USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR
USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective: 1. Improve calculator skills needed in a multiple choice statistical examination where the exam allows the student to use a scientific calculator. 2. Helps students identify where and what their problem may be. Target group: This worksheet is for students who do not have a sound mathematical background and are doing a statistics module; students who did maths literacy at school; students who have a supplementary for a statistics exam. FUNCTION CASIO fx-991ES PLUS MY NOTES FROM QL WORKSHOP SHARP EL-532WH (normal mode) Proper fractions 1 1 4 Improper fractions 10 3 10 4 =1 4 3=3 1 1 3 ab 10 4=1 4 c ab 3=3 c 1 3 1 (the answer is 3 ) 3 Mixed fractions 4 1 3 4 1 3=4 1 3 4 ab c 2ndF SHIFT SD = 13 Converting between fractions and decimals 1 ab ab c c 3=4 1 3 answer 13 3 3 Follows from above Follows from above 13 3 SD answer 4.333 13 3 a b c 4.333 SD answer 4 1 3 4.3333 a b answer 4.3333 c answer 4 1 3 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 1 4 without the use of a calculator? Did you know it is much quicker to do 12 Having difficulty simplifying it mentally than reach for a calculator?! Try doing a worksheet or QL workshop on “FRACTIONS” n! n factorial 5 SHIFT n! n n 1 2 1 5! 5 4 3 2 1 The factorial notation is used to represent the product of the first n natural numbers, where n x 1 5 2ndF = 120 The button with orange SHIFT above it engages the orange colour stats functions above each numerical pad button. 4 =120 The orange 2ndF button engages the orange colour stats functions in the left hand corner of each numerical pad button. 0 Fractions using n! 5! 3! Fractions with two factorials in the denominator 10! 7!3! 5 SHIFT x = 20 1 10 SHIFT x x 1 120 3 SHIFT x 1 1 ( 7 SHIFT 3 SHIFT x 1 ) = Brackets NB! Why? 5 2ndF = 20 4 10 2ndF 4 4 3 2ndF ( 7 2ndF 3 2ndF 4 ) = 120 Brackets NB! Why? Permutations nPr 9 SHIFT (no repeats; order is important) Before pushing equal the screen will have 9P6 . After pushing equal the screen will have 9P6 . Else Else nPr 9P6 n! (n r )! 9! (9 6)! 6=60480 9 SHIFT x SHIFT x 1 1 ( 9–6 ) = 60480 9 2ndF 6 6=60480 9 2ndF 4 2ndF 4 ( 9–6 ) = 60480 Combinations nCr 9 SHIFT (no repeats; order is NOT important) Before pushing equal the screen will have 9C6 . After pushing equal the screen will have 9C6 . Else Else 6=84 4 9 2ndF 5 6=84 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 2 nCr n! r !(n r )! 9 SHIFT x x 9C 6 9! 6!(9 6)! 1 1 ( 6 SHIFT ( 9 – 6 ) SHIFT x 1 9 2ndF 4 ( 6 2ndF 4 ( 9 – 6 ) 2ndF 4 ) = 84 ) = 84 Both sets of brackets NB! Why? Both sets of Brackets NB! Why? 5 x 6 ) = 15625 5 y x 6 = 15625 Counting r objects in n different ways n r (repeats/repetition) 56 See diagram 1 for determining the differences between PERMUTATIONS & COMBINATIONS. (Appendix A attached at the end of this QL worksheet) Exponents in a bracket 0.75(9 0.75(9 0.75 x 9 – 6 ) = 0.422 6) 6) 2 The calculator already opens with a bracket, it just needs to be closed. Why and when? You get the answer without closing the bracket, BUT what would happen if you multiplied the expression by 2? 0.75 x 9 – 6 ) 0.75 y x ( 9 – 6 ) = 0.422 Multiply the expression by 2 before pressing the equal sign, 0.75 y x ( 9 – 6 ) 2 =0.844 Brackets NB! Why? 2 = 0.844 Now closing the bracket is important. Why? Powers with base e ex SHIFT ln 1 3 ) = 1.396 2ndF ln 1 ab c 3 = 1.396 1 e3 Alternatively, Alternatively, Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 3 SHIFT ln 1 3 e 3 ) = 1.396 SHIFT ln ( ) 3 ) = 0.05 2ndF ln ( 1 2ndF ln 3 ) = 1.396 3 = 0.05 The calculator automatically puts brackets around the first term. Why is there no need to put brackets in this situation? 1 e3 1 SHIFT ln 3 ) = 0.05 Alternatively, 1 1 SHIFT ln 3 ) = 0.05 But: e 2 3 2ndF ln ( 3 ) = 0.05 Alternatively, 1 e3 Note: 4e 1 1 3 ( ) 4 ab c 2ndF ln ( 3 ) = 0.05 e 3 . Why? 1 e3 Note: e 3. SHIFT ln ( ) 2 But: e 1 3 e 3 . Why? e 3. 3 ) = -2.054 4 Alternatively replace with . 