Okay where this is a top 10 topic number two for statistics, hypothesis testing, I think it's one of the more difficult of topics for students to it to do it to an important idea in use to all the time in research and decision making for managers to a sore on the main business trip to help page for click on exam review will come down here to statistics to click on statistics revealed and will pull up the PowerPoint presentation that the SOM faculty have kindly generated to assist students in preparing further their study list for the exam and will slide down here to slide 1919 is where the start of top 10 topic number two begins to pay hypothesis testing a first of all, what is hypothesis I processed this in your idea about something that that that view of a hypothesis you have a unit you want to make an absurd station and a pain in answer to patient. I have I've a hunch about something a butt to be able to demonstrate support for that hunch you have to at least show that it didn't occur by chance. So we have to reject the idea that I could ever occurred by chance, we call that the null hypothesis contains a slight number 28 70 as it sometimes remakes of old, but it's really a sub zero means the smallest hypothesis that could possibly be true geeks of zero or the zero or the no thing hypothesis is the per visit is the hypothesis that a court of her curtly chance and we would like to reject that idea we can reject the null hypothesis that we can accept your hypothesis sometimes called the alternate hypothesis so bullet point number one population mean his meal population proportion is Potter lowercase pints third bullet point is a statement height no hypothesis is a statement about the value of the population parameter. To obtain out about the sample by about of the population that the sample was drawn from a low point member for never include a sample statistics such as ex-bar in my office is wide because the hypothesis about a population of the hypothesis isn't about the sample. Yes I know that the sample statistic like ex-bar or sample standard deviation are things that we can use to evaluate whether or not to know hypothesis is sure not but than the hypothesis itself is about the is about the population so don't include the sample statistics to make sure that the don't include the simple statistics in the statements about the null or alternate hypothesis always used the Greek letters, which means that we're talking about the population rather than the sample case, line 201H A., which means the alternate hypothesis is sometimes called each wind to distinguish it from age 0 is the alternative hypothesis okay in there one tale sample one tale alternatives and to tailor programs remember the tale refers to the endpoint in the normal distribution either the endpoint to the right or the endpoint to the left depending on whether and not it's greater than or less than sometimes we have to tail test or looking to see whether or not he could occurred in either of the lower tail or the effort to. So no one tail our time and if you're on slide 21 the right tale is a mu is greater than four example of the number of smog checks that went okay okay of the per high up there in the proportion is greater than the fraction for example, the percent effectiveness of these are just examples left tail mu is less than a number for example, the weight of a box of crackers to pay another example pop a proportion pie is less than a fraction for example of president's approval rating is lower is low. That's that's a proportion of that that's less of a particular fraction so that's those are examples between right tales and left fails to one tailed test. Here's an example, a test is a one tailed when the alternate hypothesis and each day usually derided as H. Juan Buddy Chase states a direction such as H. won the mean the mean's yearly salaries turn by full-time employees is more than $45,000 in that particular case here this bullet point number one on slide 22 the meet the population mean is greater than $45,000 will point number 2HH won the average speed of cars traveling on the freeway is less than 75 miles an hour some new population mean is less than 75. Another example, one final example and slide 22 less than 20% of the customers pay cash for the gasoline purchases to map particular case it's the up population proportion. Pie is less than .02. Excuse me is less than .220% case like 22 years it took to tail the alternative of the population mean is not equal to a particular number in other words. It's too hot or it's too cold to hot would be safe to far to the right as temperature goes up to cold is too far to the left as temperature goes down for example kind from another example are on slide 23 is the population proportion is not equal to a fraction for example for percent alcohol in a particular bottle is either too weak or too strong and women were testing things like this. We don't know if it's too weak or too strong a less forgiving particular information that said, for example, it can only be too strong. It could never ever be too weak. That kind of information is torrential information and that will be given in a problem if it's there if it's not there is a too tailed