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Hypothesis Tests for Mean
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0:00
In the name of of Allah the Merciful. Peace be upon you and God's mercy and blessings be with you Dr. Muhammad Al-Ashhab. And our date today is with the tenth lecture of the engineering statistics and probability theory course, which is titled “Homework Tests .” We will talk, God willing, through this lecture, on an introduction to the homework tests. We will apply hypothesis tests to the average of the population , and
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we will deal with two cases - the case of large samples and the case of small samples. We named it because it is often greater than or equal to 30 , or the population follows the normal distribution and the sigma is known . In this case , as we agreed, we use the (normal) Z distribution. The second is small samples, where the sample size is smaller than 30, and of course the population must follow the normal distribution, and in this case
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the sigma will be unknown and we have to use the T distribution
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. On the average of the society , the parameter here is what is meant by Mio, and statics is what is meant by the X-Bar . It simply means that I want to test a hypothesis related to the Mio, so I take a sample and according to it the X-Bar , and I can judge whether this hypothesis is correct or not . For example, if I have a company that manufactures cars And this company said that the average distance for its car is fifty miles per gallon
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, which means simply that the company claims that the miles are equal to 50 , so I want to test this hypothesis or the claim made For a company that claimed him and see if he is healthy or not, then of course I have to take a sample as we agreed here and on the basis of the sample average, a decision can be made whether the claim is valid or not , but of course it is possible when I come to take a sample, I mean, we assume that I took a sample and calculated the average for it, and
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so be it For example, 49.99 , so is this claim considered valid or not? Of course, many people may say that 49.99 is very close to 50. So we can estimate that the claim is valid. Well, if we say that it came out, for example, 49.9 only. There are people who can say, I mean, we will say that the claim is valid, and others will say no. Well, if we assume that it came out, for example, 48
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, which means it is close to the average, which is fifty. Of course, he did not say that every time it would be fifty , and he said that the average means that in some cases you can find 49, and you can find 48, and you can find 50, do we understand? Ok. I want to test this hypothesis and say whether it is acceptable or not. In this case, we must specify a specific point or a specific value.
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We say if the average of the sample was less than it. So we can say that the number of miles is less than 50, how many points can this be? It could be, for example, 45. It could be 40, depending on the case that I have , so we will learn today, God willing, how to determine this number that I can judge by, or whether it is acceptable or not . If he was claiming that the mew is equal to fifty , then I will name this first hypothesis for the one in which the equal sign is the equality sign
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. This sign can be less than or equal to, it can be equal to, and it can be greater than or equal to. Whoever has an equals sign, we call it. The second or the second individual we call him and he is the complement. meaning what? That I said here that it is a word that a mio equals fifty. Then it will be that the mio is not worth fifty. Therefore, they must feel that one of them must be correct, and what is the other? And the second one is wrong. Or one of them must be
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accepted and the other rejected. Ok, now we will learn how to do? Here he tells me.
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So the first issue here, he tells me
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, is the first thing , or what is the claim? What does it mean to buy it, how much? Of course, the word here means mew. And here he says that he did not say that he had no beeches or more beeches, so what does that mean? How many? Since there is a sign, this remains along, which is what we named it. Ok , what is his remains? Mew is not worth it. So what will we call it the HA ? who is. Ok here he told me to have a pal. Then I can say that Balf is for me a statement about what I want to
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judge by what is acceptable or not. In this case , it would be like this , that a meo would be equal to a thousand, that a meo would not be equal to a thousand. Of course, here we agreed that it would be ok to take another issue. It's the same issue. Just here he told me.
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The word means at least means is greater than or equal to greater than or equal to. We agreed that what has an equal sign is what? It remains here that is greater than or equal to a thousand. Therefore, it remains or will be that it belongs to her who is younger than. Of course, here is his claim that this is the case. The first case is the one that bothers me here. Ok let's take a third issue. He tells me here.
