Ladies and gentlemen the story you are about to hear is true. The names have been changed to protect the innocent Your detective sergeant you're assigned a robbery detail a pair of holdout men have been operating in your city You know their method of operation. You know what they look like your job get them The documented drama of an actual crime For the next 30 minutes in cooperation with the Los Angeles police department You will travel step-by-step on the side of the law through an actual case transcribed from official police violence From beginning to end from crime to punishment Dragnet is the story of your police force in action It was Wednesday June 8th was warm in Los Angeles. We were working today watch out a robbery detail My partners Frank Smith the boss of chief of detectives that brown my name's Friday We're on our way out from the office, and it was 10 14 a.m. When we got to the corner of Roberta and Mercury the after hours cafe Actually, Harry Prentice. Yeah, I'm him. What do you want police officers? It's my partner Frank Smith My name is Friday. All right by last night. Yeah, were you here? Yeah, sure was I'm the only one's got a key places open I'm here. Yes, sir now about the robbery. Yeah, sure. What do you want to know? What would you tell us what happened? Should have read the papers this morning all about it in there got a couple of copies if you want to see him No, that won't be nice. Oh, no trouble though. I'm gonna have a couple of copies right here Yeah, take a look right here. Yeah, here it is. Yeah, I'll read it to you Last night at the after hours cafe And we should give any address at the after hours cafe two gunmen held up and be Jeffrey Hanks The victim was approached outside the bar and when asked for his money. Yes, sir Now we saw that we just like to hear your version if we could. Oh Well, okay, what do you want to know? Did you see the man the gun? That's right. Yeah, sure I told the other cops they did gave me a description everything Notice anything about their speech an accent maybe no I asked them what they wanted one big one. He ordered a scotch and a rocks Other one asked for plain grapefruit juice. No booze. Just the juice. Mm-hmm And lucky I had it got a couple of cans in for vodka drinks. Don't get a lot of call for it I drank all I had one of the victim come in now. You mean Hanks. Yeah Almost been about 830 someplace in here Walked in sat down. I just seen him before no brand new face Nice looking guy though kind you like to have a regular. That's all yeah quiet good drinker doesn't bother the singles at the bar Nice, you know what I mean?  I wouldn't have no worries How do you happen to get mixed up with the other duty? No, yeah, you mean Hanks. That's right I don't know. I wasn't paying a lot of attention portable martini left him alone Mm-hmm a friend of mine lady type was here for talking to her and her boyfriend been having trouble He belted around a little bit not bad couple of scratches. You know how mean I think so Well, she was asking me what to do We've been friends for a long time since I opened a place Anyway, we was talking so I didn't see what happened. Yeah, all I know is that the three of them got up and moved to the booth in the back Ordered another round. Uh-huh I hardly got the drinks back to the table and they was ready for another one Go ahead kept running from the bar to their table for the next couple hours. You hear anything they were saying No, not much the one who was drinking scotch. You got a little plastered and loud told him a couple of times He shut up said after 86 him if they didn't yeah, kept it up So I told him to leave didn't take it kindly Not even that I'll start to get rough You know telling me that they didn't have to take no guff from a barkeep No, I told him I'm wondering in a grapefruit juice. I told him to get his friends out He says we're not causing any trouble and I said that's the way I want to keep So he said for me to leave Malone and I said to the grapefruit. I said for him to leave right then I don't want you in my place. I said yeah Well, and so Scotch on a rocks He said tell his cat to go hide his head like that He said and turn the bear loose and I grabbed the both of them threw him out through the door. Oh, you see yeah And I went back for the other guy nice looking one. It was pretty gas by then had about eight martinis asked him to leave He did have any trouble with it. No, sir. You just asked he walked out With the other two outside when he left. Yeah, stand on the sidewalk I told him to shove off. I'd call a cops old Scotch in a rocks He said for me to go soak my head when I was going after him and I figured it wouldn't do no good So I came back inside Couple minutes later the nice guy banks. Thanks.  