Probability and statistics(assign 7 and 8)

PROBABILITY AND STATISTICS BY ENGR. JORGE P. BAUTISTA

COURSE OUTLINE Introduction to Statistics Tabular and Graphical representation of Data Measures of Central Tendencies, Locations and Variations Measure of Dispersion and Correlation Probability and Combinatorics Discrete and Continuous Distributions Hypothesis Testing

Text and References Statistics: a simplified approach by Punsalan and Uriarte, 1998, Rex Texbook Probability and Statistics by Johnson, 2008, Wiley Counterexamples in Probability and Statistics by Romano and Siegel, 1986, Chapman and Hall

Introduction to Statistics Definition In its plural sense, statistics is a set of numerical data e.g. Vital statistics, monthly sales, exchange rates, etc. In its singular sense, statistics is a branch of science that deals with the collection, presentation, analysis and interpretation of data.

General uses of Statistics Aids in decision making by providing comparison of data, explains action that has taken place, justify a claim or assertion, predicts future outcome and estimates un known quantities Summarizes data for public use

Examples on the role of Statistics In Biological and medical sciences, it helps researchers discover relationship worthy of further attention. Ex. A doctor can use statistics to determine to what extent is an increase in blood pressure dependent upon age - In social sciences, it guides researchers and helps them support theories and models that cannot stand on rationale alone. Ex. Empirical studies are using statistics to obtain socio-economic profile of the middle class to form new socio-political theories.

Con’t In business, a company can use statistics to forecast sales, design products, and produce goods more efficiently. Ex. A pharmaceutical company can apply statistical procedures to find out if the new formula is indeed more effective than the one being used. In Engineering, it can be used to test properties of various materials, Ex. A quality controller can use statistics to estimate the average lifetime of the products produced by their current equipment.

Fields of Statistics Statistical Methods of Applied Statistics: Descriptive-comprise those methods concerned with the collection, description, and analysis of a set of data without drawing conclusions or inferences about a larger set. Inferential-comprise those methods concerned with making predictions or inferences about a larger set of data using only the information gathered from a subset of this larger set.

con’t b. Statistical theory of mathematical statistics- deals with the development and exposition of theories that serve as a basis of statistical methods

Descriptive VS Inferential DESCRIPTIVE A bowler wants to find his bowling average for the past 12 months A housewife wants to determine the average weekly amount she spent on groceries in the past 3 months A politician wants to know the exact number of votes he receives in the last election INFERENTIAL A bowler wants to estimate his chance of winning a game based on his current season averages and the average of his opponents. A housewife would like to predict based on last year’s grocery bills, the average weekly amount she will spend on groceries for this year. A politician would like to estimate based on opinion polls, his chance for winning in the upcoming election.

Population as Differrentiated from Sample The word population refers to groups or aggregates of people, animals, objects, materials, happenings or things of any form, this means that there are populations of students, teachers, supervisors, principals, laboratory animals, trees, manufactured articles, birds and many others. If your interest is on few members of the population to represent their characteristics or traits, these members constitute a sample. The measures of the population are called parameters, while those of the sample are called estimates or statistics.

The Variable It refers to a characteristic or property whereby the members of the group or set vary or differ from one another. However, a constant refers to a property whereby the members of the group do not differ one another. Variables can be according to functional relationship which is classified as independent and dependent. If you treat variable y as a function of variable z, then z is your independent variable and y is your dependent variable. This means that the value of y, say academic achievement depends on the value of z.

Con’t Variables according to continuity of values. 1. Continuous variable – these are variables whose levels can take continuous values. Examples are height, weight, length and width. 2. Discrete variables – these are variables whose values or levels can not take the form of a decimal. An example is the size of a particular family.

Con’t Variables according to scale of measurements: 1. Nominal – this refers to a property of the members of a group defined by an operation which allows making of statements only of equality or difference. For example, individuals can be classified according to thier sex or skin color. Color is an example of nominal variable.

Con’t 2. Ordinal – it is defined by an operation whereby members of a particular group are ranked. In this operation, we can state that one member is greater or less that the others in a criterion rather than saying that he/it is only equal or different from the others such as what is meant by the nominal variable. 3. Interval – this refers to a property defined by an operation which permits making statement of equality of intervals rather than just statement of sameness of difference and greater than or less than. An interval variable does not have a “true” zero point.; althought for convenience, a zero point may be assigned.

Con’t 4. Ratio – is defined by the operation which permits making statements of equality of ratios in addition to statements of sameness or difference, greater than or less than and equality or inequality of differences. This means that one level or value may be thought of or said as double, triple or five times another and so on.

Assignment no. 1 Make a list of at least 5 mathematician or scientist that contributes in the field of statistics. State their contributions With your knowledge of statistics, give a real life situation how statistics is applied. Expand your answer. When can a variable be considered independent and dependent? Give an example for your answer.

Con’t IV. Enumerate some uses of statistics. Do you think that any science will develop without test of the hypothesis? Why?

Examples of Scales of Measurement 1.Nominal Level Ex. Sex: M-Male F-Female Marital Status: 1-single 2- married 3- widowed 4- separated 2. Ordinal Level Ex. Teaching Ratings: 1-poor 2-fair 3- good 4- excellent

Con’t 3. Interval Level Ex. IQ, temperature 4. Ratio Level Ex. Age, no. of correct answers in exam

Data Collection Methods Survey Method – questions are asked to obtain information, either through self administered questionnaire or personal interview. Observation Method – makes possible the recording of behavior but only at the time of occurrence (ex. Traffic count, reactions to a particular stimulus)

Con’t 3. Experimental method – a method designed for collecting data under controlled conditions. An experiment is an operation where there is actual human interference with the conditions that can affect the variable under study. 4. Use of existing studies – that is census, health statistics, weather reports. 5. Registration method – that is car registration, student registration, hospital admission and ticket sales.

