GC Skills for H2 Statistics

7/27/2019 GC Skills for H2 Statistics

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TI Graphic Calculator Skills for H1/H2 Statistics

Section 1: Getting Started

1.1 Storing Data in a List

1.2 Creating a New List1.3 Editing a List1.4 Clearing a List1.5 Removing a List Name and Data1.6 Making Calculations on a List of Data1.7 Sorting Data in a List1.8 Defining and Displaying a Histogram

Section 2 : Probability Distributions

2.1 The Binomial Distribution

2.2 The Poisson Distribution2.3 The Normal Distribution

Section 3: Hypothesis Testing

3.1 Making Calculations on Grouped Data3.2 Performing a One-samplez-test3.3 Performing a One-sample t-test

Section 4: Regression

4.1 Constructing a Scatter Diagram4.2 Constructing a Least Squares Regression Line4.3 Finding the Linear Product-moment Correlation Coefficient

References:

1 A-Level Statistics with TI-83 Plus by Ho FH, Ho KF & Neo CB2 Course notes of Teaching and Learning of A-Level Mathematics: A Graphical Calculator

Approach (Statistics) by Dr Rosalind Phang3 A Concise Course in Advanced Level Statistics by J Crawshaw & J Chambers4 TI-83 Plus Graphing Calculator Guidebook


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Section 1: Getting Started

You will make use of 9 features:

5{ and6

and9

L

r

,

q


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1.1 Storing Data in a List

The following steps store the data:

1, 2, 3, 4, 5 and 6 in the list L1, 10, 20, 30, 40, 50 and 60 in the list L2.1 Press to select 1:Edit. You will see a spreadsheet

with three columns L1, L2 and L3.

2 Place the cursor at the first row ofL1 and key in the data 1 to 6,

pressing after each number. Use the cursor control keys

|,},~, to navigate and correct any mistakes wherenecessary.

3 Do the same for the list L2.

1.2 Creating a New List

The following steps store the data 100, 200, 300, 400, 500 and 600 in a new list named MULTI.

1 Highlight L1 and pressy6 to insert an empty column

before L1.

2 Type MULTI as the name of the new list.Note: you can use up to 5 characters.

3 Press to move the cursor to the first row ofMULTI.

4 Key in the data 100 to 600, pressing after each number.


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1.3 Editing a List

To correct a particular value:1

2

Highlight the value to be corrected, e.g. 300.

Type in the correct value, e.g. 350, and press.

To delete a value:1

2

Highlight the value to be deleted, e.g. 400.

Press{. The value is removed and the rows below will moveup.

To insert a value above another:1 Highlight the position to insert the new value, e.g. at 600.

2

3

Pressy6 so that the existing value will move down and azero will appear at that position.

Type in the value to be inserted and press.

1.4 Clearing a List

To clear data from a list L1:1 Press and select 4: ClrList.

2 Pressy and select L1, followed by.Note: a single command ClrList will clear the data in all the lists.


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1.5 Removing a List Name and Data

[Method 1] To remove a list from the spreadsheet:1 Highlight the name of the list to be removed, e.g. MULTI.

2 Press{. The list is removed from the spreadsheet but not thecalculators memory.

[Method 2] To remove a list from the calculators memory:1 PressyL and select 2: Mem Mgmt / Del.

2 To retrieve the required list, select 4: List.

3 Select the list to be deleted and press to remove it.

4 Pressy5 to return to the Home Screen.

1.6 Making Calculations from a List of Data

Store the following data in a list named MARKS:

84 17 38 45 47 53 76 54 75 2266 65 55 54 51 44 39 19 54 72

1 Create a list named MARKS and store the 20 data items into it.

2 Press and select 1: 1-Var Stats from the CALC menu.


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3 Pressy9 and select MARKS from the NAMES menu.

4 Press to obtain the essential values associated with thedata in the MARKS list.

Mean 51.5x = Standard deviation 18.20576832x = Unbiased estimate of the population standard deviation

18.67872418xs =

Lower quartile 1 41.5Q = Median = 53.5 Upper quartile 3 65.5Q = Lowest value = 17 Highest value = 84 Number of data items n = 20

1.7 Sorting Data in a List

To sort a list of data in ascending/descending order:1 Press and select 2: SortA( or 3: SortD(.

2 Pressy9 and select MARKS from the NAMES menu.


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3 Press to sort the list. To view the sorted list, pressand select 1: Edit.