Yeah, big cut right right along here for it Said he'd been robbed want me to call the cops Had you ever seen either of the two men before you mean Scotch in a rocks and gripping. That's right. No, I never did Why don't you take a look at this description see if it's right here? Yeah, sure glad to help her Yeah, yeah, that looks about right you're anything you're gonna add to it No, there's a couple of guys at the bottom and stand out Would you know him if you saw him again? Hard to say, you know, I mean yes a couple of guys came in order to drink nothing special I'm not sure I could point him out for you anybody forget him. Not quite. We won't 1130 a.m. We left the barn went back to the city hall We checked with the stats office and picked up the results of their latest run They listed four possibles, but when we read the names we were no closer to the hold-up men than we had been The names were ones that we checked and cleared before Frank and I went back to robbery. There was a message for us to see Captain Donahoe We went into his office. Yeah. Mm-hmm. No far as I know. There's nothing new on it. That's right This came in All right, I'll let you know right away. Yeah Hi skipper captain. That was cheap brown been calling all morning. Yeah, you know what he wants. What do I tell him? Well, we talked to the bartender. He can't give us anything. What about the victim? Not much there either able to come up with a name knows as he didn't hear anything. That's a big help Well sit up. Thank you. Yeah, thanks to Smith. Yes, I got it Thanks, Joe From the beginning. What do you got? Oh good ammo. Good description We got an idea the liner working and that's about it. What about stats? Just pick up the new run four names What's on them clean? They all been checked before got to be something around town on them Well, there should be but they're just isn't how about your informants? No, they haven't been able to tell us anything straight. No reason. They should lie victims and witnesses seen the mug books All we got they haven't been able to come up with an identification Burton up at CII said all the stuff he's got on the method of operation. We've checked it all through What about the places they could spend the money? Well, we've been to Gardena have a dozen times nothing out there about Vegas talks to the department over there They're giving us all the help they can but they haven't turned anything yet. We've got the racetracks covered possible. You missed something Sure, it's possible. I don't know what it could be. You knew you wouldn't have missed it. Yeah From the beginning go over. Okay first report was made on Man, man Got it here. Here it is Tuesday April 26th victim came out of the bar two men stopped them started conversation Got him into a car beat him and robbed him six weeks ago. That's right. The guys are still running Next one was on Friday April 29 Same MO same description how often has been happening since a couple of times a week. You can't find a leak No, sir. No rumbles. No, no, we haven't run down anybody in the office got anything We talked to me and strong. Well, I thought they had it made for us, but their lead didn't go any place That money they're getting isn't sitting in their pockets being spent someplace. We can't find the cash register I'll tell you this a fellow they nailed last night. What's his name Jeffrey Hanks? Yeah Well, he's got some friends in the city hall. They're pretty upset about what happened called the chief this morning He's been on me. I'm giving it to you. He won't be happy with excuses and I don't want to give him any so don't hand me Right boils down simple. I don't care how you do it. I don't care where I just get it done I want those two men and I want them fast, right? I know how it is now you do too. Yes, sir I don't like the pressure any more than you do we're getting it all the way around People that are making the noise should follow us around for a couple of days. That's the difference. They don't have to huh? You're getting paid Frank and I went over every piece of information we had on the case We reread the reports that had been filed by the officers at the scene Statements from witnesses and victims were gone over again. The names the stats office had given us were run through our And I again None of it did me good. We still didn't have a lead to the identity of the thieves Three days passed while we talked to informants and questioned other sources of information On Sunday June 12th. We got back to the office from talking to a bartender out on Olympic Boulevard Well, that's another one didn't go any place. Yeah, you want to get the book? Yeah, I'll take one. I got it robbery Friday, huh? Yeah, Jack Yeah, where is a place? Hmm? I Got it. Yeah, I guess the word's all over Yeah, okay, we'll check it out. Thanks boy. All right. Bye There's nothing in the book. Thank all want to know if you could make it for dinner tonight. It's got a barbecue plan You better call it off. You got other plan. That was McCready on the phone He just got a call from a bartender out in Wilshire. Yeah, woman crying in her beer says he figured we want to check it out What is it? She says her boyfriend is one of our suspects The bar sergeant Jack McCready had mentioned was on the corner of Wilshire Boulevard in Ingram Street There were two people in the place when we arrived One an elderly man sitting near the door and the other a middle-aged woman at the rear of the bar We checked with a bartender and he told us that he'd made the call He went on to say that the woman had come in about 10 that morning Since that time she'd been crying and mumbling something about going to the police Frank and I went back to talk to her Understand that you want to talk to the police I got to do with you. My name is Friday. This is my partner Frank Smith. How you doing?  