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Margin of error 1 | Inferential statistics | Probability and Statistics | Khan Academy

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Finding the 95% confidence interval for the proportion of a population voting for a candidate.
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{'title': 'Margin of error 1 | Inferential statistics | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 481.709, 'start': 428.976, 'weight': 0.744}, {'end': 639.46, 'start': 592.293, 'weight': 0.729}], 'summary': 'Discusses voting distribution, bernoulli distribution with 100 million people, parameter estimation through random sampling, calculation of sample mean and variance, and confidence interval estimation for the true mean voting percentage targeting a 95% confidence level.', 'chapters': [{'end': 103.676, 'segs': [{'end': 40.295, 'src': 'embed', 'start': 18.004, 'weight': 1, 'content': [{'end': 27.089, 'text': "Let's say I live in a very decisive country and everyone is going to vote for either and everyone participates in the election and everyone is going to vote for either candidate A or candidate B.", 'start': 18.004, 'duration': 9.085}, {'end': 34.253, 'text': "And so there's some percentage, there's some reality there that p, let me write it over here, maybe 1 minus p percent.", 'start': 27.731, 'duration': 6.522}, {'end': 36.514, 'text': 'Let me do the p first.', 'start': 34.273, 'duration': 2.241}, {'end': 40.295, 'text': "There's some reality that maybe p percent will vote for b.", 'start': 37.054, 'duration': 3.241}], 'summary': 'In a decisive country, all vote for candidate a or b, with p% supporting b.', 'duration': 22.291, 'max_score': 18.004, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw18004.jpg'}, {'end': 103.676, 'src': 'embed', 'start': 60.216, 'weight': 0, 'content': [{'end': 66.219, 'text': "And right here, the values I said you're either voting for candidate A or you're voting for candidate B, it's very hard to deal with those values.", 'start': 60.216, 'duration': 6.003}, {'end': 68.98, 'text': "You can't calculate a mean between A and B and all of that.", 'start': 66.239, 'duration': 2.741}, {'end': 69.641, 'text': 'Those are letters.', 'start': 69.04, 'duration': 0.601}, {'end': 70.401, 'text': "They're not numbers.", 'start': 69.661, 'duration': 0.74}, {'end': 79.726, 'text': "So to make it manipulatable mathematically, we're going to say sampling someone who's going to vote for A is equivalent to sampling a 0.", 'start': 70.781, 'duration': 8.945}, {'end': 85.987, 'text': "And sampling someone who's going to vote for B is equivalent to sampling a 1.", 'start': 79.726, 'duration': 6.261}, {'end': 89.309, 'text': 'And if you do that with a Bernoulli distribution?', 'start': 85.987, 'duration': 3.322}, {'end': 100.794, 'text': 'we learned in the video on Bernoulli distributions that the mean of this distribution right here is going to be equal to p.', 'start': 89.309, 'duration': 11.485}, {'end': 103.676, 'text': "And it's a pretty straightforward proof for how we got that.", 'start': 100.794, 'duration': 2.882}], 'summary': 'Transforming voting choices into numerical values to calculate mean and bernoulli distribution.', 'duration': 43.46, 'max_score': 60.216, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw60216.jpg'}], 'start': 0.555, 'title': 'Voting distribution and bernoulli', 'summary': 'Discusses the voting distribution in a country with 100 million people, where p percent will vote for candidate b and 1 minus p percent will vote for candidate a, showcasing the bernoulli distribution.', 'chapters': [{'end': 103.676, 'start': 0.555, 'title': 'Voting distribution and bernoulli', 'summary': 'Discusses the voting distribution in a country with 100 million people, where p percent will vote for candidate b and 1 minus p percent will vote for candidate a, showcasing the bernoulli distribution.', 'duration': 103.121, 'highlights': ['In