 As-salamu alaikum. Welcome to Virtual University of Pakistan. My name is Ms. Swaleha Nagmi Habibullah and, inshallah, I will be with you in this course for a span of about 15 weeks. This is the third semester of your BCS program. And in this semester, I will be delivering to you the course on statistics and probability. The objective of this course is to inculcate in you an attitude of statistical and probabilistic thinking and to give you some very basic techniques in order to apply statistical analyses to real-world situations. Statistics haikyachis. It is that science which enables you to draw conclusions about various phenomena on the basis of real-life data. Real-life data that you collect in a scientific manner on a sample basis. It is a tool for data-based research. It can also be called quantitative analysis. Statistics students haik asha subject hai jiski pohot zyada application hai. Duniya kya koi shoba aap lele je. Koi shoba jisme aapne scientific tarike se conclusions draw karni ho based on real-life data. You will be applying statistical techniques. Agriculture, anthropology, astronomy, biology, economics, engineering, environment sciences, geology, genetics, medicine, physics, psychology, sociology, zoology. Virtually every single subject from anthropology to zoology, A to Z. Statistics can be divided into two broad areas. Descriptive statistics and inferential statistics. Descriptive statistics wo hai jisme aap, wo data jo aap initially collect karthi hai usko aap describe karthi hai. Usse data ko describe karthi hai. Sbse pehle aap usko summarize karthi hai. Ek ambar hota hai. Humare paas numbers ka, figures ka, data ka. Usko hum, sbse pehle compile karthi hai aur summarize karthi hai. Humare paas tables banjati hai. Hum uski diagrammatic aur graphic representation karthi hai. Humare paas mukhlif kasam ke bohot se diagrams available hai. Bar charts hai, pi charts hai, histogram. And we will be discussing this at length in our upcoming lectures. Aap ussi agar bo quantitative data hai to aap uski, uski average value, average compute karthi hai. Usme jo variability pae jati hai, jo sari values ek baraabar nahi hoti, vari kar rahi hoti hai. Usse aap measure karthi hai. Ye sab kuch descriptive statistics ke under aata hai. Inferential statistics wo hai ki jisme aap iss data ko, jisko aap ne collect kia aur compile kia aur summarize kia. Usse aap istimal karthi hai to draw conclusions about that phenomenon about which you have collected that data. You see basically the data that you have collected that is only a sample out of a large population. And inferential statistics gives you a set of techniques, a set of tools by which you can utilize the sample data to draw conclusions about the population. Ye jo conclusions aap draw, chunke ye real life phenomena ke baare me hai. Isliye they are going to be probabilistic in nature. They are not going to be deterministic. Iski andar chunke randomness ka element bohat zyada act kartha hai. Isliye aap jo kuch bhi kahenge, aap you will be saying that in terms of probability. Aap yeh kah sakenge ke iss baat ka imkaan pachanwe fiisad hai. Ke aisa hai, jo mai conclusion draw kar rahi ho ya kar raho hoon. Ye aisa hone ka imkaan pachanwe fiisad hai. Lekin aap yeh nahi kah sakenge ke jo kuch hum kah rahe hai, we can be absolutely certain about it. Miri iss baat se aapko andaza ho gya hoga ke ye jo subject hai statistics. Iska probability theory ke saath, chole davan ka saath hai. Probability is the foundation on which the entire structure of inferential statistics rests. Aaye, aap me aapko iss course ki text book ke baare me batadhoon. The book that we will be using is Introduction to Statistical Theory by Prof. Sher Mohammad Chaudhry and Dr. Shahid Kamal. This book is in two volumes. Volume 1 contains chapters 1 to 13 and it covers the various concepts of descriptive statistics as well as probability theory and volume 2 which starts from chapter 14. It will be giving you various concepts relating to inferential statistics. Yeh to aapki text book hogi jo aap throughout this course isthimal karenge. Lekin iss ke ilawa mai recommend karungi ke aap aur bhi bohot sii kitab me jo ke bazaar mein dastiyab hai. Both local and foreign aap unko bhi zorur consult kareng. Kyunke jitni zyada kitab me aap parenge inshallah utna hi aapka horizon widen hoega. Let us now start our discussion of the subject of statistics. Yeh love statistics kahaan se aaya? Mai aapko thoda size ka ek background dena chaati. Actually this word is derived from the Latin word status S-T-A-T-U-S meaning a political