Wiley Statistical Analysis with Excel For Dummies, 2nd Edition
Contents
1. Figure 1 4 is a worksheet of expenditures for R amp D in science and engineer ing at colleges and universities for the years shown The data taken from a U S National Science Foundation report are in millions of dollars Column H holds the total for each field and row 11 holds the total for each year More about column I in a moment Chapter 1 Evaluating Data in the Real World 2 SESS Figure 1 4 Expenditures for R amp D in science and engineering as A Aut F pag A B c D E F G l j K E 1 Field 1990 1995 2000 2001 Total Proportion 2 Physical Scienca 1807 2254 2708 2800 3569 3 Environme 1069 1433 1763 1827 6092 4 222 279 341 357 1199 5 515 682 875 954 3026 8726 12185 17460 19189 57560 253 370 516 582 1721 8 Social Sciences 73 1018 1297 1436 4454 3 Other Sciences 226 426 524 579 1875 10 Engineering 2656 3515 ASAT 4999 15717 u Total 16287 22162 20041 22723 101213 2 12 1B 14 15 16 7 18 19 20 2 23 ja X H ar Sheett Sheet Sheet3 J j m i I started with column H blank and with row 11 blank How did I get the totals into column H and row 11 If I want to create a formula to calculate the first row total for Physical Sciences one way among several is to enter Dy ah ine Se ne 23 ew into cell H2 A formula always begins with Press Enter and the total appears in H2 Now to put that formula into cells H3 through H10 the trick is to position the cursor o2. Excel A Marriage Made in Heaven
3. place This line of thinking leads us to conditional probability the probability that an event occurs given that some other event occurs For example suppose I roll a die take a look at it so that you can t see it and I tell you that I ve rolled an even number What s the prob ability that I ve rolled a 2 Ordinarily the probability of a 2 is s but I ve narrowed the field I ve eliminated the three odd numbers 1 3 and 5 as pos sibilities In this case only the three even numbers 2 4 and 6 are possible so now the probability of rolling a 2 is s Exactly how does conditional probability plays into statistical analysis Read on Inferential Statistics Testing Hypotheses In advance of doing a study a statistician draws up a tentative explanation a hypothesis as to why the data might come out a certain way After the study is complete and the sample data are all tabulated he or she faces the essential decision a statistician has to make whether or not to reject the hypothesis Chapter 1 Evaluating Data in the Real World 15 That decision is wrapped in a conditional probability question what s the probability of obtaining the data given that this hypothesis is correct Statistical analysis provides tools to calculate the probability If the probabil ity turns out to be low the statistician rejects the hypothesis Here s an example Suppose you re interested in whether or not a p
4. twice as hot as 50 degrees Kelvin This is because the Kelvin zero point is absolute zero where all molecular motion the basis of heat stops Another example is a ruler Eight inches is twice as long as four inches A length of zero means a complete absence of length Any of these types can form the basis for an independent variable or a depen dent variable The analytical tools you use depend on the type of data you re dealing with A little probability When statisticians make decisions they express their confidence about those decisions in terms of probability They can never be certain about what they decide They can only tell you how probable their conclusions are So what is probability The best way to attack this is with a few examples If you toss a coin what s the probability that it comes up heads Intuitively you know that if the coin is fair you have a 50 50 chance of heads and a 50 50 chance of tails In terms of the kinds of numbers associated with probability that s 2 How about rolling a die One member of a pair of dice What s the prob ability that you roll a 3 Hmmm a die has six faces and one of them is 3 so that ought to be s right Right Here s one more You have a standard deck of playing cards You select one card at random What s the probability that it s a club Well a deck of cards has four suits so that answer is 4 I think you re getting the picture If you wan
5. Chapter 1 Evaluating Data in the Real World In This Chapter Introducing statistical concepts Generalizing from samples to populations Getting into probability Making decisions New features in Excel 2007 Understanding important Excel Fundamentals New features in this edition J field of statistics is all about decision making decision making based on groups of numbers Statisticians constantly ask questions What do the numbers tell us What are the trends What predictions can we make What conclusions can we draw To answer these questions statisticians have developed an impressive array of analytical tools These tools help us to make sense of the mountains of data that are out there waiting for us to delve into and to understand the numbers we generate in the course of our own work The Statistical And Related Notions Vou Just Have to Know Because intensive calculation is often part and parcel of the statistician s toolset many people have the misconception that statistics is about number crunching Number crunching is just one small part of the path to sound deci sions however 0 Part I Statistics and Excel A Marriage Made in Heaven By shouldering the number crunching load software increases our speed of traveling down that path Some software packages are specialized for statisti cal analysis and contain many of the tools that statisticians use Although not marketed specifically as a statistical packag