Sorting a list in ascending order will help you to draw a stem and leaf diagram or a stemploteasily.

Stem Leaf

123456

78

7 928 94 5 71 3 4 4 4 55 6

2 5 64

1.8 Defining and Displaying a Histogram

To obtain a histogram that represents the data in MARKS:1 Pressy, and select 1.

2 Select On and choose (histogram). Move the cursor to Xlist,

pressy9 and select MARKS from the NAMES menu.


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3 Pressq and select 9 ZoomStat.Note:ZoomStat defines the viewing window so that all statisticaldata points are displayed.

4 Pressr to display the histogram.Note:

The first bar shows the interval 17 30.4, having afrequency of 3 (n = 3), i.e. 17, 19, 22.

The second bar shows the interval 30.4 43.8, having afrequency of 2 (n = 2), i.e. 38, 39.

Exercise: The lifetimes (in years) of 30 batteries of a particular brand are given as follow.

5.1 7.3 6.9 4.7 4.66.2 6.4 5.5 4.9 6.96.0 4.8 4.1 5.3 8.16.3 7.5 5.0 5.7 9.33.3 3.1 4.3 5.9 6.65.8 5.0 6.1 4.6 5.7

(a) Enter data into a list named LIVES.

(b) Write down all the values of the measures that you get from the 1-Var Stats option.

(c) Construct a histogram plot. Trace and understand the values displayed on the plot.

(d) Construct a stem and leaf plot.


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Section 2: Probability Distributions

You will make use of 7 features.

and9

r

,

=

s

p


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2.1 The Binomial Distribution

( )~ B ,X n p

p is the probability of a successful outcomeXis the number of successful outcomes in n independent trials

( )P for 0,1, 2, ..., , where 1n x n xxX x C p q x n q p= = = =

( ) ( )E and VarX np X npq= =

You will use the graphical calculator to calculate Binomial probabilities for this example:

Given that ( )~ B 16,0 .4X , find (a) ( )P 2X= , (b) ( )P 5X , (c) ( )P 7X> , (d) ( )P 6 10X< .

(a) To find ( )P 2X= :

1 Pressy= and select 0: binompdf(.

2 The function binompdf(n, p, x) calculates ( )P X x= . Enter the

values ofn,p andx using the keystrokes

.

( )P 2 0.0150 (to 3 s. f.)X = =

(b) To find ( )P 5X :

1 Pressy= and select A: binomcdf(.

2 The function binomcdf(n, p, x) calculates ( )P X x . Enter the

values ofn,p andx using the keystrokes.

( )P 5 0.329 (to 3 s. f.)X =


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(c) To find ( ) ( )P 7 1 P 7X X> = :

1 Press and select A: binomcdf(.

2 Enter the values ofn,p andx.

( )P 7 0.284 (to 3 s. f.)X > =

(d) To find ( ) ( ) ( )P 6 10 P 10 P 6X X X< = [Method 1]:

1 Select the function binomcdf( and enter the values ofn,p andxaccordingly.

( )P 6 10 0.454 (to 3 s. f.)X < =

(d) To find ( ) ( ) ( ) ( ) ( )P 6 10 P 7 P 8 P 9 P 10X X X X X< = = + = + = + = [Method 2]:

1 Pressy9 and select 5: sum( from the MATH menu.

2 Pressy= and select 0: binompdf(.

3 Enter the values ofn,p andx accordingly.Note: In the function binompdf (n, p, x),x can be a list of numbers.The function sum adds up the values computed by binompdf.

You will graph the Binomial distribution for the same example.

1 Create a list L1 to contain the values 0, 1, , 16. Pressy9and select 5: seq( under the OPS menu.


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2 Key in the values for seq, then pressy9 and selectL1.Note: The function seq(X, X, 0, 16, 1) returns a list created byevaluatingXwith respect toX, from 0 to 16 by 1.

3 Create a list L2 to contain the Binomial probabilities. Pressy

= and select 0: binompdf(.

4 Key in the values ofn andp, then pressy9 andselect L2.

Note: Whenx is not specified, all values of ( )P X x= , where

0,1, 2, ..., ,x n= will be displayed in a list.

5 To graph the Binomial distribution, pressy,.

6 Select On under Plot 1. Press and select (histogram).

7 Press to Xlist. Pressy9 to select L1 i.e.x. Press to

Freq. Pressy9 to select L2 i.e. ( )P X x= .