state. And as such statistics this word statistics originally meant information that was useful for the state. For example, information about the sizes of the population and the armed forces. Lekin aap aaj ke doar mein yeh jo loves hai statistics iss ke teen waze mani hai. In the first instance statistics means data, numerical facts which have been systematically arranged. Yani adado shumar iss sense mein it is used in the plural sense. Statistics of births and deaths, statistics of crime, unemployment statistics and so on. In the second sense the word statistics pertains to that subject that we are going to discuss in this course. That science, that discipline which enables us to draw conclusions about various phenomena on the basis of real data collected on sample basis. Jaisa ke main aapko pehle bataya yeh wo subject hai wo science hai. Jiske zariye hum real life phenomena ke baare mein bohot se conclusions draw kar sakti hai in a scientific way. And then there is a third meaning of the word statistics. Yaha pe istra samjiye ke ek love hai statistic. Yani eske baare sirf statistic. And that means any quantity such as mean, median, mode, proportion, any quantity that you compute from a sample. Ek sample aapne draw kiya. For example, students ki heights aapne collect ki. Aur aap interested hai ke mean height, yeh jo bhi students hain ki mean height kya hai. Wo jo mean aapne kaalingi that is a statistic. Phir aap kahte hain ke hum yeh bhi jaana chahate hain ke in students me se, kitne urdu medium schools se aaye hain aur kitne English medium schools se to aap uska proportion nikal na chahate hain. Ke proportion of students who come from urdu medium schools. To yeh jo proportion aapne kaalingi that is another statistic. To iss tara se jo aap mukh talif statistics compute keringi. This is the third meaning of this word in this sense. Jitni guftabhu main abhi tak ki usse aapko andaza hogya hogya. Ke statistics ek aisa subject hai which deals with aggregates. It deals with samples, populations. It deals with collection of individuals and objects. It is not concerned with just any one particular individual. For example, agar aap ek college ki baat karein. Agar aap yeh jaana chahate hain ke iss college me jo students admit hote hain. Unka jo background hain, wo kya hain. Unka jo urdu medium schools se hain, bohat zyada English medium se. To aap wo jo proportion nikalingi that will be giving you an idea of this phenomenon as a whole. Kisi ek particular student ki baat nahi ho rahe ho ghi. That will be an overall indicator. Or yeh ek statistics ka ek baat basic characteristic hai. That it is that science which deals with aggregates and not about particular observations. Another very important point is that statistics deals with those aggregates which are subject to a number of random causes. For example, the height of an individual, agar aap height ki baat karte hain. It is affected by so many things. The genetic element, the diet, the kind of diet this person takes, the climate of that country and so many other things. Isi tara agar aap marks ki baat karein, marks of a student that will have to do with his IQ, the kind of training, the kind of education that he has had proper unki tutoring hui hai kaini hui hai. And so many other factors. So, this is a very important point. Ki chunke hain, hain real life phenomena se deal karte hain. Isli hain kisi bhi, kisi bhi statistical problem mein, jis variable ke saar deal kar rahe hote hain, ush cheez, wo cheez, bohat se aur cheezon ke saath related hote hain aur depend kar rahe hote hain aur cheezon ke upar. Students, I have been using the word data in this discussion. I think I have used it many times. Let us see what this word implies. Data kya cheez hain? Actually, I think it is used in different senses, in different context. Lekin pehle hum dekthin ke iska origin kya hain? It is Latin for those that are given. Latin mein iska mafum hain, those that are given. Iska jo singular form hain, that is datum. Aap kahinge ki ye kya hain? Ye to hum bilkul use nahi karte. Lekin it is data actually is the plural of datum. Datum hoega ek piece of observation aur data, a number of pieces of observation. Jaisa ke main hai pehle kaha, ke ek sample, ek population, not just one individual. So, lukbe lubab sari baat ka ye ke hum ye kaisakte hain ki data can be thought of as the results of observation. Let us consider some examples. Statements given to a police officer or a physician or a psychologist during an interview, this is data. The correct and incorrect answers given by a student on a