6. articular coin is fair whether it has an equal chance of coming up heads or tails To study this issue you d take the coin and toss it a number of times say a hundred These 100 tosses make up your sample data Starting from the hypothesis that the coin is fair you d expect that the data in your sample of 100 tosses would show 50 heads and 50 tails If it turns out to be 99 heads and 1 tail you d undoubtedly reject the fair coin hypothesis Why The conditional probability of getting 99 heads and 1 tail given a fair coin is very low Wait a second The coin could still be fair and you just happened to get a 99 1 split right Absolutely In fact you never really know You have to gather the sample data the results from 100 tosses and make a decision Your decision might be right or it might not Juries face this all the time They have to decide among competing hypoth eses that explain the evidence in a trial Think of the evidence as data One hypothesis is that the defendant is guilty The other is that the defendant is not guilty Jury members have to consider the evidence and in effect answer a conditional probability question What s the probability of the evidence given that the defendant is not guilty The answer to this question deter mines the verdict Null and alternative hypotheses Consider once again that coin tossing study I just mentioned The sample data are the results from the 100 tosses Before tossin
7. cular attribute than a lower number One example is Moh s Scale Used since 1822 it s a scale whose values are 1 through 10 Mineralogists use this scale to rate the hardness of substances Diamond rated at 10 is the hardest Talc rated at 1 is the softest A sub stance that has a given rating can scratch any substance that has a lower rating What s missing from Moh s Scale and from all ordinal data is the idea of equal intervals and equal differences The difference between a hardness of 10 and a hardness of 8 is not the same as the difference between a hardness of 6 and a hardness of 4 Chapter 1 Evaluating Data in the Real World MBER es Interval data provides equal differences Fahrenheit temperatures provide an example of interval data The difference between 60 degrees and 70 degrees is the same as the difference between 80 degrees and 90 degrees Here s something that might surprise you about Fahrenheit temperatures A temperature of 100 degrees is not twice as hot as a temperature of 50 degrees For ratio statements twice as much as half as much as to be valid zero has to mean the complete absence of the attribute you re measuring A temperature of 0 degrees F doesn t mean the absence of heat it s just an arbitrary point on the Fahrenheit scale The last data type ratio data includes a meaningful zero point For tempera tures the Kelvin scale gives us ratio data One hundred degrees Kelvin is
8. ding speeds of all the children in this study What happens next I get to that in the upcoming section entitled Inferential Statistics Testing Hypotheses For now understand that the independent variable here is Method of Instruction The two possible values of this variable are New and Traditional The dependent variable is reading speed which we might measure in words per minute In general the idea is to try and find out if changes in the independent variable are associated with changes in the dependent variable In the examples that appear throughout the book I show you how to use Excel to calculate various characteristics of groups of scores Keep in mind that each time I show you a group of scores I m really talking about the values of a dependent variable Types of data Data come in four kinds When you work with a variable the way you work with it depends on what kind of data it is The first variety is called nominal data If a number is a piece of nominal data it s just a name Its value doesn t signify anything A good example is the number on an athlete s jersey It s just a way of identifying the athlete and distinguishing him or her from teammates The number doesn t indicate the athlete s level of skill Next comes ordinal data Ordinal data are all about order and numbers begin to take on meaning over and above just being identifiers A higher number indicates the presence of more of a parti
9. e Excel provides a number of these tools which is why I wrote this book I said that number crunching is a small part of the path to sound decisions The most important part is the concepts statisticians work with and that s what I talk about for most of the rest of this chapter Samples and populations On election night TV commentators routinely predict the outcome of elec tions before the polls close Most of the time they re right How do they do that The trick is to interview a sample of voters after they cast their ballots Assuming the voters tell the truth about whom they voted for and assuming the sample truly represents the population network analysts use the sample data to generalize to the population of voters This is the job of a statistician to use the findings from a sample to make a decision about the population from which the sample comes But sometimes those decisions don t turn out the way the numbers predicted History buffs are probably familiar with the memorable picture of President Harry Truman holding up a copy of the Chicago Daily Tribune with the famous but wrong headline Dewey Defeats Truman after the 1948 election Part of the statisti cian s job is to express how much confidence he or she has in the decision Another election related example speaks to the idea of the confidence in the decision Pre election polls again assuming a representative sample of voters tell you the percen
