Calculator Section Rewrite

Overview is unchanged All the rest of topic including the worked solutions to be replaced by the following.

Using your calculator in statistics 1.

Some keys you need to be familiar with

All calculator steps shown here are for the Casio fx 82TL If your calculator is not a casio fx 82 TL, follow the same steps and where necessary the differences will be indicated.

AC

Turns the calculator on, then clears the screen.

(-)

This is the key you must use when you have a negative number to enter. On some calculators this key looks like +/. Example 1.1: -3 + 4 Calculator steps:

(-)

3 + 4 =

Or on some calculators 3

+ 4 =

+/-

Answer = 1 ( (

and

)

These are the brackets keys and are used in the same order as they appear in a sum. On some calculators they appear as &

[(..

..)]

This is the key that allows you to work with fractions.

a bc

Example 1.2: a.

1 3  5 7

Calculator steps:

1

a bc

5 + 3

a bc

7 =

The screen display will look like 1  5. The final answer will be shown as 22  35 which reads as

22 35

4 2  5 3

b.

The answer will show as 1  1  5 which reads as 1

1 5

Although fractions are often used in probability questions, most calculations in statistics are done as decimals. Your calculator will normally work in decimals unless the fraction key is used. To work in decimal form, instead of pressing the fraction key use the divide (  ) key instead. So 1

5 is the same as 1  5

a bc

This key will square a number

x2

Example 1.3: (this means 56  56)

56 2

Calculator steps:

56

x2

= Answer = 3136

Some older calculators have the x 2 key above the the steps are 56 shift x 2

key. On these calculators

This key will find the square root of a number

(

Example 1.4: 16 a. Calculator step:

(

16 = Answer = 4

Some older calculators require you to enter the number first. On these calculators the steps are b.

16

(

3.45

Answer = 1.857417562

xy

This key is the power key Example 1.5: (this means 8  8  8  8 )

84

Calculator steps:

8

4 =

xy

Answer = 4096 On some older calculators the function may be above a key so use 8 shift x y 4 = On sharp calculators the function is y x This is a key that you will use in the Poisson distribution formula

ex

Here the x can take on any nominated value. The x can be a positive number or a negative number. This function shares the key with the ln function. It is found on the top first, second or third row of the calculator. Example 1.6: a.

e3

Calculator steps:

shift

ex

3 = Answer = 20.08553692

Older calculators will use 3 shift e x b.

e 5

Calculator steps:

shift

ex

(-) 5 = Answer = 0.006737946

Or shift e x 5 +/- = Or 5 +/- shift e x

(for other newer VPAM casios and DAL sharps) (for older calculators)

Note: if you have just bought a new casio calculator and the answer for b shows as 6.737946999 03 then you need to change the default mode from exponent form to decimal number form. To do this press the mode key three times, then select 3, then select 2

ie mode mode mode 3 2 Your calculator will now give you all answers in the correct decimal format. x!

This function reads as x factorial. It is usually above another key. Example 1.7: 5!

(this means what is 5  4  3  2  1 )

Calculator steps:

5 shift

x!

= Answer = 120

This is a function that you will meet in the binomial distribution.

nCr

It allows you to calculate how many ways you can choose r items from a total of n possibilities. It is shown in your textbook not as nCr n but instead as C x or as  nx  Example 1.8: 5

C 2 or

C 25

Calculator steps:

(how many ways can you choose 2 items from a total of 5) 5

nCr

2 = Answer = 10

On some calculators the function may be above another key so use 5 shift nCr 2 =

2.

Performing calculations

Scientific calculators have in built orders of operations. This means that the calculator will automatically work out operations in questions in the following order: Brackets Powers and square roots Multiplications and divisions from left to right Additions and subtractions from left to right Sometimes you need to add a set of brackets so that the calculator is working out the sum in the desired order. Example 2.1: 2  3 4

calculator steps:

2  3 4 

(the calculator will automatically multiply before it adds) answer = 14 Example 2.2: ( 2  3)  4

calculator steps:

( 2 + 3)  4 = answer = 20

Example 2.3: 523 

27 2 9

calculator steps:

523 - 27

x2

 9 = answer = 442

Example 2.4: 5 3  e 5  6!

calculator steps:

5 x y 3  shift e x 5  6 shift x ! 

or on older calculators use 5 x y 3  5 shift e x  6 shift x !  answer = 13 357 184.32

The next four examples that have multiple steps on the top or the bottom of a fraction or have multiple steps under a square root sign need to have brackets added in order that the calculator can work out the sum in the correct order.

Example 2.5: 1 25  34

this must be worked out as

calculator steps:

1 (25  34)

1  ( 25  34 ) 

answer = 0.016949152 Example 2.6:

16.46  18.37 5.27

this must be worked out as

calculator steps:

(16.46  18.37) 5.27

( 16.47 - 18.37 )  5.27 

answer = -0.362428842 Example 2.7: 34  89

this must be worked out as

calculator steps:

(34  89)

( 34  89 ) 

for older calculators use

( 34  89 )

answer = 11.09053651 Example 2.8 (18.32  15.69)  6.2  this must be worked out as    19 

18.32  15.69 6.2 19

calculator steps:

(18.32  15.69)  (6.2 

for older calculators use:

19) 

(18.32  15.69)  (6.2  19

)

Answer = 1.84901681

Review exercise set 1: Using your calculator find the value for: 1) 482 + 753

2)

305  156

3)

16  4 8  12  5  62

4) Add together -5, -3, -1, 4, 6, 8 and 12 then square the total.