What do you want to see us about, lady? That bum, Derwin. That's what it's about. Bum. That's his name, is it? Derwin? Yeah, I think we all know good He cried down to me all the time, hadn't me. I think we was married all the time What's his last name, do you know? Holbrook Derwin Holbrook You like to buy me a drink? I don't think you had enough of that, lady. That's where you're wrong. I was supposed to show you that you don't know what you're talking about And then you went to me and all you haven't caught, Derwin, done? What's he done? Hold ups Ma'am Hold ups That's what he's done. All over town. Old Derwin's been holding people up So you're husband? Me? Derwin's wife? That's right That's real funny You know how that is I mean Derwin married I don't think I wouldn't do it For men don't think it But it ain't true I mean I don't I'm not sure Derwin's just a friend I've been a friend for 12 years, that's a long time That's right I've been holding hands for 12 years for the way I'm married You don't want to Will you tell us about the hold ups that he's been pulling? Been all the papers You haven't read it? All the papers Are you sure these are your cops? Why don't you tell us about Derwin? He's lousy All the time Talking about how someday we'd get married all the time Talking like that I got a couple of bucks put away and we'll take the big step That's why he was all the time saying it was a big step So what happens? He has a couple of bucks and he takes off with another girl The lousy bum there was just miserable Now where'd he get the money? Some of the hold ups Like on the paper, I told you that before He's the one you're after He's been picking up all those people in front of bars He's pretty good at beating people up, you know He's good at that Not much else, but he's real good at that You ever talked about the hold ups to you? Oh, no, that'd be kind of silly Talkin' to anybody, but a little me Well, how do you know he's involved? Well, just because I know, that's all What's his fellow look like? Derwin? That's right I guess the person who says he was drawing nice looking fog I guess that's what you'd say I would He's got dark hair Kind of a little wave right here Right here Yeah In front of the wave How tall is he?  About six feet How much does he weigh? I don't know I guess a couple hundred pounds Is he fat or thin? Kind of thin How old is he? 38 38 years old Two years old and mixed Two years What color is eyes? Brown Has he ever been arrested? Hmm? I say has he ever been arrested? I don't know Maybe he has, if he said nothin' to me No word Wanna run through? Yeah, I'll check You're gonna arrest him? We'll talk to him Yeah You do that You tell him from me, I think he's bum But 12 years has been tellin' me When he gets a couple of bites, we take a step When he runs out with another girl A little lousy Derwin You know where we can find him? I don't know, I guess he's home I don't know a place else he'd be He's home Do you have the address? Sure It's one eight through some half Margo Street What's that? Can't you hear very well? One eight to seven and a half Margo Street Oh yeah, I got it Hillside 75321, that's where he lives Hillside 75321 That's what I said On the second floor Margo Street Do you have any close friends that you know of? Sure, a girl he walked out with She's a close friend Any men? Nope Derwin don't like nobody and he's back there I don't know You know fellas, he's close to none Joe Yeah Could you give me a minute, lady?  Oh, I checked R and I on Holbrook How's he staying? I found a couple that fit the description One of them might be good Huh Done time for robbery Do you owe anything? No, get out clean Wonder if he stayed that way Frank and I left the bar and drove over to the Margo Street address We checked with a landlady I don't know where he was I don't know I don't know I don't know I don't know I don't know When we left the address We checked with a landlady She told us that Holbrook had moved in approximately six weeks before She went on to say That he had been behind his rent But that he had paid what he owed And had given her the money For an additional two months She said that she didn't know What he did for her living He had few visitors In the landlady couldn't tell us If he owned a car According to her he was in his room Frank and I went up to see him Here he is How's that? How's that? Derwin Holbrook Just a minute What do you want? Please, I'll see you Hey You got no right to bust in here Then you won't mind if we look around, will you? Oh, go ahead. You won't find anything. I'll check it, Joe.  