a country of 100 million people, p percent will vote for candidate B and 1 minus p percent will vote for candidate A', 'The distribution of people voting for candidate A or B can be represented using the Bernoulli distribution, where the mean is equal to p', 'Sampling someone who will vote for A is equivalent to sampling a 0, while sampling someone who will vote for B is equivalent to sampling a 1']}], 'duration': 103.121, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw555.jpg', 'highlights': ['The distribution of people voting for candidate A or B can be represented using the Bernoulli distribution, where the mean is equal to p', 'In a country of 100 million people, p percent will vote for candidate B and 1 minus p percent will vote for candidate A', 'Sampling someone who will vote for A is equivalent to sampling a 0, while sampling someone who will vote for B is equivalent to sampling a 1']}, {'end': 435.422, 'segs': [{'end': 131.326, 'src': 'embed', 'start': 103.716, 'weight': 1, 'content': [{'end': 111.179, 'text': 'So the mean of this distribution, which will actually be not a value that this distribution can take on, is going to be someplace over here.', 'start': 103.716, 'duration': 7.463}, {'end': 113.3, 'text': 'And it is going to be equal to p.', 'start': 111.439, 'duration': 1.861}, {'end': 116.758, 'text': 'My country has 100 million people.', 'start': 114.997, 'duration': 1.761}, {'end': 123.882, 'text': 'It is practically or it is definitely impossible for me to be able to go and ask all 100 million people who are they going to vote for.', 'start': 116.938, 'duration': 6.944}, {'end': 131.326, 'text': "So I won't be able to exactly figure out what these parameters are going to be, what my mean is, what p is going to be.", 'start': 124.242, 'duration': 7.084}], 'summary': "The distribution's mean, p, is impossible to obtain with 100 million people.", 'duration': 27.61, 'max_score': 103.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw103716.jpg'}, {'end': 179.03, 'src': 'embed', 'start': 145.711, 'weight': 0, 'content': [{'end': 147.913, 'text': "So I'm going to try to estimate p with a sample.", 'start': 145.711, 'duration': 2.202}, {'end': 151.756, 'text': "And then we're also going to think about how good of an estimate that is.", 'start': 148.233, 'duration': 3.523}, {'end': 156.58, 'text': "So let's say I am going to randomly survey 100 people.", 'start': 152.517, 'duration': 4.063}, {'end': 161.864, 'text': "And let's say I got the following results.", 'start': 156.7, 'duration': 5.164}, {'end': 177.809, 'text': "Let's say that 57 people say that they were going to vote for person A.", 'start': 161.884, 'duration': 15.925}, {'end': 179.03, 'text': 'Let me write it this way.', 'start': 177.809, 'duration': 1.221}], 'summary': 'Estimating p with a sample of 100 people, 57% vote for person a.', 'duration': 33.319, 'max_score': 145.711, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw145711.jpg'}, {'end': 241.506, 'src': 'embed', 'start': 212.505, 'weight': 4, 'content': [{'end': 216.887, 'text': 'My sample mean right here.', 'start': 212.505, 'duration': 4.382}, {'end': 220.248, 'text': "well, that's just going to be the average of these 0's and 1's.", 'start': 216.887, 'duration': 3.361}, {'end': 221.369, 'text': "So I got 57 0's.", 'start': 220.268, 'duration': 1.101}, {'end': 225.796, 'text': "So it's going to be 57 times 0.", 'start': 221.449, 'duration': 4.347}, {'end': 229.859, 'text': 'plus my 43 ones, so the sum of all of my samples.', 'start': 225.796, 'duration': 4.063}, {'end': 237.944, 'text': "So it's 43 ones plus 43 times 1 over the total number of samples I took, over 100.", 'start': 230.099, 'duration': 7.845}, {'end': 240.145, 'text': 'So what does this get me? So 57 times 0 is 0.', 'start': 237.944, 'duration': 2.201}, {'end': 241.506, 'text': '43 times 1 divided by 100 is 0.43.', 'start': 240.145, 'duration': 1.361}], 'summary': "Average of 0's and 1's: 