Tabular Representation Frequency Distribution is defined as the arrangement of the gathered data by categories plus their corresponding frequencies and class marks or midpoint. It has a class frequency containing the number of observations belonging to a class interval. Its class interval contain a grouping defined by the limits called the lower and the upper limit. Between these limits are called class boundaries.

Frequency of a Nominal Data Male and Female College students Major in Chemistry SEX FREQUENCY MALE 23 FEMALE 107 TOTAL 130

Frequency of Ordinal Data Ex. Frequency distribution of Employee Perception on the Behavior of their Administrators Perception Frequency Strongly favorable 10 favorable 11 Slightly favorable 12 Slightly unfavorable 14 Unfavorable 22 Strongly unfavorable 31 total 100

Frequency Distribution Table Definition: Raw data – is the set of data in its original form Array – an arrangement of observations according to their magnitude, wither in increasing or decreasing order. Advantages: easier to detect the smallest and largest value and easy to find the measures of position

Grouped Frequency of Interval Data Given the following raw scores in Algebra Examination, 56 42 28 56 41 56 55 59 50 55 57 38 62 52 66 65 33 34 37 47 42 68 62 54 68 48 56 39 77 80 62 71 57 52 60 70

Con’t Compute the range: R = H – L and the number of classes by K = 1 + 3.322log n where n = number of observations. Divide the range by 10 to 15 to determine the acceptable size of the interval. Hint: most frequency distribution have odd numbers as the size of the interval. The advantage is that the midpoints of the intervals will be whole number. Organize the class interval. See to it that the lowest interval begins with a number that is multiple of the interval size.

Con’t 4. Tally each score to the category of class interval it belongs to. 5. Count the tally columns and summarizes it under column (f). Then add the frequency which is the total number of the cases (N). 6. Determine the class boundaries. UCB and LCB.(upper and lower class boundary) 7. Compute the midpoint for each class interval and put it in the column (M). M = (LS + HS) / 2

Con’t 8. Compute the cumulative distribution for less than and greater than and put them in column cf< and cf>. (you can now interpret the data). cf = cumulative frequency 9. Compute the relative frequency distribution. This can be obtained by RF% = CF/TF x 100% CF = CLASS FREQUENCY TF = TOTAL FREQUENCY

Graphical Representation The data can be graphically presented according to their scale or level of measurements. 1. Pie chart or circle graph. The pie chart at the right is the enrollment from elementary to master’s degree of a certain university. The total population is 4350 students

Con’t 2. Histogram or bar graph- this graphical representation can be used in nominal, ordinal or interval. For nominal bar graph, the bars are far apart rather than connected since the categories are not continuous. For ordinal and interval data, the bars should be joined to emphasize the degree of differences

Given the bar graph of how students rate their library. A-strongly favorable, 90 B-favorable, 48 C-slightly favorable, 88 D-slightly unfavorable, 48 E-unfavorable, 15 F-strongly unfavorable, 25

The Histogram of Person’s Age with Frequency of Travel age freq RF 19-20 20 39.2% 21-22 21 41.2% 23-24 4 7.8% 25-26 4 7.8% 27-28 2 3.9% total 51 100%

Exercises From the previous grouped data on algebra scores, Draw its histogram using the frequency in the y axis and midpoints in the x axis. Draw the line graph or frequency polygon using frequency in the y axis and midpoints in the x axis. Draw the less than and greater than ogives of the data. Ogives is a cumulation of frequencies by class intervals. Let the y axis be the CF> and x axis be LCB while y axis be CF< and x axis be UCB

Con’t d. Plot the relative frequency using the y axis as the relative frequency in percent value while in the x axis the midpoints.

Assignment No. 2 Given the score in a statistics examinations, 38 56 35 70 44 81 44 80 45 72 45 50 51 51 52 66 54 53 56 84 58 56 57 70 56 39 56 59 72 63 89 63 69 65 61 62 64 64 69 60 53 66 66 67 67 68 68 69 66 67 70 59 40 71 73 60 73 73 73 73 73 74 73 73 79 74 74 70 73 46 74 74 74 75 75 76 55 77 78 73 48 81 44 84 77 88 63 85 73

Con’t Construct the class interval, frequency table, class midpoint(use a whole number midpoint), less than and greater than cumulative frequency, upper and lower boundary and relative frequency. Plot the histogram, frequency polygon, and ogives

Con’t 3. Draw the pie chart and bar graph of the plans of computer science students with respect to attending a seminar. Compute for the Relative frequency of each. A-will not attend=45 B-probably will not attend=30 C-probably will attend=40 D-will attend=25

Measures of Centrality and Location Mean for Ungrouped Data X’ = Σ X / N where X’ = the mean Σ X = the sum of all scores/data N = the total number of cases Mean for Grouped Data X’ = Σ fM / N where X’ = the mean M = the midpoint fM = the product of the frequency and each midpoint N = total number of cases

Con’t Ex. Find the mean of 10, 20, 25,30, 30, 35, 40 and 50. Given the grades of 50 students in a statistics class Class interval f 10-14 4 15-19 3 20-24 12 25-29 10 30-34 6 35-39 6 40-44 6 45-49 3

Con’t The weighted mean. The weighted arithmetic mean of given groups of data is the average of the means of all groups WX’ = Σ Xw / N where WX’ = the weighted mean w = the weight of X Σ Xw = the sum of the weight of X’s N = Σ w = the sum of the weight of X