You just sit still there. Sure. I got nothing to worry about. Tell the place apart. Just make sure you put it all back. Right where you found it. Let's clean, Joe. I could have told you that. Sure you could. Why, let me know why you're rousing me. You work for a living? Why? Do you? No. Why, we hear it's you're doing pretty well with the money. I guess you'd say that.  Why is investment? That's it, huh? Sure. I read the financial page all the time. A couple of bucks here, a couple more there. First thing you know, you're on your way to a millionaire. Well, the way it came to us, you might have made the money the hard way.  Is that so? Sure. I got a pretty good idea who's got the big mouth. I'll make any difference. You can't tag me for anything. Why don't you tell us? I will. You're beefing about the money. Well, you can't say I didn't make it from investments. I don't have to prove it. You do. You gotta prove I didn't make it from investing. I think we might be able to. Go ahead. I'll give you something else. What? I don't think you're gonna like the interest. We took Durwin Holbrook back to the city hall and talked to him. For over an hour, he refused to admit any part in the hold-ups. We made arrangements for the victims to come down and see him. Witnesses were asked to special show-ups, but they failed to give us a positive identification. Holbrook's name came up on the overtime sheet, and he was released from custody. There was nothing we could do. Another week went by. Holbrook was kept under constant surveillance. Monday, June 20th, Frank and I were in the squad room. Friday? Smith? Yeah, Skipper, we're here. Hi, Captain. What do you got? Your boys did another job last night out in West LA. We didn't see the reports. Just came in. Got a meeting with Orman this afternoon. Be there. I'm aside. Yeah. Victim died without regaining consciousness. Any description of the suspect? Enough to know it's the two fellas you're after. How's your hot suspect? What's his name? Holbrook? That's all right. You able to keep him this time? No, I don't think so. What's the matter? Reports says the hold-up took place at 10.15. Yeah. Holbrook was asleep at 9. We rechecked with a team of men who had been keeping Durwin Holbrook under surveillance. They verified the fact that the suspect had been in his room at the time of the robbery, and that there was no way for him to have left the apartment. We went out to the hospital and talked to the officer that had been with the latest victim. He said that the dead man hadn't said anything to help identify his assailant. We talked with a bartender at the place where the beating had occurred. He remembered two men in the bar, and he remembered them talking with the victim, but he was unable to tell us anything more. Additional broadcasts were gotten out. At the meeting with Captain Orman, it was decided that from that point, the investigation would be a joint operation between robbery and homicide. After the meeting, Frank, Captain Donahoe, and I met back in room 27. Come on in. Sit down. Yeah. Well, you heard it. Yes. How are you going to work it? There's only one way of rolling steak out. I thought you tried that, Rob. This time we'll use more cars. Better work. This latest one's really got the corner pocket screaming. How are you going to set it up? Well, that job out in West L.A. is the first time he's moved out of Central. I think we can figure he'll be back. That's playing it the long way. There's nothing left. I'll buy it. How's it going to work? We'll take a look at the map over here. He's been hitting in this area along here. Up to now, we've had it covered in through here and here. Now, we can put additional cars on these streets and maybe over here, huh? How about a check system? It'll be kind of hard to work out, Skepper. We'll be in a three-way car. McCreedy and his partner will be in one. And that way, we can keep tabs on what's going on. When do you start? Now, if we can swing it tonight. Talk to McCreedy? No, not the whole thing. Went over part of it. What's he think? He goes with us. It's the only way. Then it better come through for you. You don't get another chance. At 7.20 that night, the rolling stakeout began. It continued for another week. Several arrests were made, but when the suspects were checked, they were ruled out as the ones that we were looking for. In the meantime, the holdup men hit twice more in the area under surveillance. Bar tenders started closing their places early. Pressure from citizens groups and newspapers got a little worse. Apparently, there was nothing we could do to stop the thieves. Tuesday, June 28th. We stopped for something to eat. That's going to be another long night. I hope we get something out of it. Yeah. Howdy, fellas. What'll it be? Well, I'll have a couple of eggs over easy. I get it. A side order of bacon. Well, thank you. How about you? That sounds good, Joe. I'll have the same. Only make my eggs straight up. Well, you know, I'll have a side order of ham. Righto. Two coffees. We'll bring them right away. Sure thing. I'll get them for you right now. Mr. Nagat, get home. I'll have a cup of coffee.  