1– p ) n – x ( x = 0, 1, ·······, n ) p : success probabili[...]

20010101 Calculation Result Output Example p .................................. binomial probability u u u u u Binomial Cumulative Density Binomial cumulative density calculates a cumulative probability at a specified v alue f or the discrete binomial distribution with the specified number of tr ials and probability of success on each tr ial. Pe rf[...]

20010101 After setting all the parameters, align the cursor with [Execute] and then press the function key shown belo w to perform the calculation. • 1 (CALC) ... P erforms the calculation. Calculation Result Output Example p ......................................... probability of success 1-4-18 Distribution (DIST) 20011101[...]

20010101 k k k k k Po isson Distribution u u u u u Po isson Probability Po isson probability calculates a probability at a specified value for the discrete P oisson distribution with the specified mean. f ( x ) = x! e – x µ µ ( x = 0, 1, 2, ···) µ :m ean ( µ > 0) Pe rf or m the f ollowing key operations from the statistical data list. 5[...]

20010101 u u u u u P oisson Cumulative Density Po isson cumulativ e density calculates a cumulativ e probability at specified value for the discrete Poisson distribution with the specified mean. Pe rf or m the f ollowing key operations from the statistical data list. 5 (DIST) g (P oissn) c (C .D) The following shows the meaning of each item when da[...]

20010101 k k k k k Geometric Distrib ution u u u u u Geometric Probability Geometr ic probability calculates the probability at a specified v alue, and the number of the trial on which the first success occurs, for the geometr ic distrib ution with a specified probability of success. f ( x ) = p (1– p ) x – 1 ( x = 1, 2, 3, ···) Pe rf or m t[...]

20010101 u u u u u Geometric Cumulative Density Geometr ic cumulativ e density calculates a cumulative probability at specified value , the nu mber of the trial on which the first success occurs, f or the discrete geometr ic distr ib ution with the specified probability of success. Pe rf or m the f ollowing key operations from the statistical data [...]