10. g the coin you might start with the hypothesis that the coin is a fair one so that you expect an equal number of heads and tails This starting point is called the null hypoth esis The statistical notation for the null hypothesis is H According to this hypothesis any heads tails split in the data is consistent with a fair coin Think of it as the idea that nothing in the results of the study is out of the ordinary An alternative hypothesis is possible that the coin isn t a fair one and it s loaded to produce an unequal number of heads and tails This hypothesis says that any heads tails split is consistent with an unfair coin The alterna tive hypothesis is called believe it or not the alternative hypothesis The sta tistical notation for the alternative hypothesis is H 6 Part I Statistics and Excel A Marriage Made in Heaven EMBER D With the hypotheses in place toss the coin 100 times and note the number of heads and tails If the results are something like 90 heads and 10 tails it s a good idea to reject H If the results are around 50 heads and 50 tails don t reject H Similar ideas apply to the reading speed example I gave earlier One sample of children receives reading instruction under a new method designed to increase reading speed the other learns via a traditional method Measure the children s reading speeds before and after instruction and tabulate the improvement for each child The null h
11. he function 20 Part I Statistics and Excel A Marriage Made in Heaven 2 Select a function from the Statistical Functions menu to open a dialog box for that function 3 Enter the required information into the dialog box 4 Close the dialog box Again I ve oversimplified and again not by much as you see throughout the book Statistical Functions menu Yep This time around you have a Statistical Functions menu that wasn t in the earlier incarnations It s buried under Formulas More Functions Statistical In Chapter 2 I show you how to make that menu more accessible Excel 2007 s statistical functionality is by and large the same as in previous versions The new version adds three statistical functions COUNTIFS counts the number of cells that meet a set of conditions AVERAGEIF finds the aver age of cells that meet a condition AVERAGEIFS finds the average of cells that meet a set of conditions Some Things about Excel Vou Absolutely Have to Know Although I m assuming you re not new to Excel I think it s wise to take a little time and space up front to discuss a few Excel fundamentals that figure prominently in statistical work Knowing these fundamentals helps you work efficiently with Excel formulas Autofilling cells The first is autofill Excel s capability for repeating a calculation throughout a worksheet Insert a formula into a cell and you can drag that formula into adjoining cells
12. ith earlier versions of Excel LY ew Series Series in Type Rows Linear Growth Date Auto ill Columns Trend Step value Stop value OK Cancel I bring this up because statistical analysis often involves repeating a formula from cell to cell The formulas are usually more complex than the ones in this section and you might have to repeat them many times so it pays to know how to autofill Referencing cells The second important fundamental is the way Excel references worksheet cells Consider again the worksheet in Figure 1 4 Each autofilled formula is slightly different from the original This remember is the formula in cell H2 Chapter 1 Evaluating Data in the Real World 23 SE Figure 1 7 Whoops Incorrect autofill Eyes D2 E2 F2 G2 After autofill the formula in H3 is IDS sb 3 sp BS EB and the formula in H4 is well you get the picture This is perfectly appropriate I want the total in each row so Excel adjusts the formula accordingly as it automatically inserts it into each cell This is called relative referencing the reference the cell label gets adjusted rela tive to where it is in the worksheet Here the formula directs Excel to total up the numbers in the cells in the four columns immediately to the left Now for another possibility Suppose I want to know each row total s propor tion of the grand total the number in H11 That
