ALGEBRA FX 2.0 PLUS FX 1.0 PLUS User’s Guide 2 (Additional Functions) E http://world.casio.
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1 Contents Contents Chapter 1 Advanced Statistics Application 1-1 1-2 1-3 1-4 Advanced Statistics (STAT) .............................................................. 1-1-1 Tests (TEST) .................................................................................... 1-2-1 Confidence Interval (INTR) ............................................................... 1-3-1 Distribution (DIST) ............................................................................
Chapter 1 Advanced Statistics Application 1-1 1-2 1-3 1-4 Advanced Statistics (STAT) Tests (TEST) Confidence Interval (INTR) Distribution (DIST) 20010101
1-1-1 Advanced Statistics (STAT) 1-1 Advanced Statistics (STAT) u Function Menu The following shows the function menus for the STAT 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) ... Test (page1-2-1) • 4(INTR) ... Confidence interval (page1-3-1) • 5(DIST) ... Distribution (page1-4-1) SORT and JUMP functions are located in the TOOL menu (6(g)1(TOOL)).
1-1-2 Advanced Statistics (STAT) • Logarithmic Regression ... MSE = • Exponential Repression ... MSE = • Power Regression ... • Sin Regression ... • Logistic Regression ...
1-1-3 Advanced Statistics (STAT) 4. After you are finished, press i to clear the coordinate values and the pointer from the display. · The pointer does not appear if the calculated coordinates are not within the display range. · The 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 a graph drawn by using DefG feature.
1-1-4 Advanced Statistics (STAT) u Common Functions • The symbol “■” appears in the upper right corner of the screen while execution 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 on the display returns to the parameter setting screen. Pressing ! i(QUIT) returns to the top of list input screen.
1-2-1 Tests (TEST) 1-2 Tests (TEST) The Z Test provides a variety of different standardization-based tests. They make it possible 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 for market research and public opinion research, that need to be performed repeatedly.
1-2-2 Tests (TEST) The following pages explain various statistical calculation methods based on the principles described above. Details concerning statistical principles and terminology can be found in any standard statistics textbook. On the initial STAT Mode screen, press 3(TEST) to display the test menu, which contains the following items. • 3(TEST)b(Z) ... Z Tests (p. 1-2-2) c(T) ... t Tests (p. 1-2-10) d(χ2 ) ... χ2 Test (p. 1-2-18) e(F) ... 2-Sample F Test (p. 1-2-20) f(ANOVA) ... ANOVA (p.
1-2-3 Tests (TEST) Perform the following 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 value test conditions (“G µ0 ” specifies two-tail test, “< µ0” specifies lower one-tail test, “> µ0” specifies upper one-tail test.) µ0 ................................. assumed population mean σ ....................
1-2-4 Tests (TEST) Calculation Result Output Example µG11.4 ........................ direction of test z .................................. p .................................. o .................................. xσn-1 ............................. z score p-value mean of sample sample standard deviation (Displayed only for Data: List setting.) n .................................. size of sample # [Save Res] does not save the µ condition in line 2.
1-2-5 Tests (TEST) u 2-Sample Z Test This test is used when the standard deviations for two populations are known to test the hypothesis. The 2-Sample Z Test is applied to the normal distribution. Z= o 1 – o2 σ12 σ22 n1 + n2 o1 : mean of sample 1 o2 : mean of sample 2 σ1 : population standard deviation of sample 1 σ2 : population standard deviation of sample 2 n1 : size of sample 1 n2 : size of sample 2 Perform the following key operations from the statistical data list.
1-2-6 Tests (TEST) o1 ................................. mean of sample 1 n1 ................................. size (positive integer) of sample 1 o2 ................................. mean of sample 2 n2 ................................. size (positive integer) of sample 2 After setting all the parameters, align the cursor with [Execute] and then press one of the function keys shown below to perform the calculation or draw the graph. • 1(CALC) ... Performs the calculation. • 6(DRAW) ... Draws the graph.
1-2-7 Tests (TEST) u 1-Prop Z Test This test is used to test for an unknown proportion of successes. The 1-Prop Z Test is applied to the normal distribution. Z= p0 : expected sample proportion n : size of sample x n – p0 p0 (1– p0) n Perform the following key operations from the statistical data list. 3(TEST) b(Z) d(1-Prop) Prop ............................ sample proportion test conditions (“G p0” specifies two-tail test, “< p 0” specifies lower one-tail test, “> p0” specifies upper one-tail test.