8 Pressp and enter the following values.Note: These settings will allow all 17x-values and theirprobabilities to be displayed within the screen area. Ymax hasbeen set to .2 since the highest probability is 0.19833 if you scanthrough L2.

9 Presss to display the histogram. Pressr to see theindividual probabilities.Note: The second screen shows the cursor positioned at the bar

( )4 and P 4 0.101420x X= = = .


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2 A biased coin is tossed four times and the number of heads noted. The experiment isperformed 500 times in all and the results are summarised in the table:

Number of heads 0 1 2 3 4

Frequency 12 50 151 200 87

Illustrate, with appropriate calculations and use of the graphical calculator, how this set ofdata compares with a binomial distribution with suitable n andp values.


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2.2 The Poisson Distribution

( )~ PoX

Xis the number of occurrences of an event in a given interval of time or space, when the meannumber of occurrences in the given interval is .

( )P for 0,1, 2, 3, ..., to infinity!

x

X x e xx

= = =

( ) ( )E and VarX X = =

You will use the graphical calculator to calculate Poisson probabilities for this example:

Given that ( )~ Po 7X , find (a) ( )P 2X= , (b) ( )P 4X , (c) ( )P 7X> , (d) ( )P 5 11X< .

(a) To find ( )P 2X= :

1 Pressy= and select B: poissonpdf(.

2 The function poissonpdf(, x) calculates ( )P X x= . Enter the

values ofandx using the keystrokes.

( )P 2 0.0223 (to 3 s. f.)X = =


1 Pressy= and select C: poissoncdf(.

2 The function poissoncdf(, x) calculates ( )P X x . Enter the

values of andx using the keystrokes.

( )P 4 0.173 (to 3 s. f.)X =


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(c) To find ( ) ( )P 7 1 P 7X X> = :

1 Press and select C: poissoncdf(.

2 Enter the values of andx.

( )P 7 0.401 (to 3 s. f.)X > =

(d) To find ( ) ( ) ( )P 5 11 P 11 P 5X X X< = [Method 1]:

1 Select the function poissoncdf( and enter the values of andxaccordingly.

( )P 5 11 0.646 (to 3 s. f.)X < =

(d) To find ( ) ( ) ( ) ( ) ( )P 5 11 P 6 P 7 P 8 P 9X X X X X< = = + = + = + =

( ) ( )P 10 P 11X X+ = + = [Method 2]:

1 Pressy9 and select 5: sum( from the MATH menu.

2 Pressy= and select B: poissonpdf(.

3 Enter the values of andx accordingly.Note: In the function poissonpdf (, x),x can be a list ofnumbers. The function sum adds up the values computed bypoissonpdf.

You will now graph the Poisson distribution for the same example.1 Create a list L1 to contain the values 0, 1, , 14. Pressy9

and select 5: seq( under the OPS menu.

Note: 14 is chosen because ( )P 14 0.994X =


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2 Key in the values for seq, then pressy9 and selectL1.Note: The function seq(X, X, 0, 14, 1) returns a list created byevaluatingXwith respect toX, from 0 to 14 by 1.

3 Create a list L2 to contain the Poisson probabilities. Pressy

= and select B: poissonpdf(.

4 Key in the value of .Pressy9 and select L1, then press

y9 and select L2.

Note: Whenx is specified as a list, all values of ( )P X x= ,wherex takes the values in the list will be displayed in a list.

5 To graph the Poisson distribution, pressy,.

6 Select On under Plot 1. Press and select (histogram).

7 Press to Xlist. Pressy9 to select L1 i.e.x. Press to

Freq. Pressy9 to select L2 i.e. ( )P X x= .

8 Pressp and enter the following values.Note: These settings will allow all 15x-values and theirprobabilities to be displayed within the screen area. Ymax hasbeen set to .15 since the highest probability is 0.149 if you scanthrough L2.

9 Presss to display the histogram. Pressr to see theindividual probabilities.Note: The second screen shows the cursor positioned at the bar

( )4 and P 4 0.091226x X= = =

.


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Exercises:

1 Given that the random variable ( )~ Po 1.6X :

(a) find the following probabilities, correct to 3 significant figures;

( )P 5X= ( )P 6X ( )P 3X ( )P 2 9X< <

(b) graph the probability histogram forXover a suitable range;

(c) graph the cumulative probability histogram forXover the same range, and use it to find

the smallest integer n such that ( )P 0.01X n> < .[Hint: generate a new list containing cumulative probabilities]

2 Does this set of data fit a Poisson distribution with a suitable mean?

x 0 1 2 3 4 5

f 110 50 20 12 7 1


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2.3 The Normal Distribution

( )2~ N ,X

( ) ( ) 2E and VarX X = =

You will use the graphical calculator to calculate Normal probabilities for this example:

Given that ( )2~ N 8, 2.5X , find (a) ( )P 0 4X , (b) ( )P 6X , (c) ( )P 9.2X> .