final exam, that is data. The time required by a runner to complete a marathon, that is data. The number of errors committed by a baseball team in nine innings of play, this is going to be another set of data. And of course, data is obtained in the course of scientific inquiry. The positions of artefacts and fossils in an archaeological site. The number of interactions between two members of an animal colony during a period of observation. The spectral composition of light emitted by a star. Ye mukhli f ja main examples aapko diye, iss se shayad aapko andaza hoa hoga ke there are different types of data. And actually, it is extremely important that we be able to classify that what type of data is it that we are going to deal with in any particular research problem. Isliye aaye abis tisko ham zara formally discuss karte hain. Data, that is basically of two types. Quantitative and qualitative. Quantitative data se murad hain ke jis main aapki jo ris jo result se aapki observation ke wo numerical form mein. For example, height, weight, blood pressure, marks, income aur koi bhi aasee situation ke jis main aapki jo observations hain wo number form mein. Aapki height, 5 for 10 inch ho sakti hain. Aapka weight 150 pounds ho sakti hain. Ye jo numbers hain, 150, 5, 4 to 10 inch. This constitutes quantitative data. Qualitative data wo hain jo iss tari ke se numerically express nahi ho sakti hain. For example, agar aap kisi shak se yeh puchein ke are you married? Now, you know he or she can give you different answers. Yes, I am married. No, I am not married. Or he can say I am divorced. Aap ye jo jawabat hain, inko aap us tari ke se number form mein express nahi kar sakti hain. Jis tara se aap height yeh weight ko kar sakti hain. Ishi tara eye color, hair color, color of your dress. Color ko aap 1.7 ya 2.3 is tari ke se to express nahi kar sakti hain. Ye jo main aapko misaalein deen, in main aapko bakhubi andaza hain. Ke when you are trying to collect this kind of data for a sample of people, you will be getting different replies from different people. Yani wo jo jawab aapko masool ho raha ho ga, wo veri kar raha ho ga. Har shaksto ekhi jawab nahi dega. Kisi ke height 5 foot 10 inch hain, kisi ke 5 foot 5 inch hain. Ye jo variability hain, ishi ke bhaja se jo bohot important concept yaha pe aata hain. That is the concept of variable. A variable is that quantity which varies from individual to individual. Variables are of basically two types according to what I just said. The quantitative variable and the qualitative variable. Quantitative variable jaisa ke main pehle bataya, height, weight, blood pressure, body temperature, atmospheric pressure, income and so on and so forth. Qualitative variable, as I said earlier, color of your dress, the amount of satisfaction you have with a certain facility that the government is providing to you. Satisfaction ko bhi aap us tarika se to numerically express nahi kar sakte na, jis tra ke aap height your weight ko kar sakte hain. Aap yeh kahenge ke I am very satisfied. Ye aap kahenge ke I am not that satisfied. I am not that satisfied ke ye matlab to nahi hain ke satisfaction is 1.7. Sari guftu ko sevazhe hua ke it is extremely important that we categorize and classify the kind of data that we are going to collect and the kind of variable that we are going to deal with. Jaisa ke aap screen peh dekh rahe hain. Variables ki doh kisme hain. Qualitative variable and quantitative variable. Qualitative variable to jaisa ke main patahi chukhi hoon. Ishi kisam ki cheese hain. Jisme aap usko numerically express nahi kar sakte. Quantitative variable jo hain uski mazeed doh kisme hain. The continuous variable and the discrete variable. Aaye humin ko ek ek karke discuss karte hain. Continuous variable wo hain jaha pe aap measurement karte hain. Jab aap kisi ki height measure karte hain. Ya kisi ka weight measure karte hain. Ya temperature measure karte hain. In tamam situations mein aap ka jo variable hain that is a continuous variable. Isko continuous variable kyu kahte hain. Aaye ish baat ko ek example ki through samajne ki koshish karte hain. Supposed karein ke aap ek shaks ki height measure karna chaat hain. Ab uski jo measurement hain ke how accurately you are measuring his height. That depends on the accuracy of your measuring instrument. Agar aap ka measuring instrument bohat hi crud sa hain toh hosakta hain ke aap you may be able to measure only correct to the nearest half of an inch. For example, aap kahe ke iski height 5 feet 6 and a half inches hain. Lekin in reality wo usse thoda sa tall