CASIO FX-CG50 CALCULATOR

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Solving Samples of Statistics Problems

Using

Casio Middle East - GAKUHAN

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This booklet aims to help you through the Statistical methods using Casio’s FX-CG50. As the FX-CG50 is a

powerful and rich tool all in one calculator. It will help you tremendously in performing a large number of

operations.

The booklet assumes some basic skills in working with the FX-CG50.

Please note that there may be other methods to attain the same results. The methods presented here are not

necessarily the finest or the simplest of the choices available.

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Basic Commands

Consider the data set: {15, 22, 32, 31, 52, 41, 11}

Entering Data:

Enter the data in Lists on the calculator.

Use your arrow keys to move between lists

Clearing Data:

To clear all data from a list: (use

to change options at the bottom of the screen)

To clear an individual entry: Select the value and press DEL.

To edit an individual entry: Select the value and press w Edit.

Sorting Data: (helpful when finding the mode)

Ascending order (lowest to highest) Or Descending order (highest to lowest).

Tools q then Ascending order q Or Descending w

One Variable Statistical Calculations:

For the previous information:

• Press u button, Then Choose w CALC . Select 1-Var Stats q.

• Use the down arrow N to view all the information.

p215l22l32l31l52l41l11l

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Mean, Mode, Median

Example: Given the data set {13, 3, 10, 9, 7, 10, 12, 8, 6, 3, 9, 6, 11, 5, 9, 13, 8, 7, 7}

find the mean, median and mode.

• Go to Statistics application p2 then enter the data into a list.

(See Basic Commands for entering data.)

• Clear old data and enter the new data into the lists u!rq

• Press uuwq 1-Var Stats.

• Arrow up and down the screen to see the statistical information about the

data.

Mean   

Median (Med) = 8

Mode = 7,9

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Create a histogram for previous data

• Go back and choose graph then set to select Histogram

ddquNuq

• Draw the graph dq1ll

Example: From a Frequency Table:

Number

Frequency

• Clear old data and enter the new data into the lists p2ddu!rq

• enter the data values in L1. enter their frequencies in L2.

• Draw the histogram. Press uuquNNNw2l to choose list 2 as frequency, then press

dql to see the graph.

• To see the statistics calculation, press q 1-Var.

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Box and Whisker Plots

Example: given the data set

{85, 100, 97, 84, 73, 89, 73, 65, 50, 83, 79, 92, 78, 10},

• Clear old data and enter the new data into the lists p2ddu!rq.

• Enter the data into the lists.

• Change the functions to see GRAPH by using u then quNuwNNqdq.

• Seeing the graph: Press Lq the TRACE key to see on-screen data about the box-and-whisker plot.

The box itself is defined

• by Q1, the median and Q3.

• The spider will jump from the minimum value to Q1, to median, to Q3 and to the maximum value.

Pi Chart

Example: suppose one of the questions asked on a survey was “What type of cars do you have?”, and the results

from 44 people are shown in this table. Construct a pie chart and a bar chart of these

data.

Car

Toyota

Lexus

Mercedes

BMW

Ferrari

Kia

GMC

Frequency

• Clear old data and enter the new data into the lists

p2ddu!rq

• .

• Enter the data into the lists.

• To draw the graph quNrlq

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Scatter Plots

A scatter plot is a graph used to determine whether there is a relationship between paired data.

In many real-life situations, scatter plots follow patterns that are approximately linear. If y tends to increase as x

increases, then the paired data are said to be a positive correlation. If y tends to decrease as x increases, the

paired data are said to be a negative correlation. If the points show no linear pattern, the paired data are said to

have relatively no correlation.

To set up a scatter plot:

• Clear old data and enter the new data into the lists p2ddu!rq

• Enter the X data values in L1. Enter the Y data values in L2, being careful that each X data value and its

matching Y data value are entered on the same horizontal line.

• Change the functions to see GRAPH by using u then

• Activate the scatter plot quNq.

• To see the scatter plot, dq

The linear based regression models on the graphing calculator:

• Linear (LinReg)

y = ax + b

The graph of x versus y is linear.

Fits Linear by Transformations:

• Logarithmic (LnReg)

y = a + b ln(x)

The graph of ln(x) versus y is linear. Calculates a and b using

linear least squares on lists of ln(x) and y instead of x and y.

