Every data set has an average and a standard deviation, given by the following formulas,

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1 Discrete Data Sets A data set is any collection of data. For example, the set of test scores on the class s first test would comprise a data set. If we collect a sample from the population we are interested and then measure the characteristic we are interested such as height or weight, then we have created a data set. We can also create a data in the following manner. Say we roll a pair of dice 10 times and count the number of times a seven came up. We record that number and repeat this experiment nine more times. We have now created a data of ten numbers, each number being the number of times a seven came up. Note that although there are ten numbers in the data, some of the numbers might be the same. Data can be discrete or continuous in nature. A discrete number is a number that can be counted. For example, we can count the number of pennies in a jar, leave on a tree or the number of times we rolled a seven. A continuous number results when we measure something. We can never give the exact value of the measurement because we are limited by the accuracy of our measuring device. For example, let s say the we wish to measure the distance that a bullet travels. We might be able to measure the distance to the nearest tenth of a meter, but we could never determine the distance to seven decimal places. That s why any continuous measurement is just an approximation of the actual result. Once again, anytime we use a measuring device, e.g. a ruler, clock, scale, etc. we are creating continuous data. In this section we are going to study discrete data sets. Next week, we will study continuous data sets. Every data set has an average and a standard deviation, given by the following formulas, ( ) Fortunately for us, we do not need to know how to use these formulas. We don t even have to know the formulas. We have Excel to do that for us. Type the data into Excel in a column, say column A. Then, from the stat menu, select AVERAGE: 14

2 How to actually do this in Excel will be explained in class. In a similar fashion, we can calculate the standard deviation by using STDEV.S, As an example, let s say that upon rolling the dice ten times we got the following number of sevens, The average number of times we got a seven would be 2 and the standard deviation would be , 15

3 Look at the numbers in the spreadsheet above. Notice that we did not get numbers like 5 times, 6 times, etc. even though we rolled the dice 10 times. That s because the probability of getting such numbers is very small. For example, the probability of getting a seven six or more times would be Very small indeed. Later we will see how this number was determined. A table showing all the possible number of times a seven can occur when rolling the dice 10 times, and a second set of numbers showing the probability of getting exactly that number of sevens is called a probability distribution. Here it is for the case we have been examining, X P(X) E E E-08 The numbers in the second have been rounded to four decimal places. The last three numbers require an explanation. They are written using scientific notation. Scientific notation is a shorthand way of writing either very big or very small numbers. If the sign following the E is negative then it s a very small number. Let s take a look at the last entry, 1.65E-08. First you take the number part, 1.65 and write eight zeros in front of it, , eight because that s the number at the end of the character string. Then you move the decimal point until it s to the right of the leading zero, That s the decimal equivalent of 1.65E-08. Pretty small, isn t it? The first row of numbers are called random variables. A random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure. In other words, given the example above, until we have rolled the dice ten times, we don t know how many sevens we are going to get. The actual number will be determined by chance. For example, according to the table above, the probability that we will get exactly 4 sevens is The first thing we need to understand is the difference between finding the probability that X equals some number versus X equaling a range of possible numbers. Let s consider ( ) Here we are 16

4 asking, what is the probability that out of ten rolls, we get exactly 3 sevens. The answer is Now contrast that with ( ) Here we are asking, what is the probability that X is some number between 1 and 3. Another ways of saying this is asking what is the probability that X = 1 or X = 2 or X = 3, i.e. if we roll the dice ten times, what is the probability that we get 1 seven, 2 sevens or 3 sevens. The answer is obtained by simply adding up the corresponding probability for each of the numbers, This is a good time to make sure we understand English. Yes, I know, we all think we understand it, but when it comes to mathematics, many of us really don t have as strong a grasp as we think. First, consider the phrase at least. At least means greater than or equal to, aka no less than. For example, given the set of possibilities above for the dice rolls, at least six means the set,. At most means less than or equal to, aka no more than. At most four means the set Greater than or more than means greater than some number but not including that number. For example, greeater than six would be the set Less than means less than some number but not including that number. Less than four is the set Finally, equals or is, is just one number. Equals 4 is the set This isn t going to mean much to you until you encounter these words in a homework exercise and proceed to get it wrong, so keep this page handy somewhere so you can refer back to it easily. Binomial Distribution The probability distribution shown above has a special name. It is called the Binomial Distribution. In the example above, we rolled the dice 10 times. Each time we rolled the dice was a single trial, and altogether there were ten trials. The event we were looking for was rolling a seven. The binomial distribution arises whenever the following four conditions are met: 1. The outcome of a single trial has only two choices, black or white, yes or no, seven or not a seven. 2. The probability of getting a success, e.g. getting a yes answer or rolling a seven, does not change from one trial to the next. Every time we roll the dice, the probability of getting a seven (p = 1/6) does not change. 3. The result of each trial is independent of the other trials. Getting a seven on one roll does not change the chances of getting a seven on a subsequent roll. 4. Finally, there are to be a fixed number of trials, in this case,

5 If these conditions are met then we are dealing with a binomial distribution and the Excel stat tool, BINOM.DIST can be used to find probabilities. From the stat menu, you select BINOM.DIST and here it what it looks like, Using the tool properly is not trivial so we will break it down case by case. Case 1: We want to find the probability that X is exactly equal to some number. For example, let s say that we want to know the probability of getting exactly 4 sevens out of ten rolls. We have the following: Number_s: 4 Cumulative: 0 pay attention here, this is the only case in which this parameter is 0. Case 2: We want to find the probability that we will roll at most 4 sevens. Number_s: 4 we are finding the probability of rolling 0, 1, 2, 3, or 4 sevens Cumulative: 1 note, this is the only difference from above. Case 3:`We want to find the probability of rolling less than 4 sevens. Number_s: 3 we are dealing with discrete data here, and the next number strictly less than 4 is 3, so we are now finding the probability of 0, 1, 2, or 3 sevens. Cumulative: 1 Case 4 We want to find the probability of rolling at least 4 sevens. This means find the probability of rolling 4, 5, 6, 7, 8, 9, or 10 sevens. This requires special handling. We first use BINOM.DIST to find the probability of the complement of what we are trying to find. Recall from Probability that the complement are all those possibilities that are not part of the event whose probability we are trying to find. For example, the complete sample space for how many sevens we can get out of 10 rolls is, 18

6 The event we are trying find the probability of consists of the following possibilities, These numbers are all at least as large as 4. The complement of this set is, We first find the probability of this complement, i.e. we find ( ) Number_s: 3 Probability_s: 1/6 Cumulative: 1 this is the probability of rolling a seven on any one roll or trial This will give the probability of the complement of what we wanted to find, so recalling from our study of Probability, the probability of an event is one minus the probability of the complement. Therefore, to find the probability of the event, at least 4, we first find the probability of at most 3 and subtract that result from 1.0. Worked Example We ll run through the example above. To find the probability of rolling at least four sevens, we first find the probability of its complement, the probability of rolling at most 3 sevens: We see that this probability is To find the probability of rolling at least four sevens, we simply subtract this result from 1.0, ( ) ( ) 19

7 Case 2: We want to find the probability that we will roll more than (strictly greater than) 4 sevens. This case is very similar to Case 4 above. The only difference is that here we are actually asking for the probability of rolling 5 sevens or more, because that amount is strictly greater than 4. As above we first find the probability of the complement, 4 sevens or less, Number_s: 4 we are finding the probability of rolling 0, 1, 2, 3, or 4 sevens Cumulative: 1 note, this is the only difference from above. We then subtract this result from 1.0 to get P(X > 4). This is the end of Week 2. You will get more practice by doing the exercises in MyMathLab. 20