13. n the lower right corner of H2 until a appears hold down the left mouse button and drag the mouse through the cells That is called the cell s fill handle When you finish dragging release the mouse button and the row totals appear This saves huge amounts of time because you don t have to reenter the formula eight times Same thing with the column totals One way to create the formula that sums up the numbers in the first column 1990 is to enter D2 D3 D4 D5 D6 D7 D8 D9 D10 into cell D11 Position the cursor on D11 s fill handle drag through row 11 and release in column H and you autofill the totals into E11 through H11 22 heel Figure 1 5 The Fill pop up menu a D Figure 1 6 The Series dialog box E Part I Statistics and Excel A Marriage Made in Heaven Dragging isn t the only way to do it Another way is to select the array of cells you want to autofill including the one that contains the formula and click the down arrow next to Home Fill This opens the Fill pop up menu see Figure 1 5 Select Down and you accomplish the same thing as dragging and dropping Still another way is to select Series from the Fill pop up menu Doing this opens the Series dialog box see Figure 1 6 In this dialog box click the AutoFill radio button click OK and you re all set This does take one more step but the Series dialog box is a bit more compatible w
14. ns you d have to keep tossing the coin until the end of time Because you re never certain about your decisions it s possible to make an error regardless of what you decide As I mentioned before the coin could be fair and you just happen to get 99 heads in 100 tosses That s not likely and that s why you reject H It s also possible that the coin is biased and yet you just happen to toss 50 heads in 100 tosses Again that s not likely and you don t reject H in that case Chapter 1 Evaluating Data in the Real World Although not likely those errors are possible They lurk in every study that involves inferential statistics Statisticians have named them Type I and Type II If you reject H and you shouldn t that s a Type I error In the coin example that s rejecting the hypothesis that the coin is fair when in reality it is a fair coin If you don t reject H and you should have that s a Type II error That hap pens if you don t reject the hypothesis that the coin is fair and in reality it s biased How do you know if you ve made either type of error You don t at least not right after you make your decision to reject or not reject H If it s pos sible to know you wouldn t make the error in the first place All you can do is gather more data and see if the additional data are consistent with your decision If you think of H as a tendency to maintain the status quo and not inter
15. pret anything as being out of the ordinary no matter how it looks a Type II error means you missed out on something big Looked at in that way Type II errors form the basis of many historical ironies Here s what I mean In the 1950s a particular TV show gave talented young entertainers a few minutes to perform on stage and a chance to compete for a prize The audience voted to determine the winner The producers held audi tions around the country to find people for the show Many years after the show went off the air the producer was interviewed The interviewer asked him if he had ever turned down anyone at an audition that he shouldn t have Well said the producer once a young singer auditioned for us and he seemed really odd In what way asked the interviewer In a couple of ways said the producer He sang really loud gyrated his body and his legs when he played the guitar and he had these long side burns We figured this kid would never make it in show business so we thanked him for showing up but we sent him on his way Wait a minute are you telling me you turned down That s right We actually said no to Elvis Presley Now that s a Type II error 17 18 Part I Statistics and Excel A Marriage Made in Heaven What s New in Excel PCS Figure 1 2 The Insert Tab in the Ribbon in Excel 2007 PCs The big news in Excel 2007 througho
16. r a notation convention you have to know Statisticians use Greek let ters u p to stand for parameters and English letters X s r to stand for statistics Figure 1 1 summarizes the relationship between populations and samples and parameters and statistics Population Parameters Select Make individuals inferences about C sample Statistics Variables Dependent and independent Simply put a variable is something that can take on more than one value Something that can have only one value is called a constant Some variables you might be familiar with are today s temperature the Dow Jones Industrial Average your age and the value of the dollar against the euro Statisticians care about two kinds of variables independent and dependent Each kind of variable crops up in any study or experiment and statisticians assess the relationship between them For example imagine a new way of teaching reading that s intended to increase the reading speed of fifth graders Before putting this new method into schools it would be a good idea to test it To do that a researcher would randomly assign a sample of fifth grade students to one of two groups One 11 2 Part I Statistics and Excel A Marriage Made in Heaven gener Xe Re Re group receives instruction via the new method the other receives instruction via traditional methods Before and after both groups receive instruction the researcher measures the rea