• Exponential (ExpReg)

y = a (b

)

The graph of x versus ln(y) is linear.

Calculates A and B using linear least squares on lists of x and

ln(y) instead of x and y, and then

a = e

and b = e

• Power (PwrReg)

y = a ( x

)

The graph of ln(x) versus ln(y) is linear.

Calculates A and b using liner least squares on list of ln(x) and

ln(y) instead of x and y, and then

a = e

120

130

148

155

167

180

200

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Other models available on the graphing calculator:

• Quadratic (QuadReg)

For three points, fits a polynomial to the

data. For more than three points, fits a

polynomial regression.

• Cubic (CubicReg)

For four points, fits a polynomial to the data. For

more than four points, fits a polynomial

regression.

• Quartic (QuartReg)

For five points, fits a polynomial to the data. For

more than five points, fits a polynomial

regression.

• Logistic (Logistic)

Fits equation to data using iterative least-squares

fit.

• Sinusoidal (SinReg)

Fits sine wave to data using iterative least-squares

fit.

Example: determine a linear regression model equation to represent this data.

• Clear old data and enter the new data into the lists p2ddu!rq

• Choose Linear Regression Model from CALC uuweqw

• Create a scatter plot (GRAPH) of the data to graph the regression.

ddddquNqdq

• Draw the regression qwqu

Hours

Spent

Studying

Math

Score

390

580

650

730

410

530

600

790

350

400

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Exponential Regression Model Example

• Clear old data and enter the new data into the lists p2ddu!rq

• Create a scatter plot of the data uuquNqdq.

• Choose Exponential Regression quew

• Graph the Exponential Regression u

Logarithmic Regression Model Example

• Clear old data and enter the new data into the lists p2ddu!rq

• Create a scatter plot of the data uuquNqdq.

• Choose Logarithmic Regression quw

• Graph the Logarithmic Regression u

Time

(mins)

Temp

(F)

179

168

158

149

141

134

125

123

116

Age of

Tree

Height

9.5

16.5

17.5

18.5

19.5

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Quadratic Regression Model Example

• Clear old data and enter the new data into the lists. p2ddu!rq

• Create a scatter plot of the data uuquNqdq.

• Choose Quadratic Regression qr

• Graph the Quadratic Regression u

Sine Regression Model Example

Example: The table below shows the highest daily temperatures (in degrees Fahrenheit) averaged over the

month.

Angle

Distance

(feet

10°

115

15°

157

20°

189

24°

220

30°

253

34°

269

40°

284

45°

285

48°

277

50°

269

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Tokyo

Hiroshima

Nagasaki

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• Clear old data and enter the new data into the lists(list1, list2, list3, list4). p2ddu!rq

• Create a scatter plot of the data for all cities (list1 with list2 , list1 with list3, list1 with list4)

uuquNNNq ( 234 )d. To choose color lNNNNq

• Choose Sin Regression for each pair of lists qquy

• Draw the Sin Regression u (do these steps for all pairs of lists list1 with list2, list1 with list3, list1 with list4).

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Normal Probability Distribution

The Distribution functions:

1. pdf = Probability Density Function

This function returns the probability of a single value of the random variable x. Use this to graph a normal

curve. Using this function returns the y-coordinates of the normal curve.

normal pdf (x, mean, standard deviation)

2. cdf = Cumulative Distribution Function

This function returns the cumulative probability from zero up to some input value of the random

variable x. Technically, it returns the percentage of area under a continuous distribution curve from

negative infinity to the x. You can, however, set the lower bound.

normal cdf (lower bound, upper bound, mean, standard deviation)

3. inv = Inverse Normal Probability Distribution Function

This function returns the x-value given the probability region to the left of the x-value.

(0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value

at a given percent based upon the mean and standard deviation.

invNorm (probability, mean, standard deviation)

Example: calculate the normal probability density for a specific parameter value when x = 36, σ = 2 and μ = 35.

• Use the following steps p2ddyqqwN36l2l35ll

• To draw dNNNu

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Example: given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find:

a) the probability that a value is between 65 and 80, inclusive.

b) the probability that a value is greater than or equal to 75.

c) the probability that a value is less than 62.

d) the 90

percentile for this distribution.

a) p2ddyqwN65l80l4.5l70lqldNNu

b) The upper boundary in this problem will be positive infinity. Type 10^99 to represent positive infinity

dBBBB75l1z99lldNNNNu

c) The lower boundary in this problem will be negative infinity -1 x 10

dBB-1z99l62lldNNNNu

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d) Given a probability region to the left of a value determine the value using invNorm.

ddyqeNN0.9ll

T - Distribution

Example: calculate Student-t probability density for a specific parameter value when x = 1 and degrees of

freedom = 2.