hain. He is a little taller than 5 feet 6 and a half inches. Lekin jo ka aap ka wo measuring scale jo hain. Usse sref aadhe aade inch pe wo nishan hain. Isli aap kaheenge ke it is 6 and a half inches. Agar ek aap usse bahter instrument le hain. Jispe wo nishan jo hain wo half inch pe nahin hain. Balke har one tenth of an inch pe hain. Then you will be able to measure his height correct to the nearest tenth of an inch. So, you might then say that his height is 6.7 inches, i.e. 5 feet and 6.7 inches. Jab ke se pehle jab aap ke paas wo instrument tha jisme sref aade inch ke faas lipe wo nishan tha. Uswakta aap karethe 6 and a half inches i.e. 6.5 inches. Aap ek bahter instrument le hain to uski wajah se wo zyada refined. Aap ka jo measurement hain wo zyada refined hoge. Aap aap tasavur karein ke aap usse zyada bahter ek aur usse bhi bahter instrument le hain. Jisme wo jo nishan hain wo har tenth pe bhi nahin. Balke it is such an instrument that you are able to measure correct to two decimals. Goya aap kaise kain ke his height is 5 feet and 6.73 inches. Then you can think of another measurement instrument by which you are able to say that his height is 6.731 inches. Is tarike se agar aap theoretically is tis pe gaur karein to aap note karein ke there is no end to it. A person's height can be 6.73 inches. 5 feet to of course that goes without saying. But then it can be 6.731 inches. But then it can also be 6.73129479 inches. There can be an infinite number of decimal points, decimal places after the decimal point. Is hawale se agar haan dekhayin mukhtle flogon ki heights aap mejar karein hain. Aur possible hain ke koi 2 heights ke darmean jo gap hain poh pohth hi minimal ho. Po gap itna kum ho ke wo sathmi yaatmi ya bismi decimal place pe jake wo gap aah hain ho. Jo ke 2 askhaz ki heights ke darmean ho. In other words, height is varying in a continuous manner. Real line ke upar pehle shakhs ki height agar ek particular point pe hai to usre shakhs ki height usre agle point pe aapko nazar aaye. Is sathmiyaar aaye is sathmiyaar aaye ke aapko maeshus ho ke yein donoh ke darmean koi gap nahin. This is the concept of continuous variable. Yehi wajah hain ke hain mejreable variable ko continuous variable kehte. Iske baraks, dosera variable, the other type that we will be discussing is the discrete variable as stated earlier. discrete variable means discontinuous variable. This is the variable in which numbers are not connected. There will be gaps between various values and that is why it is called a discrete variable. The easiest example of this is that in any situation in which you have to collect your data through counting, not through measurement, but through counting, that is a situation where you are dealing with a discrete or a discontinuous variable. Yes, it is like a joke, but this is exactly the concept of discrete variable. Your four variable cannot assume the value of 4.25. It has to be either 4 or 5. So, as I said just now, this is the easiest and the best example of a discrete variable. Speaking of variables, measurements and counts, there is an important concept and that is the concept of measurement scales. We have four different kinds of measurement scales, the nominal scale, the ordinal scale, the interval scale and the ratio scale. Nominal scale kya cheez hai? The classification or grouping of the observations into mutually exclusive qualitative categories or classes is said to constitute a nominal scale. For example, students are classified as male and female. Similarly, the color of your dress could be classified as white, blue, gray or black. Ab zahir hai, students, case ke andar, there is no ordering of the various categories and this kind of a scale is called nominal scale. Iske baraks, there is a property of ordering or ranking of the measurements. For example, the performance of students can be rated as excellent, good, fair or poor. Ab aap note kar rahe hain ke yaha pe, there is a relation of order. Excellent is better than good, good is better than fair and fair is better than poor. Both these scales that we have discussed just now, they do not possess a constant interval between two values. For example, I gave you the example that rainfall can be heavy or moderate or light. Ab iske matlab nahi hai ke heavy aur moderate ke darmean jitna fasla hain, utna hi fasla moderate aur light ke darmean hain. Iske baraks, ab jodho scale aap hum discuss karne wale hain, the interval scale and the ratio scale, unke andar ye khasiyat pahe jaati hain, ke aap you have constant interval between any two values of your