 तुम योग परोप्तिरट़िन सन्दी छोद तरीए होग क्या तॉछ आफ़ोग लग परिपृिशन क्या थे. if numerous samples are taken, the frequency curve made from proportions from the various samples will be approximately bell shape. यह बात हमेशा अपने यादरक नहीं है के अगर हम किसी भी population कि अंदा से n size के samples draw करें, उन सामपलस को plot करते जाएं distribution के अपर, तो shape of the curve will be approximately normal, especially if n size is greater. तो हम उसको जा पलोट करते जाते है, तो shape हमारी normal उती जाते है, इसे तना population proportion अगर find out करना है, ज़सा मैंने का क्या female proportion students का किसी भी campus के अपर find out करना चाते है, हर लगी ने 50-50 का या 20-20 का samples नहीं काला है, और हर अगने वता है कि मेंरे सामपल में 0.6 से proportion for female students, किसी ने का 0.55, किसी ने का 0.57, किसी ने का मेंरे 0.5 है, परपोर्ष्टन पर फीमेज तो हम उसको जा पलोट करते जाएंगे, तो हमारा जो है वो frequency curve के जो shape है, ताट विल भी आप्रोक्सिमेट ली नोरमल, हम कैसे उसको फींट रोड करते है, 95% confidence interval, I will show you in this table also. हम, to be exact, we would actually add the subtract 1.96. क्यों 1.96 आपने 95% area आपने कवर करना है, अगर ये 95% area है, और 95 को 2 पे divide कर दें, तो मैं क्या एड़य अगा, 47.5 आज़ाएगा, आप अगर हम नहीं यहां का Z find out करना है, तो बी exact 95% तो फिर मुझे table में जाएंगे value देकनीपगनी, सगी, सै मैं प्रप dermation तूब बाडी वाले कोलए में 0.4750 या 0.9500 देखूँँ कर आंप यह देख्ई स्थेबल की अंदर. तो यहां पे, मुझे काँपे जाक्ए, वैल्ँ मिलेगी, यह में नी टेल आगयूँ 0.0250 में अद्बताया दा गा आपे 0.0250, two standard deviation because 95% of the values for a bell-shaped curve fall within 1.96, which I have drawn and explained to you. However, is most practical application rounding 1.96 to 2, because sometimes you don't have a stable given. By the way, if you remember 1.96, that if we have to do a 95% area of the middle, then both our values will be minus 1.96 and plus 1.96. But if you don't remember the table value and you want to find out the 95% confidence interval, then you can take a rounded value of 2 and that will not make much difference. And it is also a practice and I have seen many people doing it. But to be exact, you actually take exact area and exact value. Constructing a confidence interval, applying the reasoning we used to construct the formula for the 95% confidence interval and using the information about bell-shaped curve, we can construct for instance 68% which is 1 standard deviation plus minus and we can construct 99% also which is 3 standard deviation instead of 1. For example, by simply adding and subtracting 1 standard deviation to the sample proportion instead of 2, we can remove it. Let's do an example. This is an example as a researcher reported the height of a random sample of 200 British couples and showed that in only 10 couples was the wife taller than the husband. We can construct confidence interval for the true proportion of British couples for whom that would be the case. He has drawn a sample. He studied 200 couples and out of 200 there were 10 couples in which the wife was taller than the husband. The researchers in Trusted's entire population know what the proportion would be. That is, how many couples there would be in which the wife would be taller than the husband or how much the proportion would be. So the first step is to calculate the sample proportion. The sample proportion would be 10 by 200 which is equal to 0.05 or 5%. You calculate standard deviation. For standard deviation, I have told you the formula that the standard deviation of sampling distribution of proportion will be equal to sample mean and then 1 minus sample proportion. Without using it, it means that you have the sample proportion only. Divided by n or under root. You will find out how you will plug in the value. You have taken out the proportion sample which is 0.05 and we will minus it from 1 to 0.05. That will be equal to 0.95 and we have n given which is 200 and we will under root. If we solve all this, it will be 0.0023 under root which is equal to 0.015. This is our standard deviation or standard error of the sampling distribution of proportion. Now we have to find out the confidence interval. The formula for confidence interval is sample proportion plus minus z into standard error. Confidence interval. You have given the sample proportion which is 0.05 and after that you have to minus z. You have found out 95% of the interval. We are finding 95% confidence interval. So z is our exact 1.96 and we have calculated the standard error which is equal to 0.015. If we solve this, the answer is equal to plus minus 1.96. So it is equal to 0.03 something. And when you do this, you will do 0.03 1 time plus and 1 time minus. So it is equal to 0.02 from 0.08. So look how neat it is because you have drawn a sample. And you are really interested in the bigger picture in the bigger population of the entire country. And from this 200 sample, you are not going to make a point estimation by telling that there will be 5% couples in which the wife will be taller because there is more margin of error. What will you do for that? You will find out 95% confidence interval. This means that you are 95% confident. You are 95% sure that the population parameter which is the population proportion would lie between these two values. So now we are 95% confident that your population proportion will be between 2% to 8% Just make a wife will be taller than the husband. Similarly, another example is that a random sample of 100 students is taken from the university. They are the overall proportion of female is 0.6. The example I was giving you is that you have to find out from your university that what is the proportion of female students on this new campus. So you have drawn a sample of 100 and the sample proportion is 0.6. You have to find out 95% confidence interval. What information do you need for 95% confidence interval? You know that for 95% your Z will be equal to 1.96. You have got it. And you know that the formula of confidence interval is the sample proportion plus minus Z into standard error. So you have to calculate the standard error. Standard error or standard error of sampling distribution of proportion would be equal to.. Remember the formula that we have done in the previous example is equal to sample proportion 1 minus sample proportion divided by N under root which is equal to your sample proportion is 0.6. 1 minus 0.6 will be 0.4. N is your 100 and you will take its under root. So it will come out. How much? You have to calculate it because I have multiplied it from Z. So for confidence interval the value that we will write is our sample proportion which is 0.6. Z is our 1.96 and we will multiply it with 0.6, 0.4, 100 under root. If we will multiply these two then I have the answer which is 0.096. So I will multiply 0.096 once and I will minus once to get the confidence interval. If I will minus once then the answer will be 0.504 and if I will multiply it once then it will be 0.696. Now I am 95% sure that the sample students which I have drawn which is proportion female is 60% and I want to know what will be the proportion of female students in the entire university. So I have multiplied it once and minused it once so now I am 95% sure that in the entire campus the proportion of female students is almost 70% and the population proportion would lie between these two values and I am 95% confident about it. How did I find out that I drew the distribution and then I drew this area and I found that the population parameter that is the proportion of the population would lie between this value and this value.