Life's going to be so rough. I'll make it pretty quick. Well, let her be sore. That's the trouble with women today. All the time trying to run our lives. That's right. Have your coffee. We'll meet my friend. You'll like him. I'll feel so good. Maybe I'll leave now. Look, you'll be all right. Just take it easy. Joe, mm-hmm. Call a guy. Fits the description. Looks like he might be setting one up. Let's check it.  Stay with us. Something you want? What if he'd let us take a look at you at edification? Why? Because I ask you. Who are you? A police officer. Watch out. Come on. Come on. Come on. Los cop. shove me around. That's all this is about. We didn't do nothing. Don't worry about it. You got nothing to hold me on. Nothing. We'll talk about it downtown. Come on. Let's go. What's the charge? You got to make one. Come on. You got to make a charge. You can't take me without it. All right. Try suspicion of robbery. Now for me. You keep believing that fellow. I will.  You got nothing. I'll be free in a couple hours. You got it wrong. Tell me about it. We won't have to. He refused to admit any part in the hold-ups. He wouldn't tell us who the friend was that he'd mentioned to the intended victim. A check of R and I revealed that he'd served sentences at San Quentin and at Folsom for armed robbery. We pulled his mug shot and a team of detectives went out to check with the victims. It was an hour before they got back to the office. We talked to him and then we went to see Noonan. Decided to turn me out now? No, not quite. When are you going to give up, cop? You got nothing to make it charge on. You know it. I don't know how long you want to play this game, but I'd like to get home and get some sleep. We just got word from the officers who checked your picture with the victims. Yeah, they gave us a positive identification. Nothing different about me. Lots of guys look the same.  We got you made and you know it. I don't know anything of the kind, cop. You're saying one thing, I say different. It's my word against yours. We'll see who the court believes. We got a head start. How do you figure that? That's the way you work. Any record I got, you can't mention in court. As far as they're concerned, I'm clean. You want to call it a day? We can all get some sleep. What do you do for a living, Noonan? I'm a salesman. What do you sell? Different things. Sometimes hosiery, different things. When's the last time you worked? Six months ago. How have you lived since then? I saved my money. You know, you're in for more in a robbery charge. I'm not standing that. Man, out in West L.A. died. That's too bad. I didn't see anything about it in the papers. Only way I didn't know. Who was the guy with you on the jobs? I don't know what you're talking about. You know we'll get to him when I save yourself a lot of time. I don't know nothing of the kind.  All right, Noonan. Let's go. Now you're going to unlock the door? Yeah, this one. Where are we going? Main jail. For how long? Until they pass sentence. You sound pretty sure. We are. Come on. Hey, it's too bad about you guys. What's that? How long you been working on these robberies? Couple of months. And I'm your pigeon. You're the man. Must make you feel kind of lousy. Hmm? Be so sure and then find out you're wrong. You know, if I keep my mouth shut, you aren't going to have a case.  We can identify you positively. We got a dozen people who are going to get up on a stand and point you out as a man who knocked him down and walked on him. The time we get through with you, we'll have enough on you for your operation. You're ruling out one thing, Cap. You know what's that? The jury. All I need is one person who won't go with the rest of them. One who doesn't listen. You got it all figured out. One person doesn't care. I got it made. Just one. Well, and you're in real trouble. Huh? We'll find 12 who do. The story you've just heard is true. The names were changed to protect the innocent. On November 19th, trial was held in Department 98, Superior Court of the State of California, in and for the county of Los Angeles. In a moment, the results of that trial. Further investigation resulted in the arrest of Jerome Nathan Pollock as an accomplice in the robberies. Both he and Emmett David Noonan were tried and convicted of murder in the first degree. The jury failed to return a recommendation of leniency and both men received the maximum sentence. They were put to death in the lethal gas chamber at the State Penitentiary, San Quentin, California. You have just heard dragnet, a series of authentic cases from official files. Technical advice comes from the Office of Chief of Police, W.H. Parker, Los Angeles Police Department.