(a) To find ( )P 0 4X :

1 Pressy= and select 2: normalcdf(.

2 The function normalcdf(a, b, , ) calculates ( )P a X b < .

Enter the values ofa, b,and using the keystrokes

.

( )P 0 4 0.0541 (to 3 s. f.)X =



2 Enter the values ofa, b,and using the keystrokesy

D.

( )P 6 0.212 (to 3 s. f.)X =

Note: E99 represents the large negative number .


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(c) To find ( )P 9.2X> :


2 Enter the values ofa, b,and using the keystrokes

yD.

( )P 9.2 0.316 (to 3 s. f.)X > =

Note: E99 represents the large positive number + .

You can display ( )P 0 4X graphically, by following these steps:

1 Pressy= and select 1: ShadeNorm( under the DRAWmenu.

2 Note: The function ShadeNorm(a, b, , ) draws the normal

density function specified by and and shades the area

between a and b.

Enter the values ofa, b,and using the keystrokes.

3 Pressp and enter the following values.

4 Pressy5 to return to the Home Screen with the function

ShadeNorm and press.


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Another way to graph a normal distribution is to use theo feature. To compute the area underthis graph, you can apply the calculators integration function. Follow these steps:1 Pressoy= and select 1: normalpdf(.

2 Presss.

3 To compute the probability, pressy/ and select 7:

( )f x dx .

4 Enter the lower and upper limits accordingly.


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(c) To find ( )P 50 0.4X x< = :

1 Note: ( ) ( ) ( )P 50 0.4 P P 50 0.4X x X x X< = =

( ) ( )P 0.4 P 50X x X = +

To find the value of the LHS, press theny= and

select 2: normalcdf(.

Enter the values ofa, b, and by pressingyD

.

( )0.4 P 50 0.7085X + =

2 Pressy= and select 3: invNorm(.

Enter the values of area, and by pressing

.

( )P 0.7085 64.6X x x = = (to 3 s. f.)

Exercise: The masses of lettuces sold at a market are normally distributed with mean mass 600 gand standard deviation 20 g.

(a) Find the mass exceeded by 7% of the lettuces.

(b) Estimate the interquartile range of the distribution.


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Section 3: Hypothesis Testing


and

9

and

=


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3.1 Making Calculations on Grouped Data

You will use the graphical calculator to compute statistical quantities for grouped data. Considerthe following example:

A sample of 100 apples is taken from a load. The apples have the following distribution of sizes.

Diameter tonearest cm 6 7 8 9 10

Frequency 11 21 38 17 13

Find the mean and standard deviation of these diameters.

1 Store the diameters in the list L1 and corresponding frequenciesin the list L2.

2 Press and select 1:1-Var Stats from the CALC menu.

3 Select the list L1 followed by the list L2 by pressingyL1yL2.

Mean 8x= =

Standard Deviation 1.16= (to 3 s. f.)


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3.2 Performing a One-sample z-test

Based on the data given in the previous example, test the claim that the mean diameter of applesis not equal to 7.8 cm at the 5% level of significance. Assume that 1.15 = .

1 Press and select 1:Z-Test from the TESTS menu.

2 Select Stats, enter 7.8 for 0 and 1.15 for .

3 Highlight 0 for the alternative hypothesis and select

Calculate. Press.

Since 0.0820 0.05p = > , we do not reject 0H and conclude that

there is insufficient evidence to claim that the mean diameter ofapples is not equal to 7.8 cm at the 5% level of significance.

Note: Repeat step 1 and select Draw to obtain a standard normalcurve with the shaded areas equal thep-value.

3.3 Performing a One-sample t-test

Consider this example:

The diameters of spherical ball bearings produced by a particular machine follow a normaldistribution. Seven ball bearings are randomly selected and their diameters, in cm, are measured.The readings are: 2.12, 2.10, 1.98, 2.08, 2.04, 1.97, 2.15.

Test, at the 4% level of significance, the hypothesis that the average diameter of the ball bearingsis 2.00 cm.


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1 Store the readings in the list L1.