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Of course, what does it mean? bigger. Here I will say. Remains I have in greater than. What is left of her? that is less than or equal to. In case d Whoever equates jet with this case, then this case is a statement, and this case is a statement. Of course, in this case, he told me that this first case is a statement, so this is it. Then we learned from this issue that not every time he should be the one who says. Its definition is that it has
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an equal sign. that he shall have. Of course, let's see what the issue is about? It may or may not be. Of course, it is clear from the previous issues that I have three cases. It could be that the meo is smaller than a certain value. Or the myo is greater than a certain value. Or the meo is not equal to a certain value. Therefore, if we live in this situation , we will pray . Here he tells me that the mew is younger than Ki. This, of course, is a number.
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I mean, it could be fifty mosques. Here is when he tells me that the mio is younger than fifty. It means that in order to accept it. It must be the EXP that I calculated. Smaller than the K at a great value. Of course, it is smaller than that, which means that it will be located in the northern part. We call this case. The case is that I have it says that the greater than the greater than the greater than means that it will be located on
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the right side, so we call this. The third case is that it tells me that the meo is not worth it. The word not equal, of course, means that it is likely to be greater than or less than. Hence, we call it the area of acceptance. Or a rejection area that will be on the right and on the left. Of course, we will understand this more after a while through issues. The important thing is that we determine the situation that I have. Whether she or Otilde. Depending on if it is greater
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than it, if it is less than it is left tilde. If you are going to be born. Now let's learn about the types of eros that can appear in or. But what do we say? I mean, I don't care who among them is the one who touches the kilim, because as we have seen in the previous issues, it is possible that he is the one who touches the kilim or he is the one who touches the kilim. I have important here.
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I mean, I start at the beginning sincerely and say that this is it. Therefore, since it is in, then it is the one that contains the area in the text. So if we assume that the problem is that the problem that I had means that there is a region here on the right and a region here on the left. These areas are areas of rejection. So what does it mean? If I were to assume that the mew is equal to fifty. This value is equal to fifty. Ok, I will take. This will be calculated
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for her. Exploration will have value. If this value appears close to the meme, then although this meme is acceptable, and therefore we say that he is, or why don't we say that he is? Because we agreed that he is along, so he is. It remains my decision to be either I am working or I am. Or the second case, I don't say anything about it. Because it is as we agreed. Then I will come out of the decision of two. Either I work or I work.
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Ok when do I work? I do if it was so much smaller that it entered the area here. Or so much bigger that it entered the area here. So this area we call them, so if it falls inside here or here, it will remain, but if it appears in this area, we say or consider it acceptable. Of course, since they are for each other, therefore, if I reject one, the other will definitely be acceptable, or vice versa. Ok, in this case, I have a possibility
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, what is it? I said that I am defining here what is going on from this point to this point, and I say to him that this is okay, the area after it is the area as well as the one before this point. Of course, here I am talking about this also as a region. Well, while, of course, it is possible that, as we studied before, that this is his average, he calls it what it is equal to, if you think about it, on the islands of Ann. Well, it is possible that this
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EXPAR came out in this region. So if I refused, then I made a wrong decision. Got it? There is a possibility of error here. why? Because I took up space here, I said that if the murderer appeared in it, I would reject the claim. While it is indeed possible that the person is located in this area and the claim is valid. But this possibility will be very little. Therefore, I take the possibility of a very small one in order to reduce the error that I refuse
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and that he is healthy. In this case, I have the first type that we call what it
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means correct, but I have the second type that we call it, which
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means it may be wrong. However, I do not refuse. So we call it. Ok , we will summarize what we said in this table. If I have the H Note , it may and may not be. Ok, what is mine? As we agreed. The first thing we said is equivalent to the second one. Well if it was me it was my decision. Of course, what is my decision here? Well, if I'm still wrong. And I said it was my decision. I worked as needed. So what do we call this and symbolize it with the symbol? The second case to
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be. It remains that I am supposed to work for him. If I worked here , it means that my statement is a sound decision. But if I made it or redeemed it, it means that I made a mistake, and we call this mistake ok and symbolize it with the symbol beta. Ok, so we can understand more about the alpha And the beta we can take this simple drawing. What does this simple drawing say? If I have it on the left here
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, this is his. We will assume here that I am talking about, so I will do it if it comes out greater than a specific value, and let it be this value. Therefore, the area that is in green is all about the reaction area. Then it might actually be followed, and yet I reject it, so I will have done it, and it is possible to gain what I have. Located in District D. Of course, this area that we said is the area of acceptance. Therefore, if the EXPAR came in this
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region. Which is less than the limit of the beginning of the rejection zone. I will say that it is accepted, while it may be following another, which is a phrase , then I accepted it and said that it is being followed, while all of this yellow area is being followed. It has the symbol, and we will symbolize it with the symbol alpha. Of course, this alpha is the one we will deal with during the lecture. But I am this topic
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so that you understand that what I am taking is not correct by one hundred percent . Therefore, we are trying as much as possible so that this alpha is as low as possible, and the beta is also as low as possible. Ok , in order to do this , we need to define something called Level of Significance. Level of Significance or we can call it the level of morale. What we are saying is that we said , of course, that there is an error, so we want this to be the least possible.