Use the following steps p2ddywqwN1l2lldNNu

Example: calculate Student-t distribution probability for a specific parameter value, we will calculate Student-t

distribution probability when lower boundary = –2, upper boundary = 3, and degrees of freedom = 18.

Use the following steps p2ddywwwN-2l3l18lldNNu

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Example: Find the area under a T curve with degrees of freedom 10 for P(1 ≤ X ≤ 2 ).

• Select tcd p2ddyww.

• Enter the lower and upper bounds, and the degrees of freedom. The lower bound is the lowest number

and the upper bound is the highest number: 1,2,10

• Press l the answer is .133752549, or about 13.38%.

• To draw dNNu

Example: find the T score with a value of 0.25 to the left and df of 10.

• select Invt p2ddywe.

• Enter 0.25 in the Area. N0.25ll

• Enter 10 in the Deg of Freedom, df.

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Chi-square Distribution

Example: calculate 



probability density for a specific parameter value, we will calculate 



probability

density when x = 1 and degrees of freedom = 3.

Use the following steps: p2ddyeqwN1l3ll

To draw: dNNNu

Example: calculate 



distribution probability for a specific parameter value, we will calculate 



distribution

probability when lower boundary = 0, upper boundary = 19.023, and degrees of freedom = 9.

To calculate: p2ddyewwN0l19.023l9ll

To draw: dNNu

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F- distribution probability

F distribution probability calculates the probability of F distribution data falling between two specific values.

Example: calculate F distribution probability for a specific parameter value, we will calculate F distribution

probability when lower boundary = 0, upper boundary = 1.9824, n-df = 19 and d-df = 16.

To calculate: p2ddyrwwN0l1.9824l19l16ll

To draw: dNNu

Binomial probability

Binomial probability calculates a probability at specified value for the discrete binomial distribution with the

specified number of trials and probability of success on each trial.

Example: For data = {10, 11, 12, 13, 14} when Numtrial = 15 and success probability = 0.6. calculate binomial

probability for one list of data.

• Fill the data p2dd

• Calculate Binomial P.D yyqqNN15l0.6ll

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Example: A six-sided die is rolled twelve times and the number of sixes rolled is counted.

a) What is the probability of rolling exactly two sixes?

b) What is the probability of rolling more than two sixes?

This number of sixes can be modelled as a binomial distribution: x ~ B (12,





Solution:

a) Using Bpd pdp2ddyyqwN2l12l1b6ll

b) Find P (x1 ≤ X ≤ x2) using Bcd dyywwN3l12l12ll

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Poisson probability

Poisson probability calculates a probability at specified value for the discrete Poisson distribution with the

specified mean.

Example: Customers enter a shop at an average of three per minute. The number of customers entering the shop

in a given minute can be modelled by a Poisson distribution: X ~ P(3)

• What is the probability of exactly one customer entering the shop in a minute?

• What is the probability of five or fewer customers entering the shop in a minute?

Find P(X=x) using Ppd: p2ddyuqq

Fill the required data wN1l3ll

• Using Pcd p2ddyuqw

• Fill the required data wN0l5l3ll

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Example: Calculate Poisson probability for one list of data, we will calculate Poisson probability for data = {2, 3, 4}

when λ = 6.

• Fill the list: p2ddurq2l3l4luu

• To Calculate yuqqqNN6ll

Geometric probability

Geometric probability calculates a 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.

Example: calculate geometric probability for one list of data, we will calculate geometric probability for data = {3,

4, 5} when p = 0.4.

• Fill the list: p2ddurq3l4l5l

• To Calculate uuyuwqNN0.4ll

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Tests

The Z Test provides a variety of different standardization-based tests. They make it possible to test whether 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-Sample Z Test: tests for the unknown population mean when the population standard deviation is known.

2-Sample Z Test: tests the equality of the means of two populations based on independent samples when both

population standard deviations are known.

1-Prop Z Test: tests for an unknown proportion of successes.