2ndF ln ( 2 3 ) = -2.054 Alternatively replace . ab with c Discrete probabilty distributions make use of the universal constant e . Try a worksheet or QL workshop on “DISCRETE PROBABILITY DISTRIBUTION FUNCTIONS” Square root x When to use brackets 6 6 ) = 2.45 The calculator automatically ( 6 ) = 2.45 Brackets are not required Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 4 puts brackets around the first term. when there is one term. 6 = 2.45 6 = 2.45 Why? When there is one term brackets are not necessary. Why? Two terms under the square root sign 64 20 6 4 – 2 0 ) = 6.63 ( 6 4 – 2 0 ) = 6.63 Leaving out the closing brackets gives the same answer. Can you see why brackets are important in this situation? Why the emphasis on brackets????? 64 20 2 64–20 ) ( 64–20 ) 2 = 3.32 The bracket must be closed. Why? Alternatively replace with 2 = 3.32 The brackets tell the square root sign what needs to be beneath the square root. . Square root as a denominator 1 6 ) = 0.41 Similar to 1 6 6 above. 1 Similar to is in the numerator 1 ( 0.82 6 ) 6 above. Brackets for Denominator Brackets for Denominator Square root as a fraction in the denominator, where ( 6 ) = 0.41 2 ) = The calculator automatically 1 ( 0.82 ( 6 ) 2 ) = The outer red bolded set of brackets (arrows above) is Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 5 of the fraction 1 6 2 opens the square root sign with a bracket. Now the bracket must be closed. Why? The denominator also requires brackets. Why? 1 ( 3.5 0.9035 is in the denominator of the Alternatively , 1 3.5 10 is not (see 1 3.5 0.9035 6 above). Why? Brackets for Denominator Brackets for Denominator Square root as a fraction in the denominator, where fraction necessary. (a must!) The inner 10 ) ) = 1 ( 3.5 0.9035 ( 10 ) ) = Alternatively, 1 10) = 3.5 ab c 10 = 0.9035 Using the function Using the function means that there is no need for the fraction to go into brackets. ab c means that there is no need for the fraction to go into brackets. Yes you can use but lower a b for both c one requires brackets. Why? Denominator having a square root with two terms 6 24 12 6 ( 2 4 – 12 ) = 1.73 Alternatively replace with 6 ( 2 4 – 1 2 ) = 1.73 Alternatively replace ab . with c . The calculator always opens the square root sign. In this situation it is not essential to close bracket but it creates a habit. ( 7.3 – 3.5 ) ( 3.5 Brackets required above 10 ) ) = 3.433 and below Get into the habit of putting 7.3 3.5 brackets at the top of a 3.5 division line and at the 10 bottom. Why? It is essential to open with a bracket under the square root sign. In this situation it is not essential to close bracket but it creates a habit. ( 7.3 – 3.5 ) 10 ) = 3.433 ( 3.5 Get into the habit of putting brackets at the top of a division line and at the bottom. Why? Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 6 7.3 3.5 3.5 10 7.3 3.5 3.5 10 The calculator opened the bracket beneath the square root sign, hence the double closed bracket before the equal sign. The bottom brackets can be removed ONLY if the fraction function is used to replace the second division sign. Try this! brackets are not required beneath the square root sign because there is only one term. The bottom brackets can be removed ONLY if the fraction function ab c to replace the second division sign. Try this! Having difficulty answering all or some of the ‘why?’ above ….. try a worksheet or QL workshop on “ORDER OF OPERATION (BODMAS)” percentage sign % converting from 67% to a probability 6 7 SHIFT ( = 0.67 6 7 2ndF 1 = ERROR 1 Begin by doing 6 7 1 = 67 now 2ndF 1 = 0.67 Why does this work? The calculator requires an ANS then the 2ndF 1 = 0.67 Having difficulty doing the above without a calculator, then try a worksheet or QL workshop on “PERCENTAGES” The next sections deal with using the MODE function on a scientific calculator. ONE variable of data MODE MODE x Choose option 3:STAT by pushing the keypad for 3 Choose option Choose option 1: 1-VAR by pushing the keypad for 1 ENTER Data for x : STAT by 1 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 7 16; 15; 24; 7; 9; 21; 34 pushing the keypad for 1 Enter the data as follows: 16 = 15= 