Con’t Ex. Find the weighted mean of four groups of means below: Group, i 1 2 3 4 X i 60 50 70 75 W i 10 20 40 50

Con’t Median for Ungrouped Data The median of ungrouped data is the centermost scores in a distribution. Mdn = (X N/2 + X (N + 2)/2 ) / 2 if N is even Mdn = X (1+N)/2 if N is odd Ex. Find the median of the following sets of score: Score A: 12, 15, 19, 21, 6, 4, 2 Score B: 18, 22, 31, 12, 3, 9, 11, 8

Con’t Median for Grouped Data Procedure: Compute the cumulative frequency less than. Find N/2 Locate the class interval in which the middle class falls, and determine the exact limit of this interval. Apply the formula Mdn = L + [(N/2 – F)i]/fm where L = exact lower limit interval containing the median class F = The sum of all frequencies preceeding L. fm = Frequency of interval containing the median class i = class interval N = total number of cases

Con’t Ex. Find the median of the given frequency table. class interval f cf< 25-29 3 3 30-34 5 8 35-39 10 18 40-44 15 33 45-49 15 48 50-54 15 63 55-59 21 84 60-64 8 92 65-69 6 98 70-74 2 100

Con’t Mode of Ungrouped Data It is defined as the data value or specific score which has the highest frequency. Find the mode of the following data. Data A : 10, 11, 13, 15, 17, 20 Data B: 2, 3, 4, 4, 5, 7, 8, 10 Data C: 3.5, 4.8, 5.5, 6.2, 6.2, 6.2, 7.3, 7.3, 7.3, 8.8

Mode of Grouped Data For grouped data, the mode is defined as the midpoint of the interval containing the largest number of cases. Mdo = L + [d 1 /(d 1 + d 2 )]i where L = exact lower limit interval containing the modal class. d 1 = the difference of the modal class and the frequency of the interval preceding the modal class d 2 = the difference of the modal class and the frequency of the interval after the modal class.

Ex. Find the mode of the given frequency table. class interval f cf< 25-29 3 3 30-34 5 8 35-39 10 18 40-44 15 33 45-49 15 48 50-54 15 63 55-59 21 84 60-64 8 92 65-69 6 98 70-74 2 100

Exercises Determine the mean, median and mode of the age of 15 students in a certain class. 15, 18, 17, 16, 19, 18, 23 , 24, 18, 16, 17, 20, 21, 19 2. To qualify for scholarship, a student should have garnered an average score of 2.25. determine if the a certain student is qualified for a scholarship.

Subject no. of units grade A 1 2.0 B 2 3.0 C 3 1.5 D 3 1.25 E 5 2.0

Find the mean, median and mode of the given grouped data. Classes f 11-22 2 23-34 8 35-46 11 47-58 19 59-70 14 71-82 5 83-94 1

Quartiles refer to the values that divide the distribution into four equal parts. There are 3 quartiles represented by Q 1 , Q 2 and Q 3 . The value Q 1 refers to the value in the distribution that falls on the first one fourth of the distribution arranged in magnitude. In the case of Q 2 or the second quartile, this value corresponds to the median. In the case of third quartile or Q 3 , this value corresponds to three fourths of the distribution.

For grouped data, the computing formula of the kth quartile where k = 1,2,3,4,… is given by Q k = L + [(kn/4 - F)/fm]Ii Where L = lower class boundary of the kth quartile class F = cumulative frequency before the kth quartile class fm = frequency before the kth quartile i = size of the class interval

Exercises Compute the value of the first and third quartile of the given data class interval f cf< 25-29 3 3 30-34 5 8 35-39 10 18 40-44 15 33 45-49 15 48 50-54 15 63 55-59 21 84 60-64 8 92 65-69 6 98 70-74 2 100

Decile: If the given data is divided into ten equal parts, then we have nine points of division known as deciles. It is denoted by D 1 , D 2 , D 3 , D 4 …and D 9 D k = L + [(kn/10 – F)/fm] I Where k = 1,2,3,4 …9

Exercises Compute the value of the third, fifth and seventh decile of the given data class interval f cf< 25-29 3 3 30-34 5 8 35-39 10 18 40-44 15 33 45-49 15 48 50-54 15 63 55-59 21 84 60-64 8 92 65-69 6 98 70-74 2 100

Percentile- refer to those values that divide a distribution into one hundred equal parts. There are 99 percentiles represented by P 1 , P 2 , P 3 , P 4 , P 5 , …and P 99 . when we say 55 th percentile we are referring to that value at or below 55/100 th of the data. P k = L + [(kn/100 – F)/fm]i Where k = 1,2,3,4,5,…99

Exercises Compute the value of the 30 th , 55 th , 68 th and 88 th percentile of the given data class interval f cf< 25-29 3 3 30-34 5 8 35-39 10 18 40-44 15 33 45-49 15 48 50-54 15 63 55-59 21 84 60-64 8 92 65-69 6 98 70-74 2 100

Assignment no. 3 The rate per hour in pesos of 12 employees of a certain company were taken and are shown below. 44.75, 44.75, 38.15, 39.25, 18.00, 15.75, 44.75, 39.25, 18.50, 65.25, 71.25, 77.50 Find the mean, median and mode. If the value 15.75 was incorrectly written as 45.75, what measure of central tendency will be affected? Support your answer.

II. The final grades of a student in six subjects were tabulated below. Subj units final grade Algebra 3 60 Religion 2 90 English 3 75 Pilipino 3 86 PE 1 98 History 3 70 Determine the weighted mean If the subjects were of equal number of units, what would be his average?