कोर्स on dealing with materials data presently we are going through the sessions on hypothesis testing we have so far considered the sessions our way which deals with so if we quickly write down what we have seen we have dealt with the hypothesis H0 or H0 null hypothesis that mu is equal to mu 0 we have assumed that population is normal with mean mu and variance sigma square and we consider two possibilities possibility A when sigma 0 sorry sigma square is known and other possibility we deal with is when sigma square is unknown in either case we would like to discuss or we have already discussed the three alternates the first alternative which is a two sided alternative that mu is not equal to mu 0 second alternative is that mu is greater than mu 0 and the third alternative we have discussed is mu is less than mu 0 under both the cases that sigma square is known and sigma square is unknown we found when sigma square is known we found that test statistic is z which is x bar minus mu 0 over sigma square root n here we have assumed that we have a sample x1 x2 x3 xn is a random sample from normal population with mean mu and variance sigma square so x bar is an average of x1 x2 x3 xn and we found that this is distributed as normal 0 1 it has no unknown parameters it is completely known distribution and therefore if we consider the alternative that mu is not equal to mu 0 we found that the critical region can be defined as probability or the probability of critical region can be defined as z in absolute value is greater than some value z which should be alpha where alpha is fixed is the type 1 error this is under H0 that is when the null hypothesis is true we call this as the this is the critical region and this is the probability of critical region which is equal to alpha and then we find that actually in that case probability of z greater than small z 1 minus alpha by 2 is equal to alpha by 2 and this is the in that case z 1 minus alpha by 2 is a critical value means that the decision is going to be reject H0 if this statistics z is greater than z 1 minus alpha by 2 if we take the alternate of mu greater than mu 0 then again the test statistic is same is z and the critical region let us properly define it is x1 x2 xn such that z is greater than some value z small z this is your critical region under H0 so you can say that probability of critical region under H0 is equal to probability that z is greater than small z should be alpha and therefore what we find is that it has to be probability that z greater than z 1 minus alpha is equal to alpha if this 1 minus alpha by 2 and 1 minus alpha is confusing you I would like to remind you that you take a standard normal population this is normal 0 1 this is your mean 0 then z of z sub a is defined as this probability so that this probability is a so in this case if we want this probability to be alpha then a has to be 1 minus alpha and therefore we are taking 1 minus alpha or similarly we are taking this because this is 2 sided so then when we come to the next possibility is that alternate of H that mu is less than mu 0 then our critical region is going to be x1 x2 xn such that z is less than small z sorry it should be this way curled bracket so probability of c under H0 is equal to probability of z less than small z which should be alpha and therefore we find that probability of z less than z alpha is equal to alpha and therefore the critical value this is the critical value this is the critical value in this case the decision is in the previous case the decision is if reject H0 if z is greater than z small z1 minus alpha and in this case the decision would be reject H0 if z is smaller than small z alpha so this is what we have done so far next thing what we have made change is that suppose we are taking the sigma square as unknown then please recall that the z statistics turns into a t statistic which is same as x bar minus mu 0 divided by s over square root n where s is the sample standard deviation and now this is distributed as t with n minus 1 degrees of freedom so again there is no parametric dependence on it we already know n so we know n minus 1 so we know the distribution this should be small t sorry it is a distribution that we are already aware of and therefore now you can start that your H0 which says that mu is equal to mu 0 under H1 that mu is not equal to mu 0 your critical region is going to be all x1 x2 x3 xn such that the absolute value of t is greater than some t that is if you take a t distribution which is a symmetric distribution and we say that if t lies in any of this area it is our rejection region within this area we accept it so when with a fixed alpha from the beginning we would like to have probability of critical region given H0 which is probability