3. Statistics mode Your calculator has several modes. The casio fx 82TL has 3 (press the

MODE

button to see). To store a set of scores and calculate the mean and standard deviation and other values you need to put your calculator into statistics mode. Recent casio calculators can be put into statistics mode by pressing -

SD will show on the display

Recent sharp calculators can be put into statistics mode by pressing -

2

MODE

MODE

1

STAT will show on the display.

(for other types of calculators you will need to check with the manual) MODE 1 (casios) or MODE 0 (sharps) will take the calculator back into computational mode. Students can however leave their calculators in statistics mode as all of the functions and calculations, which we looked at in sections 1 and 2, can be handled by the calculator when it is in stats mode. Entering scores To enter the scores:

55, 54, 51, 55, 53, 53, 54, 52

1.

Ensure your calculator is in statistics mode.

2.

Clear the statistics memory by pressing

shift

AC

=

(For a casio fx 100s use shift AC, for a recent sharp use 2nd F CA)

3.

Enter the scores

55

M+

54

M+

51

M+

55

M+

53

M+

53

M+

54

M+

52

M+

To check the number of scores entered (n), look for n C Press: RCL

hyp

on casio 82TL

Press: RCL 3 on casio 100s Press: shift n on some calculators Answer = 8

To find the mean (  or x ) , look for x Press: shift x = casio fx 82 TL Press: shift x casio fx 100s Answer = 53.375

To find the standard deviation if the scores are an entire population ( ) , x n or  n (casios) or x (sharps) look for Press: shift x n = Press: shift x n

casio fx82 TL casio fx 100s Answer = 1.316956719

To find the standard deviation if the scores are a sample (s ), x n 1 or  n 1 (casios) or sx (sharps) look for Press: shift x n 1 = Press: shift x n 1

casio fx82 TL casio fx 100s Answer = 1.407885953

To find the variance of a sample ( s 2 ), Press: shift x n 1 = x 2

To add up the scores, look for x

Answer = 1.982142857

B Press: RCL o , , ,

on casio 82TL

Press: RCL 2 on casio 100s Press: shift x on some calculators Answer = 427 To add up the squares of the scores, look for x 2 A Press: RCL

(-)

on casio 82TL

Press: RCL 1 on casio 100s Press: shift x 2 on some calculators Answer = 22805

Notes: 1. Some sharp calculators have the statistics functions written in green or purple. On these calculators the green or purple RCL button is used instead of the 2nd F button to access these statistical functions. 2. Some calculators tally the scores as you are entering them. If after entering the scores you find that you have entered the incorrect number of scores then you need to clear the memory (shift AC) and re-enter the scores. If you feel you made an error when entering one or more scores, clear the memory and re-enter the scores. 3. The standard deviation for a population and a sample are similar numbers, but they are not the same. You need to decide first if the scores represent a sample or the entire population and then press the appropriate standard deviation key. 4. The variance may be given or asked for rather than the standard deviation. The variance is just the standard deviation squared – hence the symbols used are s (sample standard deviation) and s 2 (sample variance) or  (population standard deviation) and  2 (population variance). 5.

 x and  x

2

will be looked at further in the Algebra topic.

Review exercise set 2:

Enter the scores 12, 15, 21, 29, 35, 55 Find

(treat these scores as a sample)

1. the sample mean ( x )

2.

x

3. the sample standard deviation (s)

Other ways to enter scores a.

When scores are repeated

To enter 2, 3, 4, 4, 5, 6, 6, 6 You can use

2

M+

6

M+

3

M+ M+

4

M+

M+

5

M+

M+

here: x  4.5 and s = 1.511857892 b.

When scores are given in a table format – each score and its frequency (how many times the score occurs)

To enter Score 12 13 14 15 16 You can use the following method 12 13 14 15 16

shift shift shift shift shift

; ; ; ; ;

2 5 7 4 1

M+ M+ M+ M+ M+

Frequency 2 5 7 4 1

For the casio fx 100s and older casios use 12  2 M+ 13  5 M+ 14  7 M+ etc For newer sharp calculators use 12 2nd F , 2 M+ 13 2nd F , 5 M+ etc Here: x  13.84210526 and s = 1.067872126

4.

Using linear regression mode

Some calculators have linear regression mode. You will know if yours has by seeing if it has a row reading y , y n and y n 1 usually found above the number keys. Linear regression enables you to determine if some relationship exists between two sets of data. Your calculator can find the strength and direction of this relationship. To put your calculator into LR mode press the mode button of your calculator to find the reg option. For the casio fx- 82TL press mode 3 1 For the casio fx 100s press mode 3 For a recent sharp calculator press mode 2 Example 4.1: Stress test score x 53 94 64 73 82 90 Data is input using the



key

On some casios use the x d y d key On some sharps use the (x,y) key To enter the data Shift AC = 53 , 70 M+

Blood pressure y 70 91 78 78 85 84

94 64 73 82 90

, , , , ,

91 78 78 85 84

M+ M+ M+ M+ M+

Shift A gives the value of the y intercept of the regression line Answer = 47.604 Shift B gives the value of the slope of the regression line Answer = 0.4394 Shift r gives the correlation coefficient Answer = 0.95156 The means, standard deviations and sums of each set of scores can also be found. (After you finish your linear regression calculations put your calculator back into statistics mode or computational mode.) Review exercise set 3: 1.

Find x for the scores above

2.

Find

y

3.

Find

x

2

Worked solutions Review exercise set 1: 1.

48 x 2  75 x y 3 

2.

( 305  156 ) 

3.

16.4  ( 8.12 - 5.62 ) 

424179 12.20655

6.56

4. (-) 5  (-) 3  (-) 1  4  6  8  12  x 2

Review exercise set 2 1. 2. 3.

x  27.8333

 x  167

s  15.82929773

Review exercise set 3 1. 2. 3.

x  76

 y  486  x  35894 2

441