24= 7= 9= 21= 34= using the toggle button REPLAY Choose option SD by 0 pushing the keypad for 0 Notice the following on the screen Stat 0 Enter the data as follows: move the cursor up to the last value, i.e. 34 on the screen then push AC 16 M+ 15 M+ 24 M+ 7 M+ 9 M+ 21 M+ 34 M+ Notice the screen says DATA SET 7 There are 7 values in the dataset. Use either the teal ALPHA button or the RCL button to recall the information SUM of x SHIFT 1 3 STAT 3:SUM 2 2: x 126 x ALPHA 126 x2 x 2 SHIFT 1 3 STAT 3:SUM 1 x2 1: 2784 Combining sum and variable functions functions x n ALPHA 2784 The ALPHA or the RCL button engages the teal colour stats functions in the right hand corner of each numerical pad button. SHIFT 1 3 STAT 3:SUM SHIFT 1 4 2 2: 1 x ALPHA 18 x n ALPHA 0 STAT 4:VAR 1:n 18 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 8 x x 2 2 Before beginning such a problem, insert the brackets n n 1 x 2 x 2 Before beginning such a problem, insert the brackets x n x 2 2 n n 1 n 1 Brackets inserted: underneath the square root sign; at the top (numerator) and the bottom (denominator) of a fraction. Brackets inserted: underneath the square root sign; at the top (numerator) and the bottom (denominator) of a fraction ALPHA SHIFT calculator opens this one 1 x 3 STAT 3:SUM 2 ALPHA 1 1: SHIFT 1 x 2 3 2 STAT 3:SUM 2: x x x2 ALPHA 0 n x2 SHIFT 1 STAT 4 1 SHIFT 4:VAR 1:n 1 4 1 n ALPHA 0 1 = 9.274 1 STAT 4:VAR 1:n = 9.274 See the importance of brackets? Brackets create steps. SIZE sample n SHIFT 1 4 1 STAT 4:VAR 1:n 7 Sample MEAN x SHIFT 1 4 2 See the importance of brackets? Brackets create steps. n ALPHA 0 7 x ALPHA 4 18 STAT 4:VAR 2:x 18 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 9 Sample STD DEVIATION sx SHIFT 1 4 STAT 4:VAR 4 4:sx 9.274 Sample VARIANCE 9.274 x ALPHA 6 is for the Option 3: x is for the POPULATION standard deviation. What is the difference between a sample and population standard deviation? POPULATION standard deviation. What is the difference between a sample and population standard deviation? SHIFT 1 ALPHA 5 x 2 4 STAT 4:VAR x2 s2 sx ALPHA 5 4 4:sx sx 86 86 Notice that when using the calculator, the sample standard deviaiton is obtained first, squaring the value gives the sample variance. standard deviation 2 variance Notice that when using the calculator, the sample standard deviaiton is obtained first, squaring the value gives the sample variance. standard deviation 2 variance It is one thing to learn to use a calculator, but a sound understanding of Descriptive Statistics is needed. Try do worksheets or QL workshops on the following: “DESCRIPTIVE STATISTICS” ONE variable of data using FREQUENCY option Example: pdf of a discrete random variable x 0 1 2 SHIFT MODE toggle down using REPLAY 4 4:STAT 3 Frequency ? 1 1:ON MODE MODE Choose option STAT by 1 pushing the keypad for 1 Choose option SD by 0 pushing the keypad for 0 P ( x) 0.25 0.4 0.2 0.15 3 1 3:STAT 1:1 VAR Enter the data as follows: Notice the following on the screen Stat 0 Enter the data as follows: 0 = 1= 2= 3= Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 10 Then toggle to the top 0 STO 0.25 M+ where x is 0 1 STO 0.4 M+ REPLAY then move to the freq column using the right arrow on the replay button. 2 STO 0.2 M+ 0.25 = 0.4 = 0.2 =0.15 =using the toggle button move the cursor up to the last line where x=3 and freq=0.15 on the screen then push AC DATA SET 4 3 STO 0.15 M+ Notice the screen says There are 4 values in the dataset. x in this situation is the x in this situation is the variables and y is the variables and freq is the probability of x , i.e. P x probability of x , i.e. P x . The expected value of SHIFT 1 4 2 x ALPHA 4 1.25 x ALPHA 6 0.9937 STAT 4:VAR 2:x x 1.25 The standard deviation of x SHIFT 1 4 3 STAT 4:VAR 3: x 0.9937 Notice that sx ALPHA 5 Notice that SHIFT 1 4 4 ERROR STAT 4:VAR 4:sx In this situation the POPULATION standard deviation is used. Why? ERROR 2 . In this situation the POPULATION standard deviation is used. Why? x ALPHA 6 x 2 The variance of x SHIFT 1 4 3 standard deviation 2 0.9875 variance STAT 4:VAR 3: x x 2 0.9875 standard deviation 2 variance Remove frequency from the screen: The ALPHA or the RCL button engages the teal colour stats functions in the right hand corner of each numerical pad button. Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 11 SHIFT MODE toggle down using REPLAY Frequency ? 2 4 4:STAT 2:OFF It is not sufficient to learn only the calculator version. Do a worksheet or QL workshop on “DISCRETE PROBABILITY DISTRIBUTION FUNCTIONS” TWO variables of data x; y MODE 3 For example: Enter the data as follows: Choose option x 2.6 2.6 3.2 3.0 2.4 3.7 3.7 2.6 = 2.6= 3.2= 3.0= 2.4= 3.7= 3.7= pushing the keypad for 1 y 5.6 5.1 5.4 5.0 4.0 5.0 5.2 3:STAT 2 2: A BX Then toggle to the top to the very first REPLAY x equal to 2.6 then move to the freq column. 5.6 = 5.1 = 5.4 = 5.0= 4.0 = 5.0= 5.2= using the toggle button move the cursor up to the last line where x=3.7 and y=5.2 on the screen then push AC Make sure that the variable x is entered underneath the x column and the y variable underneath the y column. MODE Choose option STAT by 1 LINE by 1 pushing the keypad for 1 Notice the following on the screen Stat 1 Enter the data as follows: Always the x variable first, then the y variable. NB!! 2.6 STO 5.6 M+ 2.6 STO 5.1 M+ 3.2 STO 5.4 M+ 3.0 STO 5.0 M+ 2.4 STO 4.0 M+ 3.7 STO 5.0 M+ 3.7 STO 5.2 M+ Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 12 Notice the screen says Notice that a bx is the mathematical equation for a DATA SET 7 straight line. There are 7 paired values in the dataset. SIZE sample n SHIFT 1 4 1 STAT 4:VAR 1:n n ALPHA 0 7 x ALPHA 4 3.029 y ALPHA 7 5.043 ALPHA 5 0.531 ALPHA 8 0.509 7 Sample MEAN x SHIFT 1 4 2 STAT 4:VAR 2:x 3.029 y SHIFT 1 4 5 STAT 4:VAR 5: y 5.043 sx Sample STD DEVIATION sx SHIFT 1 4 STAT 4:VAR 4 4:sx 0.531 sy sy SHIFT 1 4 7 STAT 4:VAR 7:sy The ALPHA or the RCL button engages the teal colour stats functions in the right hand corner of each numerical pad button. 0.509 Use either the teal ALPHA button or the RCL button to recall the information SUM of x SHIFT 1 3 STAT 3:SUM 21.2 2 2: x ALPHA x 21.2 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 13 SHIFT 1 y 3 STAT 3:SUM 4 4: y ALPHA 2 y 35.3 35.3 x2 x2 SHIFT 1 3 STAT 3:SUM 1 x2 1: ALPHA 65.9 65.9 y2 y2 SHIFT 1 3 STAT 3:SUM ALPHA 3 3 179.57 y2 3: 179.57 xy xy SHIFT 1 3 STAT 3:SUM 5 5: xy yˆ b0 b1 x slope intercept Subscript increases 0 to 1 yˆ Intercept: b0 a SHIFT 1 5 Intercept: b0 1 STAT 5:Re g 1: A a ALPHA ( 4.093 a 4.093 Slope: b1 On calculator 107.44 The ALPHA or the RCL button engages the teal colour stats functions in the right hand corner of each numerical pad button. 107.44 Regression coefficients ALPHA 1 Slope: b1 b b ALPHA ) 0.314 b a b x intercept slope SHIFT 1 5 STAT 5:Re g 1 2:B 0.314 Alphabet increases from a to b. Correlation coefficient: r Hence, SHIFT 1 Intercept b0 a 5 STAT 5:Re g 3 3:r Correlation coefficient: r r ALPHA 0.327 0.327 Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 14 Slope b1 b Coefficient of Coefficient of Determination: r SHIFT 1 2 5 STAT 5:Re g x 2 Determination: r 2 3 3:r r x2 ALPHA 0.107 0.107 Estimated value of y i.e. yˆ Estimated value of y i.e. yˆ when x 3.1 : SHIFT 1 5 SHIFT 1 STAT 5 2 2:B 3.1 : ALPHA ( b ALPHA ) 3.1 5.065 a 1 STAT 5:Re g 1: A 5:Re g when x 3.1 =5.065 Alternatively, Alternatively, 3.1 SHIFT 1 STAT 5 5:Re g 5 5.065 y' 3.1 2ndF ) 5.065 5: yˆ The above section requires a sound understanding of correlation and simple regression. Try a worksheet or a QL workshop on “CORRELATION AND SIMPLE REGRESSION”. Contact [email protected] or a Quantitative Literacy Facilitator in your region Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 15 Appendix A: Repetition (repeats) Yes nr Note that nCr combination No ORDER important Yes No Permutation ( nPr ) Combination ( nCr ) nPr . Why? n! r !(n r )! n! 1 permutation r !(n r )! r ! permutation r !combination combination So there will be r !permutations for every possible combination. Diagram 1: Counting: Permutations and Combinations Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 16
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