of absolute t greater than small t which has to be alpha and therefore just as we did in the previous case capital T greater than 1 minus alpha by 2 has to be alpha by 2 and this becomes your criteria for decision where you will say that H0 to be rejected if t is greater than small t 1 minus alpha by 2 sorry here I should write n minus 1 and here also I should write n minus 1 because that specifies this degrees of freedom so on the same line quickly we can say that if we consider the testing of hypothesis H1 I mean the sorry the alternate of H2 which says that mu is greater than mu 0 then do I need to give much explanation please recall our decision is reject H0 if the same t is greater than small t 1 minus alpha with n minus 1 degree of freedom and in the case of the third hypothesis mu less than mu 0 third alternate hypothesis the the decision making is going to be reject H0 if t is less than t alpha n minus 1 n minus 1 in degrees of freedom so this is what we have learnt so far in the hypothesis testing the important two statistics we have found so far is z which is x bar minus mu 0 over sigma square root n distributed as a normal 0 1 and in the other case our test statistic is t which is x bar minus mu 0 over sample standard deviation square root n which is distributed as a students tree distribution with n minus 1 degrees of freedom in the today's case what we would like to do is consider a case when you want to test that sigma square itself is equal to some predecided value sigma 0 so today what we want to consider is a case of sigma square is equal to sigma naught square where sigma square is population variance and sigma naught square is given when this is the case when this is the case we start out that we have a sample x1 x2 x3 xn from again a normal population with mean mu and sigma square and we want to sigma square is unknown and we want to test the hypothesis that sigma square is a given value sigma naught square the situation can arise just as it would arise in a any testing of hypothesis for mu is equal to mu 0 again we consider the alternate hypothesis h1 first as sigma square is not equal to sigma naught square it is a two sided alternative ok let us follow the six steps let us follow the six steps of hypothesis testing so first step is that fix alpha ok so we have fixed up alpha number two clearly write your hypothesis null hypothesis which is sigma square is equal to sigma naught square and your alternative verses you are testing against the alternative h1 which says that sigma square is not equal to sigma 1 sigma 0 square ok what is the best statistic to represent this sigma 0 well we know that s square which is equal to 1 over n minus 1 summation i is equal to 1 to n xi minus x bar whole square this is just to recall is a good statistic because expected value of s square is equal to sigma square we have shown it in the past and therefore the fourth step we can define a critical region such that x1 x2 x3 xn should be such that the s square value is larger than some let us call it x value ok this is going to be our critical region so in that case fifth step is to identify the critical region so we find that n minus 1 times s square divided by sigma square is distributed as chi square with n minus 1 degrees of freedom please recall this n minus 1 times the sample variance divided by the population variance is distributed as chi square n minus 1 and chi square n minus 1 does not depend on any parameter it is n minus 1 which is already known because you already have a sample of size n so you already know n ok so therefore your probability is that n minus 1 s square over sigma square which I call it an x this will create a confusion so let us call it a y such under the condition h0 this probability should sorry this greater than x should be greater should be equal to alpha I will rewrite this so that there is a very clarity in it y which is equal to n minus 1 s square over sigma square is greater than x under h0 this should probability should be alpha if h0 is correct this means that you are looking for probability of y which is n minus 1 s square over sigma naught square greater than x should be alpha now because the alternate is on both sides actually this is insufficient we have to say that for alternate which is both sides that which says that sigma square is not equal to sigma naught square in a chi square distribution it should lie somewhere between two limits a and b where this is also alpha by 2 and this is alpha by 2 so considering that we modify our critical region as y less than a union y greater than b such that this is a mathematical sign for such that such that probability of y less than a under h0 is equal to alpha by 2 and probability of y greater than b I think it is better that I use an eraser because it is creating a problem so here we are y greater than b under h0 should be alpha by 2 so here please remember you cannot have a single critical value because chi square