III. The ages of qualified voters in a certain barangay were taken and are shown below Class Interval Frequency 18-23 20 24-29 25 30-35 40 36-41 52 42-47 30 48-53 21 54-59 12 60-65 6 66-71 4 72-77 1

Find the mean, median and mode Find the 1 st and 3 rd quantile Find the 4 th and 6 th decile Find the 25 th and 75 th percentile

Measure of Variation The range is considered to be the simplest form of measure of variation. It is the difference between the highest and the lowest value in the distribution. R = H – L For grouped data, the3 difference between the highest upper class boundary and the lowest lower class boundary. Example: find the range of the given grouped data in slide no. 59

Semi-inter Quartile Range This value is obtained by getting one half of the difference between the third and the first quartile. Q = (Q 3 – Q 1 )/2 Example: Find the semin-interquartile range of the previous example in slide no. 59

Average Deviation The average deviation refers to the arithmetic mean of the absolute deviations of the values from the mean of the distribution. This measure is sometimes known as the mean absolute deviation. AD = Σ│ x – x’ │ / n Where x = the individual values x’ = mean of the distribution

Steps in solving for AD Arrange the values in column according to magnitude Compute for the value of the mean x’ Determine the deviations (x – x’) Convert the deviations in step 3 into positive deviations. Use the absolute value sign. Get the sum of the absolute deviations in step 4 Divide the sum in step 5 by n.

Example: Consider the following values: 16, 13, 9, 6, 15, 7, 11, 12 Find the average deviation.

For grouped data: AD = Σ f│x – x’│ / n Where f = frequency of each class x = midpoint of each class x’ = mean of the distribution n = total number of frequency

Example: Find the average deviation of the given data Classes f 11-22 2 23-34 8 35-46 11 47-58 19 59-70 14 71-82 5 83-94 1

Variance For ungrouped data s 2 = Σ (x – x’) 2 / n Example: Find the variance of 16, 13, 9, 6, 15, 7, 11, 12

For grouped data s 2 = Σ f(x – x’) 2 / n Where f = frequency of each class x = midpoint of each class interval x’ = mean of the distribution n = total number of frequency

Example: Find the variance of the given data Classes f 11-22 2 23-34 8 35-46 11 47-58 19 59-70 14 71-82 5 83-94 1

Coefficient of variation If you wish to compare the variability between different sets of scores or data, coefficient of variation would be very useful measure for interval scale data CV = s/x Where s = standard deviation x = the mean

Example: In a particular university, a researcher wishes to compare the variation in scores of the urban students with that of the scores of the rural students in their college entrance test. It is know that the urban student’s mean score is 384 with a standard deviation of 101; while among the rural students, the mean is 174, with a standard deviation of 53, which group shows more variation in scores?

Standard Deviation s = √s 2 For ungrouped data s = √ Σ (x – x’) 2 / n For grouped data s = √ Σ f(x – x’) 2 / n

Find the standard deviation of the previous examples for ungrouped and grouped data. Find the standard deviation of the given data Classes f 11-22 2 23-34 8 35-46 11 47-58 19 59-70 14 71-82 5 83-94 1

Find the standard deviation of 16, 13, 9, 6, 15, 7, 11, 12

Measure of variation for nominal data VR = 1 – fm/N Where VR = the variation ratio fm = modal class frequency N = counting of observation

Example: With the data given by a clinical psychologist on the type of therapy used, compute the variation ratios. Type of therapy no. of patients YR 1980 YR 1985 Logotherapy 20 8 Reality Therapy 60 105 Rational Therapy 42 6 Transactional analysis 39 9 Family therapy 52 5 Others 41 8

Assignment no. 4 I. Compute for the semi-interquartile range, absolute deviation, variance and standard deviation test III of assignment no. 3. II. Compute for the semi-interquartile range, absolute deviation, variance and standard deviation of test I of assignment no. 3.

SIMPLE LINEAR REGRESSION AND MEASURES OF CORRELATION In this topic, you will learn how to predict the value of one dependent variable from the corresponding given value of the independent variable.

The scatter diagram: In solving problems that concern estimation and forecasting, a scatter diagram can be used as a graphical approach. This technique consist of joining the points corresponding to the paired scores of dependent and independent variables which are commonly represented by X and Y on the X-Y coordinate system.

Example: The working experience and income of 8 employees are given below Employee years of income experience (in Thousands) X Y A 2 8 B 8 10 C 4 11 D 11 15 E 5 9 F 13 17 G 4 8 H 15 14

Using the Least Squares Linear Regression Equation: Y = a + bX Where b = [n Σ xy – Σ x Σ y] / [n Σ x 2 – ( Σ x) 2 ] a = y’ – bx’ Obtain the equation of the given data and estimate the income of an employee if the number of years experience is 20 years.

Standard Error of Estimate Se = √ [ Σ Y i 2 – a(Y i ) – b(X i Y i )] / n-2 The standard error of estimate is interpreted as the standard deviation. We will find that the same value of X will always fall between the upper and lower 3Se limits.

Measures of Correlation The degree of relationship between variables is expressed into: Perfect correlation (positive or negative) Some degree of correlation (positive or negative) No correlation

For a perfect correlation, it is either positive or negative represented by +1 and -1. correlation coefficients, positive or negative, is represented by +0.01 to +0.99 and -0.01 to -0.99. The no correlation is represented by 0.