17. should be straightforward right Create a formula for I2 and then autofill cells I3 through 110 Similar to the earlier example I d start by entering this formula into I2 H2 H11 Press Enter and the proportion appears in I2 Position the cursor on the fill handle drag through column I release in 110 and D oh Figure 1 7 shows the unhappy result the extremely ugly DIVO in I3 through 110 What s the story ay P ox A B c D E F G H I J K E 1 Field 1990 1995 2000 2001 Total Proportion 2 Physical Science 1807 2254 2708 23800 3 ntal S 1063 1433 1763 1827 4 222 279 341 257 5 515 682 875 954 26 DIv 0 6 Life Sciences 8726 12185 17460 19189 57560 DIv 0 7 Psychology 253 370 516 582 21h Div o 8 Soclal Sciences 73 1018 1297 1436 saj Div o 9 Other Sciences 336 426 524 579 875 Div o 10 Engineering 205635254999 s774 div ol u Total 16287 22162 230041 232723 101213 Ga 12 B 14 15 16 17 18 19 20 2 22 i 2 jA z H 4e Sheett Sheet Sheet J g s i os 24 Part I Statistics and Excel A Marriage Made in Heaven EE Figure 1 8 Autofill based on absolute referencing a ar The story is this unless you tell it not to Excel uses relative referencing when you autofill So the formula inserted into I3 is not S513 tell aL Instead it s H3 H12 Why does H11 become H12 Relative referencing assumes that the formula means divide the number in
18. ss H 11 absolute refer ence for the row part only H 11 and absolute reference for the column part only H11 What s New in This Edition Although Excel s statistical functions haven t changed I ve restructured the instructions for every statistical function The instructions in this edition fit in with the steps I outlined in the preceding section With the disappearance of the Chart Wizard I ve restructured the instruc tions for creating a chart too See Chapter 3 One of my points in both editions is that when you report an average you should also report variability For this reason I believe Excel 2007 should also offer the functions STDEVIF and STDEVIFS in addition to the new functions AVERAGEIF and AVERAGEIFS Unfortunately these functions do not exist in Excel 2007 To fill the void I show you how to do what these functions would do and in the process take you through some of Excel s Logical Functions See Chapter 5 It s easier to assign a name to a cell range in Excel 2007 it takes you guessed it just a few steps now So I rely much more on named cell ranges in this edition See Chapter 2 In the Part of Tens I ve added a section on importing data from the Web See Chapter 20 I pointed out in the Introduction that I ve added Appendix B and Appendix C Each one shows how to do some nifty statistical work that doesn t come pre packaged in Excel 26 Part I Statistics and
19. t to know the probability that an event occurs figure out how many ways that event can happen and divide by 13 4 Part I Statistics and Excel A Marriage Made in Heaven the total number of events that can happen In each of the three examples the event we were interested in head 3 or club only happens one way Things can get a bit more complicated When you toss a die what s the prob ability you roll a 3 or a 4 Now you re talking about two ways the event you re interested in can occur so that s 1 1 6 s s What about the probabil ity of rolling an even number That has to be 2 4 or 6 and the probability is 1 1 1 6 34 6 1 2 On to another kind of probability question Suppose you roll a die and toss a coin at the same time What s the probability you roll a 3 and the coin comes up heads Consider all the possible events that could occur when you roll a die and toss a coin at the same time Your outcome could be a head and 1 6 or a tail and 1 6 That s a total of 12 possibilities The head and 3 combination can only happen one way So the answer is 1 In general the formula for the probability that a particular event occurs is Number of ways the event can occur P tj r e Total number of possible events I began this section by saying that statisticians express their confidence about their decisions in terms of probability which is really why I brought up this topic in the first
20. tage of sampled voters who prefer each candidate The polling organization adds how accurate they believe the polls are When you hear a newscaster say something like accurate to within three percent youre hearing a judgment about confidence Here s another example Suppose you ve been assigned to find the average reading speed of all fifth grade children in the U S but you haven t got the time or the money to test them all What would you do Your best bet is to take a sample of fifth graders measure their reading speeds in words per minute and calculate the average of the reading speeds in the sample You can then use the sample average as an estimate of the population average Chapter 1 Evaluating Data in the Real World MBER S e Figure 1 1 The rela tionship between populations samples parameters and statistics Ce Estimating the population average is one kind of inference that statisticians make from sample data I discuss inference in more detail in the upcoming section Inferential Statistics Now for some terminology you have to know Characteristics of a population like the population average are called parameters and characteristics of a sample like the sample average are called statistics When you confine your field of view to samples your statistics are descriptive When you broaden your horizons and concern yourself with populations your statistics are inferential Now fo