test. Something can either be too low or too high a slight 24 to tail test a test is too detailed. When no direction is specified in the alternate hypothesis that's restating what I just said before a test is too detailed. When no direction is specified in the alternate hypothesis. So for example H. wonder same thing as a Che alternate hypothesis that's the one that you would like to show that shrew or at least the best that your hunch the mean amount of time spent for the Internet is not equal to five hours slips of the population mean is not equal to five, which means that the null hypothesis is that the population mean is equal to five the second bullet point it someone else in the office of the mean Christ for a gallon of gasoline is not equal to $2.54 population meet is not equal to 2.54. Those are examples of two tail to us than the mean price could be below 2.54 could be above 2.44 oh okay slight 25 you reject the no hypothesis that which is a subzero behalf, and there are three bullet points. The absolute value of the test statistic which you compute its derived from sample on from the sample data of the absolute value the test statistic is greater than the critical value of the critical value is the dividing point in the Inn on the line in the normal distribution between with the null hypothesis rejected and the region where it is not rejected to okay and it's usually easier to draw this on the board and the larger chunk of the fraction of the Honda of the area under the curve is the acceptance points and the smaller fraction in the tales is the rejection .so in the from going back to the first bullet here on slide 25 or reject the no hypothesis is the absolute value is the value, which means we don't corrupt a direction for moment is greater than the critical see the tanks. So the sea value is a test statistic on a witch we compute from the sample data, and we can get and we look we compare that to hope a band with a critical see value and we can look that up in a particular table. It is if I'll have to use the tea table rather than the disease and table for example. We don't know we don't know the population standard deviation will he have the sample standard deviation or a fan is less than 30 with use the tea value we do the same idea to take the absolute value of the tea value check to see whether or not it's greater than the critical team okay and if it is we can reject the null hypothesis which gives us more support to be able to accept the alternative hypothesis or at least failed to reject the null hypothesis about to happen this well second bowl we reject the null hypothesis if the probability value is less than the significance level, which is a. Note that the we change the direction of the signed the PP where checking to see if the probability is less than that of significance level significant soul might be .05 we might be able to compute the pea values .02 a week and reject them no hypothesis and take as it's a smaller part of the tail on the right-hand side of of the of the normal distribution third bullet rejoice in all hypothesis if there's a very large difference between the sample statistic and the population parameter in the null hypothesis. You won't see that very often each general in the first two bullets which also came here's an example on slide 26 of us of a smog check for example first full null hypothesis of the population mean is equal to 80. The alternate hypothesis with what you believe to be true for example, but you can't prove it at of the population mean is greater than 80 so this and so one tale test right as it has a directional sign him and are greater than if the test statistic is to point to remember, you know, you have to calculate in the thing to be given on the in for the exam and will be given in the problem some are the test statistic is 2.2 and the critical value is 1.96. And if you draw that out 1.96 is closer to the mean and 2.22.2 is to the right of 1.96 in other words, it's more in detail than it is closer to the mean closer to the part of the the middle part of the Bell curve if the test statistic is 2.2 in the critical volume was 1.96 we can reject the null hypothesis and therefore conclude that the population mean is likely to be greater than 80 okay so cute again, go through both point number three if you draw a normal distribution.
A line on the right-hand side in the vertical line is 1.9 sex that's the critical you fear for is the uncanny. And that if the computer to test statistic is 2.22.2 is to the right of the critical value in other words it's in the tale it's in the rejection region. So it's a consensus in the rejection region, we can reject the null hypothesis and in this particular case, that means that we can get close we can get closer to cook to accepting the alternate hypothesis and conclude that the population mean is likely a last bull and ominous live in page 26 for examples take the same example with but with different numbers if the test statistic was 1.6 and the critical value remains the same at 1.961.96 if he drove up the normal distribution on the right-hand side put a vertical line put the number 1.96 on the line 1.6 is close use to the left of 1.96 is closer to the