How do we then do the proportions? Testing for the proportions. When we test for the proportion, we use the Z test. And if you are not given the sample proportion, you will be given the X that satisfies the proportion. And you can calculate your sample proportion. And marketing company claims it receives 8%. So now, here is the catch. When you read the questions, they might give you a P for a proportion or a Pi for the population proportion so that you know that this is for the proportion. Sometimes it also helps because they might give it to you as a percentage. So you should know that this is a proportion that we're talking about. To test this claim, a random sample of 500 ways are made with 25 questions. Test the hypothesis at 0, 0, 5. So here, they also give us our X and our N. So it means they didn't give us the proportion so we need to calculate that. The six steps still relevant, even with the proportion. So you just need to know all the six steps of hypothesis testing for this way. Let's look at how we do that. So we know that they set 8% of the remaining. So it is equal. Therefore, it is not equal in our alternative hypothesis. This is a two-tailed test. State what we are given, our alpha of 0, 0, 5, our N of 500, and our P. Remember, it was 25 over 500. I think it was 25 divided by 500, 0, 0, 5. So that was our PX over N, which gives us 0, 0, 5. Our critical value, we know by now, alpha divided by 2 was 0, 0, 5. It's always going to be 1, 9, 6. What I would suggest, when you go write the exam, because now you're going to write online, it's easy. Not that I'm teaching you how to cheat, but I'm making your life easy for when you go write the exam, especially when you go for this type of test. So you can create yourself a table and call it the critical value table. What you do is, here you will have at 90, maybe you say confidence level. So you will create for yourself, sorry. So you will have a confidence level. So let's say it's 95%, which is 0, 0, 5, which then you say it's alpha. And then here you say Z alpha over 2, and here you say Z alpha. And then you say 1, 9, 6. And then you go find the value for this one, because it's not 1, 9, 6, it's different. And you do the same for 90 and have the 0, 1, 0. And you go find the value for it, which is 1, 6, 4, 5, and so forth and so forth. So you will have that next to your material. When you go write the exam, you don't have to go to the tables and go find, find, find, find. It will always be there. But you will need to make life easy for yourself when you go write the exam, OK? So with the critical value we've defined, we can say where our region of rejection will be. We can redefine them, because it's a two-tail test. Then we have two regions of rejection, OK? Then we go calculate the test statistic, our P. Our sample proportion was 0, 0, 5. Our sample proportion was 0, 0, 5. Population proportion is always in the hypothesis. Substitute, divide by the square root of, which is our standard error, which is our square root of our population proportion times 1 minus our population proportion divide by sample size, which is 0, 0, 8 times 1 minus 0, 0, 8 divide by 500. We take the square root, and we simplify the whole equation with it. Minus 2.47, and when minus 0.247 falls, it falls on the rejection area on the negative side. And then we can then conclude by saying, we reject the non-hypothesis at alpha 0, 0, 5, and conclude that there is sufficient evidence to reject that the company claim of 98% of the response rate exists. And that's how you do the proportion, OK? Likewise, if we use the P value, we take our test statistic, which was, remember, we found that it is minus 2.47. And remember, since it is negative, it's a two-tail. So since it's negative, and it's a two-tail, we go to the table, we go find the value on the table. The probability on the table is 0, 0, 6, 8. It's 0, 0, 6, 8. Since it's on both sides, we then have to multiply 0, 0, 8, or add 0, 0, 8 to 0, 6, 8 to itself. And then we find that the P value will be 0, 0, 1, 3, 6. And since our alpha was 0, 0, 5, therefore, we can also safely say we reject the non-hypothesis because our P value, it's less than our alpha. And we can reach the same conclusion. Let's look at this example. An airline claims that 6% of all lost luggage is never found. A random sample of 70 out of 200 luggage were found, are not found. Here's the hypothesis. At the population proportion of 0, 0, 6 against the alternative, that is more than 0, 0, 6. So it makes our life easy because they tell us this is an upper tail area, and it's a one-side test. Clip number one, state what you are given. The non-hypothesis and the alternative, they have stated it on our behalf. Non-hypothesis, the mean equals, not the mean, the proportion is its proportions. Proportion is 0, 0, 6. The alternative, they told us the proportion is greater than 0, 0, 6. State what you are given. We can just calculate our sample proportion here, which is x over n. 17 divided by 200 equals 0, 0, 8, 5. And our alpha, they didn't give the alpha and so forth and so forth. So so far, I can see that number one will be correct. Calculating the standard error, the standard error is your population proportion 1 minus the population proportion divided by n. We root of our population proportion is 0, 0, 6 times 1 minus 0, 0, 6 divided by 200. You calculate that, you find the square root of 0.06 times 0.1 minus 1 minus 0.06 divided by 200. The answer we get is 0, 0, 0, 1, 6, 6, 8. That gives us 0, 0, 1, 6, 8. Standard error, 0 comma, that is incorrect. The test statistic is equals to, so therefore it means I must calculate our z-stat, which is p minus divided by the standard error. Let's do it here at the bottom. z-stat is p minus population proportion divided by population proportion 1 minus population proportion divided by n. Our sample proportion 0.085 minus our population proportion 0, 0, 6 divided by the standard error I've calculated it 0 comma, 0, 1, 6, 8. So 0.085 minus 0.06 equals 0.025 divided by 0.0168 equals 1.4888. 1.488, which standard is not correct. The p-value, since this is positive, I need to go find the p-value. So this is positive. It means the value I find on the table. So to find the p-value, I will use 1 minus the value I find on the table. We go to the table. We go to the 1.49. So we go to 1.49 on the z-table. We need to make sure that we are on the right table. On the z-cumulative standardized table. And this was positive. So we go look for 1.49. 1.49 is equals to 0 comma, 9319. 0.93119, which then gives us 1 minus 0.93119 gives us 0.036. P-value, 0 comma, 1.4, that is not correct. The null hypothesis is rejected. So if we use the p-value, so we know that for this, my pen doesn't want to write anymore. So our p-value is 0.068, 0.068, and our alpha is 0 comma, 1.0. So our p-value, 0 comma, 0.068, and our alpha is 0 comma, 1.0. So is this less than or greater than? If the p-value, if it's less than, we reject. And here is the problem we have. The null hypothesis is rejected at alpha, 0 comma, 0, 1. So our p-value in this instance is 0 comma, 6, and our alpha is 0 comma, 1. So you will have two questions that are almost exactly the same. Any question? Okay, so if there are no questions, before I recap, remember, you can do your last exercise, which you should be able to answer question 9 and question 10. And based on the information given, so they have told you it's a two-tail test. They have calculated the sample proportion.  Use that. Then you go find the p-value. After you have calculated your test statistic, you will find the answer you're looking for. If your test statistic is negative, then you will go and find the value on the table and multiply that value by 2, and that will give you your p-value. Never say I didn't help you to answer your questions. And with that, she concludes today's session. Just to recap on what we did, we looked at how we calculate the hypothesis testing for the... Oh, we actually started with the basic concepts of the hypothesis testing. Then we looked at how we calculate the hypothesis testing for the mean. Then for the mean, when the population standard deviation is unknown. And we looked at how we calculate the hypothesis testing for the proportion. Remember, with hypothesis testing, the six steps are very important. And now, hypothesis and the alternative hypothesis, the sign under the alternative hypothesis helps you to understand whether you're doing a two-tail test or a one-tail test. It is very important to know those because if you're doing a two-tail test, you will know how to find your critical value and it defines your two areas of rejection. And also, when you do the Z test, you can use your information about the two-tail test to find your p-value. And with that, thank you guys for coming through. If you have any questions, you can please go ask. Thank you very much.