1-2-8 Tests (TEST) u 2-Prop Z Test This test is used to compare the proportion of successes. The 2-Prop Z Test is applied to the normal distribution. x1 x2 n1 – n2 Z= x1 : data value of sample 1 x2 : data value of sample 2 n1 : size of sample 1 n2 : size of sample 2 p̂ : estimated sample proportion p(1 – p ) 1 + 1 n1 n2 Perform the following key operation from the statistical data list. 3(TEST) b(Z) e(2-Prop) p1 .................................
1-2-9 Tests (TEST) p1> p2 ............................ direction of test z .................................. z score p .................................. p-value p̂1 ................................. estimated proportion of sample 1 p̂2 ................................. estimated proportion of sample 2 p̂ .................................. estimated sample proportion n1 ................................. size of sample 1 n2 .................................
1-2-10 Tests (TEST) k t Tests u t Test Common Functions You can use the following graph analysis functions after drawing a graph. • 1(T) ... Displays t score. Pressing 1 (T) displays the t score at the bottom of the display, and displays the pointer at the corresponding location in the graph (unless the location is off the graph screen). Two points are displayed in the case of a two-tail test. Use d and e to move the pointer. Press i to clear the t score. • 2(P) ... Displays p-value.
1-2-11 Tests (TEST) u 1-Sample t Test This test uses the hypothesis test for a single unknown population mean when the population standard deviation is unknown. The 1-Sample t Test is applied to t-distribution. t= o – µ0 xσ n–1 n o : mean of sample µ0 : assumed population mean xσn-1 : sample standard deviation n : size of sample Perform the following key operations from the statistical data list.
1-2-12 Tests (TEST) Calculation Result Output Example µ G 11.3 ...................... direction of test t ................................... p .................................. o .................................. xσn-1 ............................. n .................................. t score p-value mean of sample sample standard deviation size of sample # [Save Res] does not save the µ condition in line 2.
1-2-13 Tests (TEST) u 2-Sample t Test 2-Sample t Test compares the population means when the population standard deviations are unknown. The 2-Sample t Test is applied to t-distribution. The following applies when pooling is in effect.
1-2-14 Tests (TEST) The following shows the meaning of each item in the case of list data specification. Data ............................ data type µ1 ................................. sample mean value test conditions (“G µ2 ” specifies two-tail test, “< µ2” specifies one-tail test where sample 1 is smaller than sample 2, “> µ2” specifies one-tail test where sample 1 is greater than sample 2.) List(1) ..........................
1-2-15 Tests (TEST) Calculation Result Output Example µ1G µ2 ........................... direction of test t ................................... t score p .................................. p-value df ................................. degrees of freedom o1 ................................. mean of sample 1 o2 ................................. mean of sample 2 x1σn-1 ............................ standard deviation of sample 1 x2σn-1 ............................ standard deviation of sample 2 xpσn-1 ......
1-2-16 Tests (TEST) u LinearReg t Test LinearReg t Test treats paired-variable data sets as (x, y) pairs, and uses the method of least squares to determine the most appropriate a, b coefficients of the data for the regression formula y = a + bx. It also determines the correlation coefficient and t value, and calculates the extent of the relationship between x and y.
1-2-17 Tests (TEST) 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 term b .................................. coefficient s .................................. standard error r .................................. correlation coefficient r2 .................................
1-2-18 Tests (TEST) k χ2 Test χ2 Test sets up a number of independent groups and tests hypothesis related to the proportion of the sample included in each group. The χ2 Test is applied to dichotomous variables (variable with two possible values, such as yes/no). k Expected counts Σ x ×Σ x ij Fij = i=1 ij j=1 k ΣΣ x ij i=1 j=1 (xij – Fij)2 Fij i =1 j =1 k χ2 = Σ Σ Perform the following key operations from the statistical data list. 3(TEST) d(χ2) Next, specify the matrix that contains the data.
1-2-19 Tests (TEST) After setting all the parameters, align the cursor with [Execute] and then press one of the function keys shown below to perform the calculation or draw the graph. • 1(CALC) ... Performs the calculation. • 6(DRAW) ... Draws the graph. Calculation Result Output Example χ2 ................................. χ2 value p .................................. p-value df ................................. degrees of freedom You can use the following graph analysis functions after drawing a graph.
1-2-20 Tests (TEST) k 2-Sample F Test 2-Sample F Test tests the hypothesis for the ratio of sample variances. The F Test is applied to the F distribution. F= x1σn–12 x2σn–12 Perform the following 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 specification. Data ............................ data type σ1 .................................