middle of that normal distributions so it's in the acceptance region. It's not in the rejection region so since it's in the separation we cannot reject the null hypothesis we have to except the no hypothesis at this point and with that means we have to reserve judgment about the null hypothesis. Another words, we can reach you at the idea that it could have occurred by that the mean of 80 could have occurred by chance. Since we can't reject the idea that it couldn't overcome occurred by chance and we can't say would very much confidence that the mean is greater than 80 and paying you to think about that for second. That's the key IDM and hypothesis testing is to reject the idea that the current week when a reject very well but the idea that it could ever occurred by chance that the no hypothesis could ever curb a chance of the week except the bill to my office day slide 27 type 1 versus a type to air this is very important to understand as well this comes up all the time and statistics. The first book. .a is equal to the lowercase a is lowercase Greek. Symbol a, you forget what's the first letter of the alphabet. A pain in the English alphabet and. It's a, sequel to the probability appealed to the type 1 air, which is equal to the significance level, which is then the definition is the probability that chill reject the true-not a hypothesis when it is indeed true that you don't want to do that. That's a really better it's the worst kind of rare source call the type 1 error to it's the first one is the first letter in. The a button a or a opaque the other time the other type of error is beta. The than that and that's our per case being much space to the is equal to the probability of a type 2 or that's the probability that you do not reject the null hypothesis given that the null hypothesis is false of paint those of the two kinds of errors are for outcomes which are one false. It really is true, that really is false and tank and it's what you say and whether or not it is and is in fact, in reality, and so that's two times to this for outcomes two of the outcomes are not air if you say, that is true, but it really is sure that's not a manner that's if you say this false, and it really is an false. That's not a narrower so two of the outcomes or are fine, but two of the the other two outcomes are both errors one of those errors is a type 1 error and the other one of those areas is a type 2 or the worse error is called a and a is called type 1 error that's reject that the probability of future rejected the the null hypothesis when it is in fact true. In the other one and the other is called beta, and that's" type to whereabouts the probability that you do not reject the null hypothesis when its faults, and for example on page 27 the no hypothesis of a defendant is innocent. Will cram a teacher's statement in in in US logs and is innocent until proven guilty. So a is the probability that the jury convicts an innocent person. That's not very good right, we want that number to be very loved. We want the probability that a jury convicts an innocent person to be very very love to an pain what's the other or that they could make the jury brought beta are the type to where's the probability that it does the jury acquitted quits a guilty person to pay now that's an error we don't want data errors to occur either but given the choice between those two heiress. We want the lowest amount of a, and unfortunately the things that you do to reduce a sometimes can increase that the type to wear and the things that you do to reduce the type to wear can increase a type 1 error but in mode and in all cases we want to keep in the type 1 error as low as possible. So in this particular case, we don't ever want to convict an innocent person, although that certainly has happened in history just not often contain and we do that at the expense of that of the jury of quitting, which means letting go for a guilty person, which we don't want to happen either, but that certainly has happened in society as well and to the list of the slide 28 here's a nice box you'll probably have to memorize this box on slide 28 is easier to do this you puke a little better writing papers you need to take notes seeking always a jot this down to charge your memory, but after this is just a small little table of the four outcomes of the decision to no hypothesis is true. No hypothesis is false, those of the top two columns and in the rows you have the view. You reject the no hypothesis, and you do not reject the know what a officers and and those of the four outcomes remember two of the outcomes are the correct decision's one minus a is the correct decision one minus Beta is a correct decision. So four minus two is to the other two boxes are errors of cake if you reject the null hypothesis when the null hypothesis is true that some alpha error of type 1. There are too noted by the Greek a. If you do not reject them no hypothesis when the null hypothesis is false with the lower right corner of slight 28 as call the type to where it's been noted by the Greek letter beta, paying to slide 29 and paying here's an example for the smog check in again, but instead of computing critical the bills will use the pea values