distribution is not a symmetric distribution it is a skewed distribution and therefore you cannot have a solution like you had it with respect to the standard normal distribution and the t distribution with n minus 1 degrees of freedom and therefore actually you are going to have to find two critical values and these two critical values can be given as chi square n minus 1 alpha by 2 and b is going to be chi square n minus 1 1 minus alpha by 2 please remember we are always considering the probability on the left hand side of the value so these are your critical value and your decision is going to be reject h0 if either p sorry either y which is n minus 1 a square over sigma naught square is less than chi square n minus 1 alpha by 2 or y is greater than chi square n minus 1 1 minus alpha by 2 I hope this is clear that we are what we find let us quickly go through it if we follow the sixth step this is the sixth step so if we follow the sixth step we fix the first alpha value so first we fix an alpha value this is our null hypothesis and the alternate hypothesis we find that s square is the correct statistic to take to find a test statistic for this or to find a critical region for this test and therefore this test statistic we chose as a sample variance because expected value of sample variance is the population variance then we define a critical region x1, x2, x3, xN such that x square should be greater than x and then we take appropriate statistic which would be devoid of any unknown parameter so we say that it is n minus 1 I better write a parenthesis here so it is n minus 1 square over sigma square actually here itself I should have used this kind of an arrangement but we have actually continued with the same arrangement but we find that when you simplify this and you want to find the critical value we realize that this is a two sided hypothesis and therefore we must have the two values a and b and therefore we find that the critical region can be modified to this and finally we come up with the test now I believe it is not very difficult to understand as to what would be the case if you take h is 0 as sigma square is equal to sigma naught square versus h2 which says that sigma square is greater than sigma naught square then naturally you will say that your test statistic is y which is n minus 1 s square over sigma square and this is distributed as chi square with n minus 1 degrees of freedom under h0 and therefore a probability of y greater than x has to be alpha now we are correct because if you look at the region we want the sigma square to be greater than sigma naught square this is sigma naught square then you would like to have it greater than some value which is a critical value which will be a critical value which I am calling x but now you can make out that in that case x is nothing but chi square with n minus 1 degrees of freedom and 1 minus alpha please remember again if you take the alpha chi square distribution and if you wish to have this value such that this is alpha in that case this area is 1 minus alpha and therefore this value is I call it sorry chi square n minus 1 1 minus alpha so that is the value which will come so the decision is going to be reject h0 if n minus 1 s square over sigma naught square is greater than chi square value of chi square with n minus 1 degrees of freedom at 1 minus alpha probability similarly you can easily show that if you want to test the hypothesis h0 is equal to sigma square is equal to sigma naught square versus the hypothesis h3 which says that sigma square is less than sigma naught square then your test statistic is same which is the test statistic is going to be n minus 1 s square divided by sigma 0 square which will be distributed as chi square with n minus 1 degrees of freedom and we are looking for a probability that y is less than x has to be alpha under h0 so if you look at the probability curve you are looking at this value of x so that this is alpha and this is given by chi square n minus 1 alpha and therefore your decision is going to be reject h0 if n minus 1 s square over sigma naught square is less than chi square with n minus 1 degrees of freedom and alpha probability so with this we conclude this session on testing of hypothesis for testing of hypothesis for sample variance so sample variance sorry not sample variance it should be population variance sigma square the hypothesis we tested is sigma square is a given value sigma naught square and we considered three alternative one is a two sided alternative which says that sigma square is not equal to sigma naught square second one say that sigma square is greater than sigma naught square and the third hypothesis say that sigma square is less than sigma naught square in all the cases we found that test statistic y is nothing but n minus 1 sample variance divided by sigma naught square which is distributed as chi square with n minus 1 degrees of freedom and then we derived the three tests thank you