2 Press and select 2:T-Test from the TESTS menu.

3 Select Data and enter 2 for 0 .

4 Highlight 0 for the alternative hypothesis and select

Calculate. Press.

Since 0.0525 0.04p = > , we do not reject 0H and conclude that

there is sufficient evidence to claim that the average diameter ofball bearings is equal to 2.00 cm at the 4% level of significance.

Note: Repeat step 1 and select Draw to obtain a standard normal

curve with the shaded areas equal thep-value.

Exercise: A company claims that batteries produced by it have an average lifetime of 100 hrs.The lifetimes of the batteries are normally distributed and have a standard deviation of 1.2 hrs. Arandom sample of 10 batteries was tested and the lifetimes were recorded below:

98.2 101 98.7 99 101.1 100.5 98 98.5 99.4 100

(a) Test whether the company has overstated the average lifetime of its batteries at 5% level ofsignificance.

(b) Would the conclusion in (a) change, if the standard deviation was not known?


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Section 4: Regression


,

and9


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4.1 Constructing a Scatter Diagram

You will use the Graphical Calculator to store data and present it on a scatter diagram. Considerthis example:

The table shows the turnover and profit of an engineering firm from 1984 to 1989.

Year 1984 1985 1986 1987 1988 1989

Turnover, $x(in hundreds)

70 80 85 78 92 98

Profit, $y (inhundreds)

6 5 5.25 5.04 6.44 8.24

Draw scatter diagram to represent this information.

1 Create two lists X and Y and enter data accordingly.

2 Pressy, and select 1: Plot 1.

3 Select On. Select".

At Xlist, pressy9 and choose the list variable X from theNAMES menu.

At Ylist, pressy9 and choose the list variable Y from theNAMES menu.

At Mark, pick the first style of points to be plotted.

4 Pressq and select 9: ZoomStat.


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This screenshot shows that there are six points being plotted.

5 To read the coordinates of each point, pressr and use thecursor control keys to move from point to point.

4.2 Constructing a Least Squares Regression Line

Based on the same example, you will draw a least squares regression line of profit on turnover.

Note: At this point, it is not a concern whether the line drawn is a good fit.

1 Press and select 4: LinReg(ax + b) from the CALC menu.

2 Pressy9 to retrieve the list variable X followed by Y,separated by a comma.

3 Type another comma. Press and select 1: Function from theY-VARS menu.

4 Select Y1. This is to store the equation of the regression line in

Y1. You can presso to view the equation of the regression line.


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5 Press to obtain the values of the gradient and they-intercept of the regression line.

6 To view the regression line, presss orq.

7 Pressr and use the cursor control keys to move the cursoralong the regression line.

To find the profit when the turnover is 95, type 95 and press

. The value 6.9580839 will appear on the screen indicatingthat the profit based on the regression line is about $696.

Note: In step 1, you can instead select 8: LinReg(a + bx). They-intercept and gradient of the regression line remain the same. Inother words, the equation of the regression line is unaffected.


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4.3 Finding the Linear Product-moment Correlation Coefficient

The product-moment correlation coefficient, r, is a numerical value between 1 and 1 inclusive

which indicates the linear degree of scatter. The nearer the value ofr to 1 or 1, the closer thepoints on the scatter diagram are to the regression line.

Find rfor the same example.

1 PressyND and select DiagnosticOn.

2 On the Home Screen, repeat steps 1 and 2 on page 30 tocompute the gradient andy-intercept of the regression line,

together with the calculation ofr.

0.701(to 3 s. f.)r =

0.701r= indicates that there is some positive linear correlationbetweenXand Ybut it does not appear to be very strong.

Note:DiagnosticOn will remain activated until you deactivate it.

To deactivate, pressyND and select DiagnosticOff.

Exercise: A list of 100 words with varying degrees of difficulty was given to eight children. Theage of each children and the number of words he/she managed to define correctly were recordedin the table below.

Child A B C D E F G H

Age,x (in years) 2.5 3.1 4.3 5.0 5.9 7.1 8.1 9.4

Number ofwords,y

9 15 26 35 43 57 69 88

(a) Tabulate the values of lnx and lny . Plot a scatter diagram of this data.

(b) Calculate the linear product-moment correlation coefficient between lny and lnx .

Comment on the suitability of the model ln lny a b x= + .


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(c) Find the equation of the least squares regression line of lny on lnx .

(d) How many correct words can a child of age 6.5 define?

GC Skills for H2 Statistics

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