1-2-21 Tests (TEST) After setting all the parameters, align the cursor with [Execute] and then press one of the function keys shown below to perform the calculation or draw the graph. • 1(CALC) ... Performs the calculation. • 6(DRAW) ... Draws the graph. Calculation Result Output Example σ1Gσ2 .......................... direction of test F .................................. F value p .................................. p-value o1 .................................
1-2-22 Tests (TEST) k ANOVA ANOVA tests the hypothesis that the population means of the samples are equal when there are multiple samples. One-Way ANOVA is used when there is one independent variable and one dependent variable. Two-Way ANOVA is used when there are two independent variables and one dependent variable. Perform the following key operations from the statistical data list. 3(TEST) f(ANOVA) The following is the meaning of each item in the case of list data specification. How Many ..............
1-2-23 Tests (TEST) Calculation Result Output Example One-Way ANOVA Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value Line 2 (ERR) ............... Error df value, SS value, MS value Two-Way ANOVA Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value Line 2 (B) .................... Factor B df value, SS value, MS value, F value, p-value Line 3 (AB) ..................
1-2-24 Tests (TEST) k ANOVA (Two-Way) u Description The nearby table shows measurement results for a metal product produced by a heat treatment process based on two treatment levels: time (A) and temperature (B). The experiments were repeated twice each under identical conditions. B (Heat Treatment Temperature) A (Time) B1 B2 A1 113 , 116 139 , 132 A2 133 , 131 126 , 122 Perform analysis of variance on the following null hypothesis, using a significance level of 5%.
1-2-25 Tests (TEST) u Input Example u Results 20010101
1-3-1 Confidence Interval (INTR) 1-3 Confidence Interval (INTR) A confidence interval is a range (interval) that includes a statistical value, usually the population mean. A confidence interval that is too broad makes it difficult to get an idea of where the population value (true value) is located. A narrow confidence interval, on the other hand, limits the population value and makes it difficult to obtain reliable results. The most commonly used confidence levels are 95% and 99%.
1-3-2 Confidence Interval (INTR) u General Confidence Interval Precautions Inputting a value in the range of 0 < C-Level < 1 for the C-Level setting sets you value you input. Inputting a value in the range of 1 < C-Level < 100 sets a value equivalent to your input divided by 100. # Inputting a value of 100 or greater, or a negative value causes an error (Ma ERROR).
1-3-3 Confidence Interval (INTR) k Z Interval u 1-Sample Z Interval 1-Sample Z Interval calculates the confidence interval for an unknown population mean when the population standard deviation is known. The following is the confidence interval. Left = o – Z α σ 2 n Right = o + Z α σ 2 n However, α is the level of significance. The value 100 (1 – α) % is the confidence level. When the confidence level is 95%, for example, inputting 0.95 produces 1 – 0.95 = 0.05 = α.
1-3-4 Confidence Interval (INTR) After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation. • 1(CALC) ... Performs the calculation. Calculation Result Output Example Left .............................. interval lower limit (left edge) Right ............................ interval upper limit (right edge) o .................................. mean of sample xσn-1 .............................
1-3-5 Confidence Interval (INTR) The following shows the meaning of each item in the case of list data specification. Data ............................ data type C-Level ........................ confidence level (0 < C-Level < 1) σ1 ................................. population standard deviation of sample 1 (σ1 > 0) σ2 ................................. population standard deviation of sample 2 (σ2 > 0) List(1) ..........................
1-3-6 Confidence Interval (INTR) u 1-Prop Z Interval 1-Prop Z Interval uses the number of data to calculate the confidence interval for an unknown proportion of successes. The following is the confidence interval. The value 100 (1 – α) % is the confidence level. x Left = n – Z α 2 x Right = n + Z α 2 1 x x n n 1– n n : size of sample x : data 1 x x n n 1– n Perform the following key operations from the statistical data list. 4(INTR) b(Z) d(1-Prop) Data is specified using parameter specification.
1-3-7 Confidence Interval (INTR) u 2-Prop Z Interval 2-Prop Z Interval uses the number 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 value 100 (1 – α) % is the confidence level.
1-3-8 Confidence Interval (INTR) Left .............................. interval lower limit (left edge) Right ............................ interval upper limit (right edge) p̂1 ................................. estimated sample propotion for sample 1 p̂2 ................................. estimated sample propotion for sample 2 n1 ................................. size of sample 1 n2 .................................
