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Statistics and Probability Solved Assignments - Semester Fall 2008

Assignment 1 Question 1: (Marks: 16) Write the short notes on the following: Solution: i) Variable and constant Variable: A measurable quantity which can vary from one individual or object to another is called a variable. Constant: A quantity which can assume only one value is called a constant. ii) Continuous and Qualitative variable Continuous variable: A variable which can assume an infinite number of values with in a given range is called a continuous variable .e.g. weight, height, length Qualitative variable: A variable that can not be expressed in numerical form but shows the presence or absences of some attribute is called qualitative data. For examples marital status, religion, sex etc iii) Population and sample Population: The collection of individuals or objects having some common measurable characteristics. Sample: A representative part of the population is called sample. iv) Primary and secondary data Primary data: the data published or used by an organization which originally collected by them is called primary data. Secondary data: The data published or used by an organization which they did not collect originally is known as secondary data. v) Sampling errors and non-sampling errors Sampling Error: The difference between the estimate derived from the sample and the true population value (the parameter) is technically called the sampling error. Non-sampling errors: There are certain errors which are not attributable to sampling but arise in the process of data collection, even if a complete count is carried out. Such errors are known as non-sampling errors. vi) Multiple bar chart and component bar chart Multiple bar chart: This chart is simple an extension of simple bar chart. In this chart, grouped (adjacent) bars are used to represent related set of data. Each bar in a group is shaded differently for distinction. Component bar chart: This chart is consisting of horizontal or vertical bar which are subdivided into two or more parts. This chart is used when it is desired to present data which are subdivisions of totals. vii) Frequency distribution Frequency distribution: A frequency distribution is a tabular arrangement of data in which various items are arranged into classes and the number if item falling in each class (called class frequency) is stated. viii) Measure of central tendency Measure of central tendency: A single value which represents the whole data is called the average value. Since the average tends to lie in the center of data/distribution, it is also called measures of central tendency. Question 2: (Marks: 4) State which of the following represent qualitative data and which one of them represents quantitative data. i) Religion of the people of the country (qualitative data) ii) Fee of VU students (quantitative data) iii) Majority of population like Geo TV (qualitative data) iv) Inches of rainfall in Lahore city during the last year (quantitative data) Note: Question 3: (Marks: 10) The following data are the weights in pound of 42 students of Virtual University. Construct a stem-and-leaf display of the data. 135 157 152 189 135 164 146 144 154 153 150 158 168 165 140 132 140 126 146 135 144 147 138 173 161 125 136 176 142 145 149 148 163 147 135 142 150 156 145 128 154 171 Solution: The stem-and-leaf display of the data is shown below. Stem Leaf 12 5 6 8 13 8 2 6 5 5 5 5 14 4 9 6 0 7 8 4 6 2 0 5 2 5 7 15 0 7 8 2 4 3 0 6 4 16 4 8 3 5 1 17 6 3 1 18 9 Stem Leaf 12 5 6 8 13 2 5 5 5 5 6 8 14 0 0 2 2 4 4 5 5 6 6 7 7 8 9 15 0 0 2 3 4 4 6 7 8 16 1 3 4 5 8 17 1 3 6 18 9 Assignment 2 Question 1: a) What is the difference between Chebyshevs inequality and empirical rule (in terms of skweness)? Solution: Chebyshevs inequality and Empirical rule both tells us the proportion of data values that must lie within a specified number of standard deviation from mean. Chebyshevs inequality is a general rule for all symmetric and non symmetric distributions. But empirical rule is applicable only on the symmetric distributions. b) The share prices of a company in Lahore and Islamabad market during the last months are recorded below: Months Jan Feb March April May Jun July Aug Sep Oct Lahore 105 120 115 118 130 127 109 110 104 112 Islamabad 108 117 120 130 100 125 125 120 110 135 In which market, the shares prices are more stable? Solution: For the stability of market we have to check the Coefficient of variation for both cities, the city having less CV will show stability in its market. ( )( ) ( ) 33 . 8 11510132944115101150222= =||.|

\| == = = nxnxlahore Snxlahore x ( ) 24 . 7 10011533 . 8100 . . = = =xSLahore V C ( )( ) ( )( ) 48 . 8 10011909 . 10100 . .09 . 10 11910142628119101190222= = == =||.|

\| == = = ySIslamabad V CnynyIslamabad SnyIslamabad y By the comparison of coefficient of variations shows that there is more stability in Lahore stock exchange as compare to Islamabad. Question 2: a) Interpret standard deviation. Solution: The standard deviation is a very important concept that serves as a basic measure of variability. A smaller value of the standard deviation indicates that most of the observations in a data set are close to the mean while large value of S.D implies that the observations are scattered widely about the mean. b) The following data give the number of passengers traveling by airplane from one city to another in one week. 115 112 129 113 119 124 132 120 110 116 Calculate the mean and standard deviation and determine the percentage of class that lies between (i) o u (ii) o u 2 (iii) o u 3 . What percentage of data lies outside these limits? Solution: Let x represents the number of people traveling by airplane from one city to another in one week. Calculations for mean and standard deviations are given 119011910xxn= = = ( )222142096119 6.9710x xSn n| |= = = | |\ . Thus percentage of data lies between given limits: Interval Values within Interval %age of values within interval %age of values Falling Outside o u 119 6.97 125.97,112.03 = o u 2 119 2(6.97) 132.94,105.06 = ( )3119 3 6.97 139.91, 98.09u o = 113,115, 116, 119, 120, 124 110, 112,113, 115, 116, 119, 120, 124, 129, 132 All values 6100 60%10 = % 100 1001010= 100 40% nil nil Assignment 3 Question 1 Give the short answers of the following: I. What are moments? And why we use moments. II. What is meant by kurtosis? III. Lepto kurtic IV. Platy kurtic V. Normal distribution VI. Regression VII. Regressor VIII. Regressand Solution: What are moments? And why we use moments. Moments are central parameters, which are used for testing the symmetry and normality of the distribution. What is meant by kurtosis? The term kurtosis is meant to show the degree of peak ness of the distribution. Lepto kurtic: A distribution having a relatively higher peak is called Lepto kurtic distribution. Platy kurtic: A distribution, which is flat topped, is called platy distribution. Normal distribution: A distribution which is neither very peaked nor very flat, is called normal distribution or mesokurtic. Regression: It investigates the dependence of one dependent variable on the other independent variable. Regressor: The independent or the non-random variable is also referred to as the regressor, the predictor, the regression variable or the explanatory variable. Regressand: The dependent or the random variable is also referred as the regressand , the predictand , the response or the explained variable. Question 2: If distribution has mean 1403 and mode 1487, what can you say about the skewness? Solution: Mean = 1403 Mode = 1487 The distribution is negatively skewed, because Mean < Mode Question 3: a) Distinguish between permutation and combination. b) First four moments of a certain distribution about Y = 17.5 are 0.3,74,45, and 12125 respectively. Find out whether the distribution is Lepto kurtic or Platy kurtic. Solution: a. Permutation: A permutation is an arrangement of all or part of a set of objects in a definite order. The number of permutations of n distinct objects taken r at a time is !( )!nrnPn r= Combination: A combination is an arrangement of objects without regard to their order. The number of combinations of n objects taken r at a time is !!( )!nrnCr n r= b. First four moments about Y = 17.5 12340.3744512125mmmm' =' =' =' = Moments about mean: 122 2 13 3 2 1 132 44 4 3 1 2 1 10( ) 74 0.09 73.913 2( )45

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