57% 0's, 43% 1's.", 'duration': 29.001, 'max_score': 212.505, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw212505.jpg'}, {'end': 435.422, 'src': 'embed', 'start': 354.918, 'weight': 2, 'content': [{'end': 357.14, 'text': "So I'll do the numerator first.", 'start': 354.918, 'duration': 2.222}, {'end': 359.061, 'text': 'I have 57 times 0 minus 0.43 squared.', 'start': 357.16, 'duration': 1.901}, {'end': 375.413, 'text': 'plus 43 times 1 minus 0.43 squared.', 'start': 372.932, 'duration': 2.481}, {'end': 399.718, 'text': 'And then all of that divided by 100 minus 1, or 99, is equal to 0.2475.', 'start': 375.433, 'duration': 24.285}, {'end': 403.121, 'text': 'So my sample variance is equal to 0.2475.', 'start': 399.718, 'duration': 3.403}, {'end': 410.347, 'text': 'And if I want to figure out my sample standard deviation, I just take the square root of that.', 'start': 403.121, 'duration': 7.226}, {'end': 415.611, 'text': 'My sample standard deviation is just going to be the square root of my sample variance.', 'start': 410.828, 'duration': 4.783}, {'end': 420.075, 'text': 'So I take the square root of that value that I just had, which is 0.497.', 'start': 415.792, 'duration': 4.283}, {'end': 428.976, 'text': 'So actually, let me just round that up as 0.50.', 'start': 420.075, 'duration': 8.901}, {'end': 435.422, 'text': 'So my sample standard deviation is 0.50.', 'start': 428.976, 'duration': 6.446}], 'summary': 'Sample standard deviation is 0.50, calculated from sample variance of 0.2475.', 'duration': 80.504, 'max_score': 354.918, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw354918.jpg'}], 'start': 103.716, 'title': 'Sampling and parameter estimation', 'summary': 'Covers the estimation of parameter p through random sampling of 100 people, yielding 57 votes for person a and 43 for person b, along with the calculation of sample mean, variance, and standard deviation using the same sample size.', 'chapters': [{'end': 211.745, 'start': 103.716, 'title': 'Estimating parameter p with sampling', 'summary': 'Explains the process of estimating parameter p through random sampling, with a sample size of 100 people, where 57 people are likely to vote for person a and 43 for person b.', 'duration': 108.029, 'highlights': ['The mean of this distribution is going to be equal to p, and the process of estimating p through random sampling is explained.', 'A random survey of 100 people is conducted, where 57 people are likely to vote for person A and 43 for person B.', 'The sample mean and sample variance for the given sample are to be determined.']}, {'end': 278.041, 'start': 212.505, 'title': 'Sample mean and variance calculation', 'summary': 'Explains the calculation of sample mean and variance using a set of 100 data points, yielding a mean of 0.43 and detailing the formula for sample variance.', 'duration': 65.536, 'highlights': ['The sample mean is calculated as the average of the data points, yielding a value of 0.43 for the given 100 data points.', 'The formula for sample variance is explained as the sum of squared distances to the mean divided by the number of samples minus 1.']}, {'end': 435.422, 'start': 278.401, 'title': 'Calculating sample standard deviation', 'summary': 'Explains the process of calculating sample variance and sample standard deviation using a sample size of 100, resulting in a sample variance of 0.2475 and a sample standard deviation of 0.50.', 'duration': 157.021, 'highlights': ['The sample variance is calculated using the formula 57 times 0 minus 0.43 squared plus 43 times 1 minus 0.43 squared, all divided by 99, resulting in a sample variance of 0.2475.', 'The sample standard deviation is obtained by taking the square root of the sample variance, resulting in a sample standard deviation of 0.50.']