 Hello, we are on to module 7 now, which is discussing one sample hypothesis testing. So our first stop is looking at an introduction or an overview of what a hypothesis test actually is. So whenever you are performing a hypothesis test, you identify what we call the hypotheses and then you mark which one is your claim. Next you compute the appropriate test statistic, then you find the p-value or what is known as a critical value and then you write a conclusion. These are the steps that we will be going through by each one as we proceed here. So first, what is a hypothesis? So a hypothesis is a claim or statement about the property of a population. So for instance, we could be looking at a population mean or we could be looking at a population proportion and we are going to make a statement whether that mean or proportion is greater than, less than, equal to, not equal to some sort of value. So you always have two hypotheses but we will talk more about those in just a minute. So a hypothesis test is a procedure for determining whether the stated null hypothesis should or should not be rejected. So we will have two hypotheses, we will determine which one we should keep and which one we should quote reject so to speak. I will get more specific about this in just a minute because there is some really specific terminology you should use when you are performing a hypothesis test. So you have two hypotheses when you are performing a hypothesis test and it is always your first step to write these two hypotheses down. Your first hypothesis is called the null hypothesis. You often see abbreviated as H sub 0. It is a statement that the value of a population parameter like a mean or a proportion or even a standard deviation is, this is underlined and colored for a reason, is equal to some claimed value. The important thing is that equality equal to always goes with the null hypothesis. It goes with the null hypothesis, equality always goes with it and back when I was learning hypothesis testing, if I would have known this one statement here it would have saved me hours of frustration with identifying my hypotheses. So the null hypothesis is always the hypothesis that we are testing. It is always the hypothesis we assume to be true and then based on our assumption that the null hypothesis is true it allows us to reach a conclusion on whether we reject that hypothesis or fail to reject that hypothesis. So based on our sample evidence we determine do we reject the null hypothesis or do we basically keep it? The symbolic form of the null hypothesis must use one of the following symbols. Remember I said the null hypothesis has to contain equality, so less than or equal to, greater than or equal to or even just equal to. And just to let you know that it's not uncommon for some people always use equal to, always, they'll just generically always put equal to for the null hypothesis, which isn't wrong. The alternative hypothesis, the other hypothesis, which is denoted by HA or H1 depending on who your instructor is or what your textbook is, is a statement that the parameter has a value that differs from the null hypothesis. It is the opposite sign of the null hypothesis, so the symbolic form of the alternative hypothesis must use one of the following symbols. You have less than, you have greater than, you have not equal to. So I'm just going to give you a little bit of a sketch here. So if the null hypothesis uses less than or equal to, well the opposite sign that would have to be used in the alternative would be greater than. If the null hypothesis uses greater than or equal to, then the alternative hypothesis would have to be less than. And if the null hypothesis uses equal to, then the alternative would have to be, well not equal to. So they will give you, in the context of the question, they will always give you one of the hypotheses and then you use this opposite approach to find out what sign your other hypothesis should use. So also very important table. So a random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. You would like to test whether the population mean time on death row could likely be 15 years. I want to identify the hypotheses. So you have two hypotheses. You have h0, that's your null, and then you have h1, that's your alternative. And we're talking about a population mean here. So population mean, that means we're talking about mu. So based on the information and the question, what am I testing? We are testing whether the mean time on death row could likely be 15 years. Likely be 15 years. How do you translate that to math? Likely be is equal to, equal to 15. So where would this go? Would this go if the null or the alternative? Well, since it contains equality, that has to be your null hypothesis. So mu equals 15 is the null. So what does that mean the alternative would be? What is the opposite of equal to? Well, not equal to. Those are our hypotheses. Which one is the claim? Which one is the one given in the context of the question? Well, that would be your null. All right, so that's how you identify hypotheses. Let's do another one. A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with the standard deviation of 6.3 years. You would like to test whether the population mean time on death row could be at least 15 years. Identify the hypotheses. So you have your null hypothesis and your alternative hypothesis. We're going to find these. So I'm talking about a mean, so I'm talking about mu at least. At least means greater than or equal to 15 years. So where would greater than or equal to go? Would it go with the null or would it go with the alternative? Since it contained equal to, it's going to have to go with the null hypothesis. And that's our claim. That's what was given in the question. So what is the opposite of greater than or equal to? The opposite of greater than or equal to would be less than. And then you use the same number. Always have opposite signs when you're dealing with your hypotheses. So those are the hypotheses for that question. Alright, in example 7-2 which we're going to do several things with this example. We assume that 100 babies are born to 100 couples treated with a certain method of gender selection that is claimed to make girls more likely. That's very important. You observe 58 girls and 100 babies. So there's your sample data. I want us to write the hypotheses to test the claim that with the gender selection method the proportion of girls is greater than 50%. So all I'm doing for part A of 7-2 is I'm writing out my hypotheses. We have what our claim is greater than 50%. The proportion is greater than 50% or 0.5. Where would this go? Would this go with the null or would this go with the alternative? Well, does it contain equality? Do you see it equal to sign? No. So it goes to the alternative hypothesis. So as a result what is the opposite of greater than? It's less than or equal to. Or like I said some people just stay generic and they always put equal to for the null hypothesis, which is fine. You can do that. Either is correct. Alright, so which one was our claim? Well, the alternative hypothesis is our claim in this case. So either hypothesis could be your claim just to let you know. So to perform a hypothesis test we take the sample data and we calculate what is called a test statistic. If you want to have a little fun try saying that 10 times fast. The test statistic is a value used to make a decision about the null hypothesis. Do we reject it or do we keep it or fail to reject it? It is found by converting the sample statistics to a z-score or a t-score with the assumption that the null hypothesis is true. We'll basically use a formula to find this. Based on what you're trying to run a hypothesis on for instance if you're running a hypothesis test on a population proportion p then you're given a z-score formula p hat minus p divided by the square root of p times q over n. p has your sample proportion and then p is your predicted population proportion value used in your hypotheses. q is always one minus p. If you're looking for a mean mu and the sigma is not known then you have a t-score test statistic formula and then if you're looking for population mean and sigma is known you can use your basic z-score formula x bar minus mu divided by sigma over square root of n. So we pick the appropriate formula and we apply it. So consider the claim that the certain method of gender selection increases the likelihood of having a baby girl. Preliminary results from a test of the method of gender selection involved on 100 couples who gave birth to 58 girls and 42 boys. Use the given claim and the preliminary results to calculate the value of the test statistic. So remember what our hypotheses were. This is from 7 to part a. I have mu you have oops sorry p I have p equal to 0.5 and I have p greater than 0.5. Remember that's our claim. So let's calculate the value of the test statistic. So since we're messing with the population proportion there's only one formula to pick from z equals p hat minus p over the square root of pq over n. So here's our recipe or a formula let's plug in what we know. All right so I know so I know that p hat we're looking at proportion of girls. I know that p hat is well 58 from my sample 58 out of 100 were girls so that's 0.58. I know p the proportion value is 0.5 that's what we're assuming the true proportion value to be. I know that q is 1 minus 0.5 or 1 minus p and then I have my sample size n which is going to be 100 100 couples. So basically I have 0.58 minus 0.5 I'm literally going to plug and chug divided by the square root of 0.5 times 0.5 over 100. All right so when you do this calculation you basically get 0.08 divided by the square root of 0.0025 0.5 times 0.5 divided by 100 is 0.0025 and that's going to give you 0.08 divided by 0.05. Your test statistic is going to be 1.6 you're like well that's great and all but why in the world do we need a test statistic well let's find out.