21. the cell by whatever number is nine cells south of here in the same column Because H12 has nothing in it the formula is tell ing Excel to divide by zero which is a no no The idea is to tell Excel to divide all the numbers by the number in H11 not by whatever number is nine cells south of here To do this you work with absolute referencing You show absolute referencing by adding signs to the cell ID The correct formula for 2 is H2 H 11 This tells Excel not to adjust the column and not to adjust the row when you autofill Figure 1 8 shows the worksheet with the proportions G P e x A B c D E F G H j K E 1 Field 1990 1995 2000 2001 Total 2 Physical Science 1807 2254 2708 2800 3 Environmental Sciences 1069 1433 1763 1827 4 A S ces 222 279 341 357 5 515 682 875 954 6 8726 12185 17460 19189 7 sy 253 370 516 582 8 Soclal Sciences 73 1018 1297 1436 9 Other Sciences 326 426 534 579 10 Engineering 2656 3515 4SA7 4999 157174 0 155 u Total 16287 22162 20041 22723 101213 12 13 14 15 16 17 18 19 20 2 2 jA X H ri Sheet Sheet Sheet J j m m E To convert a relative reference into absolute reference format select the cell address or addresses you want to convert and press the F4 key F4 is a toggle that goes between relative reference H11 for example absolute Chapter 1 Evaluating Data in the Real World 25 reference for both the row and column in the addre
22. ut Microsoft Office 2007 in fact is the user interface Where a bar of menus once ruled you now find a tabbed band Appearing near the top of the worksheet window this band is called the Ribbon Figure 1 2 shows the appearance of the Ribbon after I select the Insert tab Bookl Microsoft Excel non commercial use The Ribbon exposes Excel s capabilities in a way that s much easier to under stand than in previous versions Each tab presents groups of icon labeled command buttons rather than menu choices Mouseover help adds still more information when you re trying to figure out the capability a particular button activates Clicking a button typically opens up a whole category of possibilities Buttons that do this are called category buttons Microsoft has developed shorthand for describing a mouse click on a com mand button in the Ribbon and I use that shorthand throughout this book The shorthand is Tab Command Button To indicate clicking on the Insert tab s Other Charts category button for example I write Insert Other Charts By the way when I click that button the gallery in Figure 1 3 appears I can extend the shorthand To select the first chart in that gallery it s called High Low Close as mouseover help would tell you I write Insert Other Charts High Low Close Chapter 1 Evaluating Data in the Real World 9 E Figure 1 3 Clicking Insert Other Charts opens this galler
23. y E At hal lied Surface C Doughnut QQ Bubble e 8s s Radar pik All Chart Types The downside to all this of course is the Ribbon s newness If you ve spent years with previous versions you ve developed an overall sense of where fre quently used capabilities reside Now you have to reorient The switch from the menu bar to the Ribbon relocates almost everything Its worth your while to reorient After you get accustomed to the Ribbon you ll see that everything takes just a few steps now Wait a second Figure 1 3 shows a gallery of charts to insert into a worksheet What happened to the Chart Wizard It s gone from Excel 2007 In keeping with everything takes just a few steps now to create a chart you 1 Select the data to include in the chart 2 Insert the chart into the worksheet 3 Use the Design tab and the Layout tab to make modifications I ve oversimplified but not by much as Chapter 3 shows Creating a chart is more intuitive than it used to be You re no longer confined to the order of steps specified in the Chart Wizard Wait another second Design tab Layout tab They re not in Figure 1 2 After you insert a chart and select it they appear Tabs that appear when needed are called contextual tabs Also in keeping with everything takes just a few steps now to use a statistical function you 1 Select a cell for the result of t
24. ypothesis H is that one method isn t different from the other If the improvements are greater with the new method than with the traditional method so much greater that it s unlikely that the methods aren t different from one another reject H If they re not don t reject H Notice that I didn t say accept H The way the logic works you never accept a hypothesis You either reject H or don t reject H Notice also that in the coin tossing example I said around 50 heads and 50 tails What does around mean Also I said if it s 90 10 reject H What about 85 15 80 20 70 30 Exactly how much different from 50 50 does the split have to be for you reject H In the reading speed example how much greater does the improvement have to be to reject H I won t answer these questions now Statisticians have formulated decision rules for situations like this and we ll explore those rules throughout the book Two types of error Whenever you evaluate the data from a study and decide to reject H or to not reject H you can never be absolutely sure You never really know what the true state of the world is In the context of the coin tossing example that means you never know for certain if the coin is fair or not All you can do is make a decision based on the sample data you gather If you want to be cer tain about the coin you d have to have the data for the entire population of tosses which mea
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