variable. The interval scale is a measurement scale possessing a constant interval size, but not a true zero point. Temperature measured on either the Celsius or the Fahrenheit scale is an outstanding example of the interval scale. Jab aap Fahrenheit me mesher karte hain, to aap kate hain ke it is 104 degrees at temperature. Jab aap centigrade me wohi temperature me mesher karte hain, to aap 40 degrees kate hain, 40 degrees centigrade is equal to 104 degrees Fahrenheit. Lekin in donohi scales me aap agar gaur karein to aap ko idea yeh hoga ke jo zero hai uska jo zero hai that is not the true zero point. Yani agar zero degree Fahrenheit ki baat ho rahe hain, to iska matlab yeh nahi hai ke haraarat ki mik daar sifar hain. Aap ka temperature minus 10 degrees bhi ho sakta hain. Aur jiswak minus 10 degrees hai uswak bhi haraarat ki ek mik daar faza me maujood hain. This goes for the Fahrenheit scale as well as the centigrade scale. So yeh jo baat hain jo main aajya se pehle kaha ke jab aap ke scale me constant interval to maujood hain. Yani 40 degrees aur 41 degrees ke darmean utna hi fasla hain, jitna ke 41 or 42 ke darmean hain. Lekin uska jo zero hain that is not a true zero, aise scale ko kate hain, interval scale. Racial scale wo hain ke jisme zero, jisko aap zero kate hain that is the true zero. Yani for example height, agar aap kaheen ke kisi cheez ki height zero hain, to iska yeh matlab hain ke uski bilkul koi height nahi hain us cheez ki. This is of course a theoretical point, kyunke aap ki koi bhi material cheez jo hogi uski toh ek particular height hogi. Chahi wo bahot hi yani come chahi kyun naho. Lekin theoretically aap is baat ko understand kr sakte hain. Ke jab aam kate hain ke zero height, it means ke it is zero and then you can have other heights. But also the other example is of age. Agar aap time of birth se start karein toh an moment of birth jo hain uswakta aap kaisakte hain that the age is zero. Zero year, zero month, zero day aur uske baat jayse jayse din guzarthe hain. You can say that the child is one day old and two days old and so on. Let us now consider an example that illustrates the various types of variables and scales that may be encountered in a real life study. Chemical and manufacturing plants sometimes discharge toxic waste materials such as DDT into nearby rivers and streams. These toxins can adversely affect the plants and animals inhabiting the river and the riverbank. A study of fish was conducted in the Tennessee River in Alabama and its three tributary creeks the Flint Creek, the Limestone Creek and the Spring Creek. A total of 144 fish were captured and the following variables measured for each one. Number one, the river or the creek from where the fish was captured. Number two, the species of the fish that is channel catfish, largemouth bass or smallmouth buffalo fish. Number three, the length of the fish measured in centimeters. Number four, the weight of the fish measured in grams. And number five, the DDT concentration in the bodily system of the fish. We would like to classify each of the five variables measured as quantitative or qualitative. Also, we would like to identify the types of measurement scales for each of the five variables. Students, the variables length, weight and DDT concentration are quantitative because each one of them is measured on a numerical scale. Length in centimeters, weight in grams and DDT in parts per million. Also, please note that all three of these variables are being measured on the ratio scale. Now, why do we say that all of them are being measured on the ratio scale? Students, whenever we speak about the weight of an object, obviously if our measuring instrument reads zero, this would mean that the object being measured has zero weight. In other words, it is weightless and in this sense the zero would be a true zero. An exactly similar argument holds for the length of an object. So, I hope that it is clear to you that weight and length of the fish are being measured on the ratio scale. As far as the DDT concentration in the bodily system of the fish is concerned, obviously if there is absolutely no DDT in the fish, then the DDT concentration reading will be zero and this particular zero reading will be a true zero. As explained, the three variables length, weight and DDT concentration are quantitative variables measured on the ratio scale. In contrast, the other two variables that is the river or the creek from which the fish were captured and the species of the fish are qualitative variables and students, I hope that you will agree with me that