in stats of the no hypothesis same no hypothesis says before the population mean a sequel to a ill turn in to the alternate hypothesis is that the population means greater than a contained no hypothesis book population musical that they'd alternate hypothesis or H1 or a shade is the population mean you is greater than a tuple of point number three or example, if the P. value is .01 and the a is .05 in this particular case, we reject the no hypothesis and conclude that the population mean is most likely to be greater than a why because the key value point to one is less than the alpha value of .05 paying. In other words, there's a small amount of the critical region on the right-hand side of the tale of day and a smoky and even smaller part of that is point of the the the the the the critical region on the right-hand side is .05 an even smaller part of that is .01 sol .01 is inside the .05 if you want to think of it that way so we get to reject the null hypothesis for a second bowl of point if the P. value is .207 probability is up .0. 70 a is .05 than we do not reject the null hypothesis and we have to reserve judgment about the no hypothesis in other words, we cannot conclude that our alternative offices are all all are in. We cannot conclude their arts or alternative hypothesis that the population may use greater than 80 is in fact true, or least likely to pay and that's unfortunate, but that's the way it is okay is to test statistic on page on a slide 30 when testing for the population mean from a large sample in the population standard deviation is known to test statistic is given by the following formula remember we know that the note and large sample means great and is greater than 30 in the and we know the population standard deviation, which usually isn't the case but if we if it's known as stated in the problem of the test statistic is given by zine is equal to the sample mean ex bar, minus the population mean, all divided by on the population standard deviation divided by the Square revamp a map to the computer a disease of the the the tests is a disease and here's an example on slide 31 the processors of best male indicates, label that the bottle contains 16 ounces of mayonnaise, and thus to the standard deviation of the process is .5. A sample of 36 bottles from last hours production showed a mean way of 16.12 ounces per bottle at the .05 significance low can we conclude that the mean amount per bottle is greater than 16 ounces. This is a classic example of a hypothesis to show scrolling down to the next slide slide 32 okay so here's what you do step never one state, the null hypothesis and the alternative hypothesis is her first job and no hypothesis is that the population mean is equal to 16 the alternate hypothesis eight sure beach a year each one is the pop is that the population Dean is greater than 16 weeks out that in the in the problem up here are because it did the the the the best male label indicates that the bottle is 16 ounces of Meme so that's the population mean faces state the null hypothesis population mean is equal to 16 that's age 0 day `we want what we want to do is be able to reject the idea that that occurred by chance of Mac gives more weight to the alternative hypothesis VH1 rates is a population Main is greater than 16 and paint like we've seen in our sample for example, but one sample is not the whole population of cake number to select the level of sick significance in this particular case, we selected the .05 Signet from low significant slow. It was given in the problem that you don't have to make it up in a number three identify the test statistic because we know the population standard deviation. It was given in the problem. The test statistic is a silly use a formula for is a companion which is on the previous to slide to go with state the decision role we reject the null hypothesis is the absolute value as he is greater than 1.6451.645 where we get that number from that's is the value 4.051.645 so slide 33, compute the value of the test statistics. See is equal to the sample mean, minus the population mean divided by this bit population standard deviation divided by Square Ravenna, which is 16.12 -16 divide about .5 m 36, which is 1.44 okay 1.44. In his and a BZ value (we were and remember the decision rule sliding back here to slide 30 to reject the null hypothesis is the absolute value of zine, though the number we just computed 1.44 is greater than 1.645, which uses the value of the of .05 level at which his .05 significance local to a all right is so that's the question on slide 33 is 1.44 greater than 1.645 on slide toward the 32 node is not want to pay 1.44 is less than that. So we cannot reject the null hypothesis so conclusions do not reject the null hypothesis we cannot conclude that the meme is greater than 16 ounces at a pay on slide 33 number six do not reject the no hypothesis because 1.44 is not greater than 1.645 oh, which means we have to reserve judgment about whether or not the mean is greater than 60 ounces we can't rule out the fact that it might have occurred by chance that our sample of 16.12 might have occurred by chance paint. And that's the example for and that's it for a hypothesis to