0 to +0.25 very small positive correlation +0.26 to +0.50 moderately small positive correlation +0.51 to +0.75 high positive correlation +0.76 to +0.99 very high positive correlation +1.00 perfect positive correlation ---------------------------------------------------------- 0 to -0.25 very small negative correlation -0.26 to -0.50 moderately small positive correlation -0.51 to -0.75 high negative correlation -0.76 to -0.99 very high negative correlation -1.00 perfect negative correlation

Anybody who wants to interpret the results of the coefficient of correlation should be guided by the following reminders: The relationship of two variables does no necessarily mean that one is the cause of the effect of the other variable. It does not imply cause-effect relationship. When the computed Pearson r is high, it does not necessarily mean that one factor is strongly dependent on the other. On the other hand, when the computed Pearson r is small it does not necessarily mean that one factor has no dependence on the other. If there is a reason to believe that the two variables are related and the computed Pearson r is high, these two variables are really meant as associated. On the other hand, if the variables correlated are low, other factors might be responsible for such small association. Lastly, the meaning of correlation coefficient just simply informs us that when two variables change there may be a strong or weak relationship taking place.

The formula for finding the Pearson r is [n Σ XY – Σ X Σ Y] r = ------------------------------ √ [n Σ X 2 – ( Σ X) 2 ] [n Σ Y 2 – ( Σ Y) 2 ]

Example: Given two sets of scores. Find the Pearson r and interpret the result. X Y 18 10 16 14 14 14 13 12 12 10 10 8 10 5 8 6 6 12 3 0

Correlation between Ordinal Data This is the Spearman Rank-Order Correlation Coefficient (Spearman Rho). For cases of 30 or less, Spearman ρ is the most widely used of the rank correlation method. 6 Σ D 2 ρ = 1 - ----------- n(n 2 – 1) Where D = (RX – RY)

Example: Individual Test X Test Y 1 18 24 2 17 28 3 14 30 4 13 26 5 12 22 6 10 18 7 8 15 8 8 12

Gamma Rank Order An alternative to the rank order correlation is the Goodman’s and Kruskal’s Gamma (G). The value of one variable can be estimated or predicted from the other variable when you have the knowledge of their values. The gamma can also be used when ties are found in the ranking of the data.

N S - N 1 G = ----------------- N S + N 1 Where N S = the number of pairs ordered in the parallel direction N 1 = the number of pairs ordered in the opposite direction

Given a segment of the Filipino Electorate according to religion and political party LAKAS LP NP Total Catholic 50 25 20 INC 34 72 21 Born Again 22 12 10 Total

Correlation between Nominal Data The Guttman’s Coefficient of predictability is the proportionate reduction in error measure which shows the index of how much an error is reduced in predicting values of one variable from the value of another. Σ FBR - MBC λ c = ------------------ N – MBC Where FBR = the biggest cell frequencies in the ith row MBC = the biggest column totals N = total observations

Σ FBC - MBR λ r = ------------------- N – MBR Where FBC = the biggest cell frequencies in the column MBR = the biggest of the row totals N = total number of observations Compute for the λ c and λ r for the segment of Filipino electorate and political parties.

Assignment no. 5 Given the average yearly cost and sales of company A for a period of 8 years. Find the pearson r and interpret the results. Year Cost Sales per P10,000 per P10,000 15 38 30 53.3 16 60 39 72 20 40 36 47.5 45 82 10 21.5

Given the grades of 10 students in statistics determine the spearman rho and interpret the result Student Q1 Q2 A 62 57 B 90 88 C 75 90 D 60 67 E 58 60 F 89 79 G 91 78 H 90 62 I 94 86 J 50 55

3. Compute for the gamma shown and interpret the result Socio-economic status EDUCATIONAL STATUS TOTAL UPPER MIDDLE LOWER TOTAL UPPER 24 19 5 MIDDLE 12 54 29 LOWER 9 26 25 TOTAL

4. Compute for the λ c and λ r for the problem no. 3.

Counting Techniques Consider the numbers 1,2,3 and 4. suppose you want to determine the total 2 digit numbers that can be formed if these are combined. First, let us assume that no digit is to be repeated. 12 21 31 41 23 32 42 24 34 43 Notice that we were able to used all the possibilities. In this example, we have 12 possible 2 digit numbers.

Now, what if the digits can be repeated? 12 13 14 22 23 24 23 33 34 42 43 44 Hence, we have 16 possible outcomes. In the first activity, we can do it in n 1 ways and after it has been done, the second activity can be done in n 2 ways, then the total number of ways in which the two activities can be done is equal to n 1 n 2 .

Example: How many two digit numbers can be formed from the numbers 1,2,3 and 4 if Repetition is not allowed? Repetition is allowed? 2. How many three digit numbers can be formed from the digits 1,2,3,4 and 5 if any of the digits can be repeated? 3. The club members are going to elect their officers. If there are 5 candidates for president, 5 candidates for vice president and 3 for secretary, then how many ways can the officers be elected?

An office executive plans to buy as laptop in which there are 5 brands available. Each of the brands has 3 models and each model has 5 colors to chose from. In how many ways can the executive choose? Consider the numbers 2,3 5 and 7. if repetition is not allowed, how many three digit numbers can be formed such that They are all odd? They are all even? They are greater that 500?

6. A pizza place offers 3 choices of salad, 20 kinds of pizza and 4 different deserts. How many different 3 course meals can one order? 7. The executive of a certain company is consist of 5 males and 2 females. How many ways can the presidents and secretary be chosen if The president must be female and the secretary must be male? The president and the secretary are of opposite sex? The president and the secretary should be male?

Permutation The term permutation refers to the arrangement of objects with reference to order. P(n,r) = n! / (n – r)! Evaluate: P(10,6) P(5,5) P(4,3) + P(4,4)

Examples: In how many ways can a president, a vice president, a secretary and a treasurer be elected from a class with 40 students? In how many ways can 7 individuals be seated in a row of 7 chairs? In how many ways can 9 individuals be seated in a row of 9 chairs if two individuals wanted to be seated side by side?