1-3-9 Confidence Interval (INTR) o .................................. mean of sample xσn-1 ............................. sample standard deviation (xσn-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 below to perform the calculation. • 1(CALC) ... Performs the calculation. Calculation Result Output Example Left ..............................
1-3-10 Confidence Interval (INTR) The following confidence interval applies when pooling is not in effect. The value 100 (1 – α) % is the confidence level. Left = (o1 – o2)– tdf α 2 Right = (o1 – o2)+ tdf α 2 df = x1σ n–12 x2 σn–12 + n n1 2 x1σ n–12 x2 σn–12 + n n1 2 1 2 C 2 + (1–C) n1–1 n2–1 x1σ n–12 n1 C= x1σ n–12 x2 σn–12 n1 + n2 Perform the following key operations from the statistical data list.
1-3-11 Confidence Interval (INTR) o1 ................................. mean of sample 1 x1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1 n1 ................................. size (positive integer) of sample 1 o2 ................................. mean of sample 2 x2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2 n2 .................................
1-4-1 Distribution (DIST) 1-4 Distribution (DIST) There is a variety of different types of distribution, but the most well-known is “normal distribution,” which is essential for performing statistical calculations. Normal distribution is a symmetrical distribution centered on the greatest occurrences of mean data (highest frequency), with the frequency decreasing as you move away from the center.
1-4-2 Distribution (DIST) u Common Distribution Functions After drawing a graph, you can use the P-CAL function to calculate an estimated p-value for a particular x value. The following is the general procedure for using the P-CAL function. 1. After drawing a graph, press 1 (P-CAL) to display the x value input dialog box. 2. Input the value you want for x and then press w. • This causes the x and p values to appear at the bottom of the display, and moves the pointer to the corresponding point on the graph.
1-4-3 Distribution (DIST) k Normal Distribution u Normal Probability Density Normal probability density calculates the probability density of nomal distribution from a specified x value. Normal probability density is applied to standard normal distribution. f (x) = 1 e– 2πσ (x – µµ) 2 2σ 2 (σ > 0) Perform the following key operations from the statistical data list. 5(DIST) b(Norm) b(P.D) Data is specified using parameter specification. The following shows the meaning of each item. x ...............
1-4-4 Distribution (DIST) u Normal Distribution Probability Normal distribution probability calculates the probability of normal distribution data falling between two specific values. p= 1 2πσ ∫ b e a – (x – µ µ) 2σ 2 a : lower boundary b : upper boundary 2 dx Perform the following key operations from the statistical data list. 5(DIST) b(Norm) c(C.D) Data is specified using parameter specification. The following shows the meaning of each item. Lower ..........................
1-4-5 Distribution (DIST) Calculation Result Output Example p .................................. normal distribution probability z:Low ........................... z:Low value (converted to standardize z score for lower value) z:Up .............................
1-4-6 Distribution (DIST) After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation. • 1(CALC) ... Performs the calculation. Calculation Result Output Examples x .......................................
1-4-7 Distribution (DIST) k Student-t Distribution u Student-t Probability Density Student- t probability density calculates t probability density from a specified x value. x2 df + 1 1+ Γ 2 df f (x) = π df df Γ 2 – df+1 2 Perform the following key operations from the statistical data list. 5(DIST) c(T) b(P.D) Data is specified using parameter specification. The following shows the meaning of each item. x .................................. data df .................................
1-4-8 Distribution (DIST) u Student-t Distribution Probability Student- t distribution probability calculates the probability of t distribution data falling between two specific values. df + 1 2 p= df Γ 2 π df Γ ∫ b a x2 1+ df – df+1 2 dx a : lower boundary b : upper boundary Perform the following key operations from the statistical data list. 5(DIST) c(T) c(C.D) Data is specified using parameter specification. The following shows the meaning of each item. Lower ..........................
1-4-9 Distribution (DIST) Calculation Result Output Example p .................................. Student- t distribution probability t:Low ........................... t:Low value (input lower value) t:Up ............................. t:Up value (input upper value) k χ2 Distribution u χ2 Probability Density χ2 probability density calculates the probability density function for the χ2 distribution at a specified x value.
1-4-10 Distribution (DIST) Calculation Result Output Example p .................................. χ2 probability density when the [Stat Wind] setting is [Auto]. # Current V-Window settings are used for graph drawing when the SET UP screen's [Stat Wind] setting is [Manual]. The VWindow settings below are set automatically Xmin = 0, Xmax = 11.5, Xscale = 2, Ymin = -0.1, Ymax = 0.5, Yscale = 0.