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So this error that we agreed to call alpha, we will use it here as. I mean, when I tell you that five percent is a percent, you immediately understand that alpha is equal to how much? five percent. And vice versa, of course. Since we agreed that this is an error, we want the error to be as low as possible, so you will find very small values of alpha. The values we use most often mean that alpha is equal to ten out of a hundred, five out of a hundred, or
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one out of a hundred. Of course, the biggest value we use is that alpha is equal to one tenth or tenth of a percent. Ok, we understood this case. If I have the issue, it may be a case of the three countries. It's a b or a hotel test. Well, we agreed that in the case of the lift, the area where I will refuse will be on the left, because it says that the mew is smaller than the kza. In the case of an area, it will be located on the right. In the Bacon area, it is
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part on the right and part on the left. Of course, we call it, and therefore it is.
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Steps to do The first thing is that I have to have data, which is, of course, the data that remains mentioned in the issue. I start with my solution that I am doing by knowing what it is? What is it about? As we learned in the first three issues. And then I choose. And we agreed that we can use or use Zed. So it remains to calculate, and it is possible to calculate the increase in some cases, or calculate that in some cases. Then I make a decision
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if I will use the Z or the T. And then, according to the value of Z or the value of that. And then the last stage is that I am either the H Note who is or. Therefore, in the event that I did, then my conclogin says that the HAA is true because it is de-registered. And vice versa here, if I am the H Note, I mean I accepted it, then, therefore, or the H A is a true note. In order to choose, we used to say before that I have three cases. The first case is that
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Taiotide and Sigmanon are used. In this case , we use Z. The second case is that it is just or greater than or equal to thirty. In this case, I also use Zed. The third case is. In this case, who do I have to use? So we learned how to do the first step that is. And the second step is what it is. The third step , we will learn how to count. Who is this. We can calculate the value of z from this equation, that z is equal to on the islands of N, or we can calculate from the equation,
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but by substituting it, because what do you use instead of it? Ace who is. Of course, in the event that now was greater than or equal to thirty. I use the same law and replace it. What is its name and what is it used for because it is easier for many people than. Ok, in the event that I used it, how will I take the rest of my decision? I say here that the decision is taken on the basis of As we agreed that if it
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is smaller than what is a specific value. In this case, it remains, and therefore what
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you calculated was smaller than. And we will learn after a while how to count. The second case if it is greater than a specified value. In this case, it remains with me and remains on the right, and therefore I refuse if z is greater than z, alpha. The third case is that it is not equal to a certain value. So I have two rejection areas. So, if z is less than minus z, add alpha over both. Or if the value of z is greater than the z of alpha over both. Of course, this means that the sample was
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in the rejection area on the right. Here, it was rejected because the sample was in the rejection area, which is located on the left. And of course I have it Case de Totilde. The same words that we said about the zd apply to the one. But instead of comparing the z to the alpha, compare the one. Ok, let's start now by taking a group of issues in order to understand the most topic.
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