4. Suppose 5 different math books and 7 different physics books shall be arranged in a shelf. In how many ways can such books be arranged if the books of the same subject be placed side by side? Determine the possible permutations of the word MISSISSIPPI. Find the total 8 digit numbers that can be formed using all the digits in the following numerals 55777115

In how many ways can 6 persons be seated around a table with 6 chairs if two individuals wanted to be seated side by side? In a local election, there are 7 people running for 3 positions. In how many ways can this be done?

Combination A combination is an arrangement of objects not in particular order. nCr = C(n,r) = n! / r!(n-r)! Evaluate: 8 C 4 5( 5 C 4 – 5 C 2 ) 7 C 5 / ( 7 C 6 – 7 C 2 )

A class is consist of 12 boys and 10 girls. In how many ways can the class elect the president, vice president, secretary and a treasurer? In how many ways can the class elect 4 members of a certain committee? In how many ways can a student answer 6 out of ten questions? In how many ways can a student answer 6 out of 10 questions if he is required to answer 2 of the first 5 questions?

In how many ways can 3 balls be drawn from a box containing 8 red and 6 green balls? A box contain 8 red and 6 green balls. In how many ways can 3 balls be drawn such that They are all green? 2 is red and 1 is green? 1 is red and 2 is green?

A shipment of 40 computers are unloaded from the van and tested. 6 of them are defective. In how many ways can we select a set of 5 computers and get at least one defective? Five letters a,b,c,d,e are to be chosen. In how many ways could you choose None of them At least two of them At most three of them

Assignment no. 6 How many possible outcomes are there if A die is rolled? A pair of dice is rolled? 2. In how many ways can 5 math teachers be assigned to 4 available subjects if each of the 5 teachers have equal chance of being assigned to any of the 4 subjects?

3. Consider the numbers 1,2,3,5,and 6. how many 3 digit numbers can be formed from these numbers if Repetition is not allowed and 0 should not be in the first digit? Repetition is allowed and 0 should not be in the first digit? 4. A college has 3 entrance gates and 2 exit gates. In how many ways can a student enter then leave the building?

In how many ways can 9 passengers be seated in a bus if there are only 5 seats available? In how many ways can 4 boys and 4 girls be seated in a row of 8 chairs if They can sit anywhere? The boys and girls are to be seated alternately? 7. In how many ways can ten participants in a race placed first, second and third?

Determine the number of distinct permutations of each of the following: STATISTICS ADRENALIN 44044999404 A class consist of 12 boys and 10 girls. In how many ways can a committee of five be formed if All members are boys? 2 are boys and 3 are girls?

10. In how many ways can a student answer an exam if out of the 6 problem, he is required to answer only 4?

Probability In the study of probability, we shall consider activities for which the outcomes cannot be predicted with certainty. These activities, called experiment , could always result in a single outcome. Although the single outcome can not be predicted before the performance of the experiment, the set of all possible outcomes can be determined. This set of all possible outcomes is referred to as sample space . Each individual element or outcome in a sample space is known as a sample point .

Definition of terms: Random experiment- any process of generating a set of data or observations that can be repeated under basically the same conditions, which lead to well defined outcomes. Sample space – set of all possible outcomes of an experiment, usually denoted by S. Sample point- an element of the sample space or outcomes.

event- any subset of the sample space usually denoted by capital letters. Null space- a subset of the sample space that contains no elements and denoted by the symbol Ø. Simple event – an event which contains only one element of the sample space. Compound event – an event that can be expressed as the union of the simple events, thus containing more than one sample points. Mutually exclusive events- two events A and B are mutually exclusive if A∩B have no elements in common.

The probability of a event A denoted by P(A) is the sum of the probabilities of mutually exclusive outcomes that constitute the event. It must satisfy the following properties: 0 ≤ P(A) ≤ 1

Example: 1. Consider the activity of rolling a die. This activity has 6 possible outcomes, that is 1,2,3,4,5 and 6. thus, S = {1,2,3,4,5,6} Any numbers 1 to 6 is a sample point of S. we can say that there are 6 sample points. If we let A be the event of getting an even number and B an event of getting a perfect square, then A = {2,4,6} and B = {1,4} Note that the elements of A are elements of the sample space S. the number of sample points in a sample space S, events A and B are usually written as n(S) = 6, n(A) = 3 and n(B) = 2.

If a pair of dice is rolled, then determine the number of sample points of the following: Sample space Event of getting a sum of 5. Event of getting a sum of at most 4. A box contains 6 red and 4 green balls. If three balls are drawn from the box, then determine the number of sample points of the following: The sample space The event of getting all green balls The event of getting 1 red and 2 green balls.

Probability is the chance that an event will happen. The probability of an event A denoted by P(A) refers to the number between 0 and 1 including the values of 0 and 1. This number can be expressed as a fraction, as a decimal or as a percent. When we assign a probability of 0 to event A, it means that it is impossible for event A to occur. When event A is assigned a probability of 1, then we say that event A will really occur.

P(A) + P(A)’ = 1 The probability of occurrence plus the probability of non-occurrence is always equal to 1. Example: A student in a statistics class was able to compute the probability of passing the subject to be equal to 0.55. Based on this information, what is the probability that he is not going to pass the subject?

Three approaches of probability: Subjective probability- it is determine by the use of intuition, personal beliefs and other indirect information. A posteriori or probability of relative frequency (empirical probability) – it is determined by repeating the experiment a large number of times using the following rule: no. of times event A occurred P(A) = --------------------------------------------------- no. of times experiment was repeated

Example: Records show that 120 out of 500 students who entered in a CS/IT programs leave the school due to financial problems. What is the probability that a freshman entering this college will leave the school due to financial problem?