1-4-11 Distribution (DIST) u χ2 Distribution Probability χ2 distribution probability calculates the probability of χ2 distribution data falling between two specific values. p= 1 df Γ 2 1 2 df 2 ∫ b df –1 – x2 e x 2 dx a : lower boundary b : upper boundary a Perform the following key operations from the statistical data list. 5(DIST) d(χ2) c(C.D) Data is specified using parameter specification. The following shows the meaning of each item. Lower ..........................
1-4-12 Distribution (DIST) Calculation Result Output Example p .................................. χ2 distribution probability k F Distribution u F Probability Density F probability density calculates the probability density function for the F distribution at a specified x value. n+d 2 f (x) = n d Γ Γ 2 2 Γ n d n 2 x n –1 2 1 + nx d – n+d 2 Perform the following key operations from the statistical data list. 5(DIST) e(F) b(P.D) Data is specified using parameter specification.
1-4-13 Distribution (DIST) Calculation Result Output Example p .................................. F probability density # V-Window settings for graph drawing are set automatically when the SET UP screen's [Stat Wind] setting is [Auto]. Current V- Window settings are used for graph drawing when the [Stat Wind] setting is [Manual].
1-4-14 Distribution (DIST) u F Distribution Probability F distribution probability calculates the probability of F distribution data falling between two specific values. n+d 2 p= n d Γ Γ 2 2 Γ n d n 2 ∫ b x n –1 2 a 1 + nx d – a : lower boundary b : upper boundary n+d 2 dx Perform the following key operations from the statistical data list. 5(DIST) e(F) c(C.D) Data is specified using parameter specification. The following shows the meaning of each item. Lower ..........................
1-4-15 Distribution (DIST) Calculation Result Output Example p ..................................
1-4-16 Distribution (DIST) k Binomial Distribution u Binomial Probability Binomial probability calculates a probability at a specified value for the discrete binomial distribution with the specified number of trials and probability of success on each trial. f (x) = n C x px (1–p) n – x (x = 0, 1, ·······, n) p : success probability (0 < p < 1) n : number of trials Perform the following key operations from the statistical data list. 5(DIST) f(Binmal) b(P.
1-4-17 Distribution (DIST) Calculation Result Output Example p .................................. binomial probability u Binomial Cumulative Density Binomial cumulative density calculates a cumulative probability at a specified value for the discrete binomial distribution with the specified number of trials and probability of success on each trial. Perform the following key operations from the statistical data list. 5 (DIST) f (Binmal) c (C.
1-4-18 Distribution (DIST) After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation. • 1(CALC) ... Performs the calculation. Calculation Result Output Example p .........................................
1-4-19 Distribution (DIST) k Poisson Distribution u Poisson Probability Poisson probability calculates a probability at a specified value for the discrete Poisson distribution with the specified mean. f (x) = e– µµ x x! (x = 0, 1, 2, ···) µ : mean (µ > 0) Perform the following key operations from the statistical data list. 5(DIST) g(Poissn) b(P.D) The following shows the meaning of each item when data is specified using list specification. Data ............................ data type List ............
1-4-20 Distribution (DIST) u Poisson Cumulative Density Poisson cumulative density calculates a cumulative probability at specified value for the discrete Poisson distribution with the specified mean. Perform the following key operations from the statistical data list. 5(DIST) g(Poissn) c(C.D) The following shows the meaning of each item when data is specified using list specification. Data ............................ data type List ..............................
1-4-21 Distribution (DIST) k Geometric Distribution u Geometric Probability Geometric probability calculates the probability at a specified value, and the number of the trial on which the first success occurs, for the geometric distribution with a specified probability of success. f (x) = p(1– p) x – 1 (x = 1, 2, 3, ···) Perform the following key operations from the statistical data list. 5(DIST) h(Geo) b(P.D) The following shows the meaning of each item when data is specified using list specification.
1-4-22 Distribution (DIST) u Geometric Cumulative Density Geometric cumulative density calculates a cumulative probability at specified value, the number of the trial on which the first success occurs, for the discrete geometric distribution with the specified probability of success. Perform the following key operations from the statistical data list. 5(DIST) h(Geo) c(C.D) The following shows the meaning of each item when data is specified using list specification. Data ............................