both of these variables are being measured on the nominal scale. Students, why do we say that both of these variables are qualitative? The reason is that in case of the river or the creek from which the fish were captured as well as in the case of the species of the fish, obviously these cannot be measured quantitatively, they can only be classified into categories. Also, obviously there is no ordering of the various categories such as the various types of species of the fish that we have and since there is no ordering present, therefore each of these two variables pertain to the nominal scale. Now, students having had a fairly detailed discussion regarding the various types of variables and scales that we encounter in real life, it should be obvious that the statistical methods for describing, reporting and analyzing data depend on the type of data that has been collected. We will be discussing a number of such methods in the forthcoming lectures of this course. A few minutes ago, when I talked about continuous variables, I told you that your measuring instrument depends on the refinement of how many decimal places you can measure it. So, from here, the concept of errors of measurement begins. When you say that a student, we have measured his weight and it is 60 kilogram, so in reality it is somewhere between 59 and a half and 60 and a half kilogram. If you say that I have measured correct to one decimal and his weight is 60.3 kilogram, so in reality it is somewhere between 60.25 and 60.35 kilogram. So, the point is that there will always be some difference between the true value and the value that you have measured. And this difference is called error of measurement. Errors, you can have biased errors or random errors. Biased errors are when your measuring instrument is faulty. You are saying that I have a guzz, but in reality it is less than one inch from a guzz. If you do not measure five guzz clothes, then at the end of it five inches will be less. These kinds of errors are cumulative. Every time, the error of one inch will be added. And there are random errors that you are doing wrong intentionally or in a faulty instrument. Your instrument is correct, you are measuring it. If one person will measure the height of a different off-road with the same instrument, then sometimes it will be a little more than its true height. So, these errors are compensating in nature. The positive errors will cancel out with the negative errors in the long run. These errors are purely due to chance factors and that is why they are also called chance errors or accidental errors. Students, as explained in the earlier part of this lecture, the goal of the science of statistics is to draw conclusions or inferences about populations on the basis of samples. So, let me now explain to you some fundamental and vitally important points pertaining to the concept of statistical inference. A statistical inference is an estimate or prediction or some other generalization about a population based on information contained in a sample. That is, we use the information contained in the sample to learn about the larger population. A population is the collection of all individuals, items or data under consideration in a statistical study and a sample is that part of the population from which information is collected. And the picture that you now have on the screen displays the relationship between a population and a sample drawn from the population. Let us now concentrate on the five elements of an inferential statistical problem. The five elements are a population, one or more variables of interest, a sample, an inference and a measure of reliability. Students, in order to explain the concept of reliability, I would like to draw your attention to the fact that making an inference about the population from the sample is only part of the story. We also need to know its reliability that is how good the inference is. A measure of reliability is a statement usually quantified about the degree of uncertainty associated with statistical inference. The point to be understood is that the only way we can be certain that an inference about a population is correct is to include the entire population in our study. However, in many, many practical situations because of the resource constraints such as insufficient time or money at our disposal, we are not able to work with the whole population and we need to base our inferences on just a portion of the population that is a sample. Consequently, whenever possible, it is important to determine and report the