2. Last year, the efficiency rating of the employees of a certain company were taken and presented in a frequency distribution below: Efficiency rating no. of employees 60-65 12 66-71 10 72-77 31 78-83 29 84-89 8 Based on the data, what can we say about the proportion of employees for this year who shall have an efficiency rating from 72-77 and 84-89?

A Priori or classical probability – it is determined even before the experiment is performed using the following rule: n(A) P(A) = -------- n(S) Where n(A) = no. of sample points in event A n(S) = no. of sample points in sample space S.

If a coin is tossed , what is the probability of getting a head? If two coins are tossed, what is the probability of getting both heads? If a die is rolled, what is the probability of getting an odd number? An even number? A perfect square? If a pair of dice is rolled, what is the probability of getting a sum of 6? A sum of 13?

5. The probability that a college student without a flu shot will get the flu is 0.42.what is the probability that a college student without the flu shot will not get the flu? 6. A box contains 7 red and 6 green balls. If 2 balls are drawn from the box, what is the probability of getting both green? 1 red and 1 green?

Addition Rule: In practice, the probability of two or more events are usually considered. If we let A and B be events then these two events can be combined to form another event. The event that at least one of the events A or B will happen is denoted by AUB. The event that both events A and B will occur is denoted by A∩B. The probability of AUB denoted by P(AUB) is given by P(AUB) = P(A) + P(B) – P(A∩B)

Two events A and B are said to be mutually exclusive if they can not occur both at the same time. This implies that the occurrence of event A excludes the occurrence of event B and vice versa. Therefore, P(A∩B) has no sample point which is equal to 0. The previous equation will be P(AUB) = P(A) + P(B)

Consider rolling a die and the events of getting an odd number, an even number and a perfect square. Determine the probability of getting An odd or an even number. An even number or a perfect square. (this implies that the two events can occur both at the same time. Therefore the two events are non-mutually exclusive events)

2. A card is drawn from an ordinary deck of 52 playing cards. Find the probability of getting An ace or a queen A queen or a face card A black card or a queen

3. You are going to rolled a pair of dice. Find the probability of getting the sum that is even or the sum that is multiple of 3. 4. A student goes to the library and checks out that 40% are work of fiction, 30% are non fiction and 20% are either fiction or non-fiction. What is the probability that the student check out a work of fiction, non-fiction or both?

5 . The probability that Anita will buy machine A is 7/11 and the probability that she will buy machine B is 5/11. If the probability of buying either machine A and B is 9/11, what is the probability of buying the two machine?

6. A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice 4 times a week. Thirty of the intermediate swimmers practice 4 times a week. Ten of the novice swimmers practice 4 times a week. Suppose one member of the swim team is randomly chosen. Answer the questions (Verify the answers):

What is the probability that the member is a novice swimmer? What is the probability that a member practice 4 times a week? What is the probability that the member is an advanced swimmer and practice 4 times a week? What is the probability that a member is an advance swimmer and an intermediate swimmer? Are they mutually exclusive?

SEATWORK 1. A BOX CONTAINS 7 RED, 3 GREEN AND 2 YELLOW BALLS. IF ONE BALL IS DRAWN FROM THE BOX, THEN WHAT IS THE PROBABILITY OF GETTING A RED? A NON-RED? A NON-GREEN? 2. SUPPOSE THAT WE ROLL A DICE, WHAT IS THE PROBABILITY OF GETTING A SUM OF 6 OR 8? 3. SUPPOSE WE PICK ONE CARD FROM A DECK OF CARDS, WHAT IS THE PROBABILITY OF GETTING A KING OR A SPADE? A KING OR NUMBER 8? 4. KLAUS IS TRYING TO CHOOSE WHERE TO GO ON VACATION. HIS CHOICES ARE A=BAGIUO AND B=TAGAYTAY. HE CAN ONLY AFFORD ONE VACATION. THE PROBABILITY OF CHOOSING A IS 0.36 AND THE PROBABILITY OF CHOOSING B IS 0.44. WHAT IS THE PROBABILITY THAT HE CHOOSES TO GO EITHER A OR B? WHAT IS THE PROBABILITY THAT HE WILL NOT CHOOSE ANY OF THE TWO DISTINATION?

Conditional Probability and Multiplication Rule It is the probability that a second event will occur if the first event already happened. Symbolically, conditional probability is written as P(A/B) and is read as the probability of event A given that B has occurred. The computing formula for the conditional probability of A given B is given by P(A/B) = P(A ∩ B)/P(B), provided P(B) is not equal to zero.

Let P(A) = 0.55 P(B) = 0.35 P(A ∩B) = 0.20 Find P(A/B) and P(B/A) 2 . A die is rolled. If the result is an even number, what is the probability that it is a perfect square? 3. A card is drawn from a deck of 52 cards. Given that the card drawn is a face card, then what is the probability of getting a king? A spade? A red card?

4 . A vendor has 35 balloons on strings. 20 balloons are yellow, 8 are red and 7 are green. A balloon was selected at random and sold. Given that the balloon selected and sold is yellow, what is the probability that the next balloon selected and sold at random is also yellow? 5. Given that 25 microwaves are on display in a certain store but 2 of them are defective. A customer wishes to buy 2 microwaves and pick them up without replacement. Find the probability that the two are defective.

6. Should women participate in combat? yes no Male 32 18 Female 8 42 Find the probability that the respondent answered YES given that the respondent was a female. Find the probability that the respondent was a male given that the respondent answered NO.

7. A box contains 3 red and 8 black balls. If two balls are drawn in succession without replacement, what is the probability that Both are red? The first ball is red and the second ball is black? 8. A box contains 3 red and 8 black balls. If 2 balls are drawn at random with replacement, what is the probability that both are red?