2-Prop Z Test: tests to compare the proportion of successes from two populations.

The t Test: tests the hypothesis when the population standard deviation is unknown. The hypothesis that is the

opposite of the hypothesis being proven is called the null hypothesis, while the hypothesis being proved is called

the alternative hypothesis. The t Test is normally applied to test the null hypothesis. Then a determination is made

whether the null hypothesis or alternative hypothesis will be adopted.

1-Sample t Test: tests the hypothesis for a single unknown population mean when the population standard

deviation is unknown.

2-Sample t Test: compares the population means when the population standard deviations are unknown.

LinearReg t Test: calculates the strength of the linear association of paired data.

The 



test, a number of independent groups are provided, and a hypothesis is tested relative to the probability of

samples being included in each group.

The 



GOF test (



one-way Test): tests whether the observed count of sample data fits a certain distribution.

For example, it can be used to determine conformance with normal distribution or binomial distribution.

The 



two-way test: creates a cross-tabulation table that structures mainly two qualitative variables (such as

“Yes” and “No”), and evaluates the independence of the variables.

2-Sample F Test: tests the hypothesis for the ratio of sample variances. It could be used, for example, to test the

carcinogenic effects of multiple suspected factors such as tobacco use, alcohol, vitamin deficiency, high coffee

intake, inactivity, poor living habits, etc.

ANOVA: tests the hypothesis that the population means of the samples are equal when there are multiple

samples. It could be used, for example, to test whether or not different combinations of materials have an effect

on the quality and life of a final product.

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.

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1-Sample Z test

Example: Perform a 1-Sample Z Test for one list of data  < 0 test for the data List1 = {11.2, 10.9, 12.5, 11.3,

11.7}, when μ = 11.5 and  = 3.

• Fill the data with list1 p2dd

• 1-sample Z eqq

• Fill the values of μ and  qNwN11.5l3l

• Draw the graph dNNNNu

2-Sample Z test

Example: Perform a 2-Sample Z Test when two lists of data are input, we will perform a 1 < 2 test for the data

List1 = {11.2, 10.9, 12.5, 11.3, 11.7} and

List2 = {0.84, 0.9, 0.14, –0.75, –0.95}, when 1 = 15.5 and 2 = 13.5.

• Clear old data and enter the new data into the lists p2ddu!rq

• Z 2-samples uueqwNN15.5l13.5ll

• To draw dNNNNNNNu

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1-Prop Z test

Example: To perform a 1-Prop Z Test for specific expected sample proportion, data value, and sample size

Perform the calculation using: p0 = 0.5, x = 2048, n = 4040.

• 1-Prop Z test p2ddeqe

• Fill the data N0.5l2048l4040ll

• To draw dNNu

2-Prop Z test

Example: To perform a p1 > p2 2-Prop Z Test for expected sample proportions, data values, and sample sizes

Perform a p1 > p2 test using: x1 = 225, n1 = 300, x2 = 230, n2 = 300.

• 2-Prop Z test p2ddeqr.

• Fill the required data e225l300l230l300ll

• To draw dNNuq

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1-Sample T test

Example: Perform a 1-Sample t Test for one list of data where  ≠ 0 , List1 = {11.2, 10.9, 12.5, 11.3, 11.7},

when 0 = 11.3.

• Clear old data and enter the new data into the lists p2ddu!rq

• 1-sample T uuewqNqN11.3ll

• To see the graph dNNNNu

2-Sample T test

Example: Perform a 2-Sample T Test when two lists of data are input for  ≠ 2 , List1 = {55, 54, 51, 55, 53, 53,

54, 53} and List2 = {55.5, 52.3,51.8, 57.2, 56.5} when pooling is not in effect.

• Clear old data and enter the new data into the lists p2ddu!rq

• 2-sample T uuewwl

• For graphing dNNNNNNNNNu

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LinearReg t Test

Example: Perform a LinearReg t Test when two lists of data are input for this example, we will perform a

LinearReg t Test for x-axis data {0.5, 1.2, 2.4, 4, 5.2} and y-axis data {–2.1, 0.3, 1.5, 5, 2.4}.

• Clear old data and enter the new data into the lists p2ddu!rq

• T test LinearReg uuewel

Chi-Square Test



Test sets up several independent groups and tests hypotheses related to the proportion of the sample

included in each group. The 

Test is applied to dichotomous variables (variable with two possible

values, such as yes/no).