reliability of each inference made. As such, reliability is the fifth element of inferential statistical problems. Let me explain some of the concepts that I just presented with the help of an example. A large paint retailer has had numerous complaints from customers about underfilled paint cans. As a result, the retailer has begun inspecting the incoming shipments of paint from suppliers. Shipments with underfilled problems will be returned to the supplier. A recent shipment contained 2,440 gallon size cans, the retailer sampled 50 cans and weighed each one on a scale capable of measuring weight up to four decimal places. Properly filled cans weigh 10 pounds. Now, in this problem students, we would like to describe the population, the variable of interest, the sample, the inference and last but not the least a measure of the uncertainty of our inference. In other words, the reliability of our inference. So, let us address the various questions one by one. First and foremost students, it is obvious that the population in this example is the set of units of interest to the retailer, which is the shipment of 2,440 cans of paint that he received. The weight of the paint cans is of course the variable the retailer which is to evaluate. The sample is a subset of the population and in this particular example, it is the set of 50 cans that were inspected by the retailer. As far as the inference is concerned students, it involves the generalization of the information contained in the sample of the paint cans to the population of the paint cans. In particular, the retailer wants to learn about the extent of the underfill problem in the population and this could be accomplished by finding the average weight of the cans in the sample and to use it to estimate the average weight of the cans in the population. Students, as far as a measure of the uncertainty of our inference is concerned, the point to be noted is that using statistical methods, we can determine a bound on the estimation error. First, what do we mean by estimation error? Students, the difference between the average weight of the sample and the average weight of the population of the cans, this difference represents the estimation error or when we bound on the estimation error, we are simply talking about a number that the estimation error is not likely to exceed. This bound is a measure of the uncertainty of our inference or in other words, the reliability of our statistical inference. With reference to this example, how are we going to measure the reliability of the inference concerning the weights of all the paint cans? To answer this question, note that the weights of the 50 paint cans which are in the sample, their average is not going to be exactly equal to the average weight of the entire population. For example, if our sample yields a mean weight of 9 pounds, it does not follow that the mean weight of the entire population of the cans is also going to be exactly 9 pounds. Nevertheless, we can use sound statistical reasoning to ensure that our sampling procedure will generate an estimate that will be almost certainly lying within a specified limit of the true mean weight of all the cans. For example, such reasoning might assure us that the estimate of the population mean from the sample is almost certainly within 1 pound of the actual population mean. This implies that the actual mean weight of the entire population of the cans is almost certain to lie between 9 minus 1 that is 8 pounds and 9 plus 1 that is 10 pounds. This interval of 8 to 10 or in other words, 9 plus minus 1 pounds represents a measure of reliability for our inference. Students, let us now review what we have discussed in today's lecture. I discussed with you the nature of the science of statistics and the importance of statistics in various fields. Also, we discussed some technical concepts such as the meaning of data, various types of variables, various types of measurement scales and the concept of errors of measurement. I will be dealing with the concept of sampling and I will be talking about the difference between random and non-random sampling. We will be talking in some detail about simple random sampling and also I will give you a brief introduction to some other techniques of random sampling. Also, we will be talking about various methods of data collection. In other words, students, inshallah next time you will begin your journey in a discipline about which it has been said that statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. Best of luck and until next time, Allah Hafiz.