Assignment no. 7 1.. A BOX CONTAINS 7 RED, 3 GREEN AND 2 YELLOW BALLS. IF ONE BALL IS DRAWN FROM THE BOX, THEN WHAT IS THE PROBABILITY OF GETTING A RED? A NON-RED? A NON-GREEN? 2. SUPPOSE THAT WE ROLL A DICE, WHAT IS THE PROBABILITY OF GETTING A SUM OF 6 OR 8? 3. SUPPOSE WE PICK ONE CARD FROM A DECK OF CARDS, WHAT IS THE PROBABILITY OF GETTING A KING OR A SPADE? A KING OR NUMBER 8? 4. KLAUS IS TRYING TO CHOOSE WHERE TO GO ON VACATION. HIS CHOICES ARE A=BAGIUO AND B=TAGAYTAY. HE CAN ONLY AFFORD ONE VACATION. THE PROBABILITY OF CHOOSING A IS 0.36 AND THE PROBABILITY OF CHOOSING B IS 0.44. WHAT IS THE PROBABILITY THAT HE CHOOSES TO GO EITHER A OR B? WHAT IS THE PROBABILITY THAT HE WILL NOT CHOOSE ANY OF THE TWO DISTINATION?

5. The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday?

Normal Distribution The normal probability curve is one of the most commonly used theoretical distributions in statistical inference. The mathematical equation of the normal curve was developed by De Moivre in 1773. the distribution is sometimes called the Gaussian distribution in honor of Gauss, who also derived the equation in the 19 th century.

Con’t A large population investigated in education and the behavioral sciences has characteristics that follow a normal distribution. If we are to study, for instance, the scholastic mental capacity of a school population N= 1500, we may find that majority of the student population will yield average scores, a small portion will yield above and below average scores and a few students will yield extremely high and low scores.

Con’t The characteristics of the Normal Curve is 1. The curve is symmetrical and bell shaped. It has its highest point at the center. The lines at both sides fall off toward the opposite directions at exactly equal distance from the center. Therefore if the curve is folded at the middle, the two sides are perfectly of the same size and shape.

Con’t 2. The number of cases, N, is infinite. This is the reason why the curve is asymptotic to the baseline which means that the curve at both sides does not touch the baseline or the axis, and that the curve may extend infinitely to both directions. 3. The three measures of central tendency, mean, median and mode coincide at one point at the center of the distribution.

Con’t 4. The height of the curve indicate the frequency of cases, expressed as probability, proportion or percentage. Hence, the total area under the normal curve is 1.0 in terms of probability or proportion and 100% in terms of percentage. Thus one half of the area is 50% 5. The basic unit of measurement is expressed in sigma units ( σ ) or standard deviations along the baseline. It is also called Z-scores.

Con’t 6. Two parameters are used to describe the curve. One is the parameter mean( μ or x’) which is equal to zero and the other is the standard deviation( σ ) which is equal to 1. 7. Standard deviations or A scores departing away from the mean ( μ or x’) towards the right of the curve is in positive while scores departing from the mean is in negative values.

From the previous curve We can say that, At least 68% of the values in the given set of data fall within plus or minus 1 standard deviation from the mean. In symbols, the interval is given by (x’ – 1 σ ) – (x’ + 1 σ ). At least 95% of the value in the given set of data fall within plus or minus 2 standard deviation from the mean. In symbol, the interval is (x’ – 2 σ ) – (x’ + 2 σ ) and so on.

To illustrate the significance of the empirical rule, consider the NCEE scores of students in a certain college whose mean score x’ or μ = 65 and the standard deviation σ or SD = 6 approximately, 68% of the students in that college have NCEE scores between 80 plus or minus 10, that is 65 – (1)(6) – 65 + (1)(6) 59 - 71

The Standard Score The standard score Z represents a normal distribution with mean x’ = 0 and SD = 1. such transformation can be obtained by using the formula below. Z = (x – x’) / SD

Normal Curve Areas The total area under the normal curve is equal to 1. since a normally distributed set of data is symmetric, then the total area from Z = 0 to the right is equal to 0.5. the area from Z = 0 to the left is also equal to 0.5. Example: Find the area under the curve from 0 <Z<1.25 -1.25<Z<0

Normal Probability Distribution Find the probability value of P(Z >1.45) P(Z<-0.4) P(-0.4<Z<1.45) P(1.15<Z<2.33) P(Z<1.28) P(Z>-1.04)

Con’t 7. The examination results of a large group of students in statistics are normally distributed with a mean of 40 and a standard deviation of 4. If a student is chosen at random, what is the probability that his score is Below 30? Above 55? Below 42? Between 35 to 45? Between 33 to 50?

Con’t The efficiency rating of 400 faculty members of a certain university were taken and resulted in a mean rating of 78 with a standard deviation of 6.75. assuming that the set of data are normally distributed, how many of the faculty members have an efficiency rating of Greater than 78? Less that 78? Greater than 85? Between 75-90?

Assignment no. 8 Find the area under the following condition Between the -2.02 and 1.01 To the right of 1.62 To the left of 0.56 Between 0.65 and 1.18 Between -2.09 and -0.78 II. In a reading ability test, with a sample of 120 cases, the mean score is 50 and the standard deviation is 5.5.

Con’t a. What percentage of the cases falls between the mean and a score of 55? b. What is the probability that a score picked at random will lie above the score of 55? c. What is the probability that a score will lie below 40? d. How many cases fall between 55 to 60? e. How many cases fall between 40 to 49?

Probability and statistics(assign 7 and 8)

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