Example: To perform a 



Test on a specific matrix cell, we will perform a 



Test for Mat A, which contains the

following data.

 

 

• 



Test -2 way p2ddeew

• Observed matrix to fill the data we2l2ll1l4l5l10l

• Calculate the value ddBqDraw the graph du

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2-Sample F Test

Example: Perform a 2-Sample F Test when two lists of data are input for this example, we will perform a 2-

Sample F Test for the data List1 = {0.5, 1.2, 2.4, 4, 5.2} and List2 = {–2.1, 0.3, 1.5, 5,2.4}.

• Clear old data and enter the new data into the lists p2ddu!rquu

• Sample F Test erqBq

• Draw the graph duw

ANOVA tests

Example: Perform one-way ANOVA (analysis of variance) when three lists of data are input for this example, we

will perform analysis of variance for the data List1 = {1,1,2,2} List2 = {90,95,84,86}.

• Clear old data and enter the new data into the lists p2ddu!rq

• Sample F Test uueyqNNq2ll

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Example: Perform two-way ANOVA (analysis of variance) when three lists of data are input For this example, we

will perform analysis of variance for the data List1 = {1,1,1,1,2,2,2,2}, List2 = {1,1,2,2,1,1,2,2,} and List3 =

{113,116,139,132,133,131,126,122}.

• Clear old data and enter the new data into the lists p2ddu!rquu

• Sample F Test eywNNNq3ll

• Draw the graph dBBBBu

Confidant Intervals

• 1-Sample Z Interval calculates the confidence interval when the population standard

deviation is known.

• 2-Sample Z Interval calculates the confidence interval when the population standard

deviations of two samples are known.

• 1-Prop Z Interval calculates the confidence interval when the proportion is not known.

• 2-Prop Z Interval calculates the confidence interval when the proportions of two samples

are not known.

• 1-Sample t Interval calculates the confidence interval for an unknown population mean

when the population standard deviation is unknown.

• 2-Sample t Interval calculates the confidence interval for the difference between two

population means when both population standard deviations are unknown.

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Example: To calculate the 1-Sample Z Interval for one list of data, we will obtain the Z Interval for the data {11,

10, 12, 11, 11,15}, when C-Level = 0.95 (95% confidence level) and σ = 3.

• Clear old data and enter the new data into the lists p2ddu!rquu

• Z-INTR 1-sample to calculate the interval rqqqN0.95l3ll

Example: To calculate the 2-Sample Z Interval when two lists of data are input for this example, we will obtain the

2-Sample Z Interval for the data 1 = {55, 54, 51, 55, 53, 53, 54, 53} and data 2 = {55.5, 52.3,51.8, 57.2, 56.5}

when C-Level = 0.95 (95% confidence level), σ1 = 15.5, and σ2 = 13.5.

• Clear old data and enter the new data into the lists p2ddu!rquu

• 2-sample Z-INTR to calculate the interval rqwNN15.5l13.5ll

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Example: To calculate the 1-Prop Z Interval using parameter value specification for this example, we will obtain

the 1-Prop Z Interval when C-Level = 0.99, x = 55, and n = 100.

• Fill the data for 1-Prop Z-INTR to calculate the interval

ddrqe0.99l55l100ll

Example: To calculate the 2-Prop Z Interval using parameter value specification for this example, we will obtain

the 2-Prop Z Interval when C-Level = 0.95, x1 = 49, n1 = 61, x2 = 38 and n2 = 62.

• Fill the data for 1-Prop Z-INTR to calculate the interval

p2ddrqr0.92l49l61l38l62ll

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Example: To calculate the 1-Sample t Interval for one list of data, we will obtain the 1-Sample t Interval for data =

{11, 10, 12, 13, 17} when C-Level = 0.95.

• Clear old data and enter the new data into the lists p2ddu!rq

• To calculate the interval (INTR) uurwqqN0.95ll

Example: To calculate the 2-Sample t Interval when two lists of data are input, we will obtain the 2-Sample t

Interval for data 1 = {55, 54, 51, 55, 53, 53, 54, 53} and data 2 = {55.5, 52.3, 51.8, 57.2, 56.5} without pooling

when C-Level = 0.95.

• Clear old data and enter the new data into the lists p2ddu!rq

• To calculate the interval (INTR) uurwwqN0.95ll