}], 'duration': 331.706, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw103716.jpg', 'highlights': ['A random survey of 100 people is conducted, where 57 people are likely to vote for person A and 43 for person B.', 'The mean of this distribution is going to be equal to p, and the process of estimating p through random sampling is explained.', 'The sample standard deviation is obtained by taking the square root of the sample variance, resulting in a sample standard deviation of 0.50.', 'The sample variance is calculated using the formula 57 times 0 minus 0.43 squared plus 43 times 1 minus 0.43 squared, all divided by 99, resulting in a sample variance of 0.2475.', 'The sample mean is calculated as the average of the data points, yielding a value of 0.43 for the given 100 data points.']}, {'end': 900.998, 'segs': [{'end': 486.057, 'src': 'embed', 'start': 458.26, 'weight': 0, 'content': [{'end': 473.306, 'text': "Let's try to think of an interval around 43% for which we are 95% reasonably confident, or roughly 95% sure that the real mean is in that interval.", 'start': 458.26, 'duration': 15.046}, {'end': 474.706, 'text': 'And let me make it very clear.', 'start': 473.486, 'duration': 1.22}, {'end': 475.507, 'text': 'Let me draw.', 'start': 474.986, 'duration': 0.521}, {'end': 481.709, 'text': 'So when we get our sample mean, we are sampling from the sampling distribution of the sampling mean.', 'start': 475.907, 'duration': 5.802}, {'end': 483.756, 'text': 'So let me draw that.', 'start': 482.876, 'duration': 0.88}, {'end': 486.057, 'text': 'The sampling distribution of the sample mean.', 'start': 483.836, 'duration': 2.221}], 'summary': 'Attempting to find a 43% interval with 95% confidence in the sampling distribution of the sample mean.', 'duration': 27.797, 'max_score': 458.26, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw458260.jpg'}, {'end': 543.404, 'src': 'embed', 'start': 512.705, 'weight': 2, 'content': [{'end': 517.166, 'text': 'The odds that your sample mean would be 1 would be very low probability.', 'start': 512.705, 'duration': 4.461}, {'end': 519.986, 'text': 'And then you would have one more bar, a bar like that, a bar like that.', 'start': 517.186, 'duration': 2.8}, {'end': 521.067, 'text': 'But that takes forever to draw.', 'start': 520.027, 'duration': 1.04}, {'end': 525.768, 'text': "So I'm just going to approximate it with this normal curve right over there.", 'start': 521.107, 'duration': 4.661}, {'end': 531.57, 'text': 'And so the sampling distribution of the sample mean, let me write it over here.', 'start': 527.069, 'duration': 4.501}, {'end': 535.231, 'text': 'So this is the sampling distribution of the sample mean.', 'start': 531.61, 'duration': 3.621}, {'end': 543.404, 'text': 'of the sample mean.', 'start': 541.682, 'duration': 1.722}], 'summary': 'The probability of a sample mean being 1 is very low, so it is approximated with a normal curve for the sampling distribution.', 'duration': 30.699, 'max_score': 512.705, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw512705.jpg'}, {'end': 639.46, 'src': 'heatmap', 'start': 592.293, 'weight': 0.729, 'content': [{'end': 602.623, 'text': 'The standard deviation of this distribution, that distance right over here, the standard deviation of the sampling distribution of the sample,', 'start': 592.293, 'duration': 10.33}, {'end': 606.987, 'text': "mean we've seen it multiple times already it's going to be this standard deviation.", 'start': 602.623, 'duration': 4.364}, {'end': 610.27, 'text': "It's going to be the standard deviation of our population distribution.", 'start': 607.447, 'duration': 2.823}, {'end': 613.97, 'text': 'So that standard deviation is going to be that distance over there.', 'start': 610.768, 'duration': 3.202}, {'end': 617.412, 'text': "So there's some standard deviation associated with this distribution.", 'start': 614.41, 'duration': 3.002}, {'end': 623.116, 'text': "It's going to be that standard deviation divided by the square root of our sample size.", 'start': 617.812, 'duration': 5.304}, {'end': 628.699, 'text': 'And we saw many videos ago why that at least experimentally makes sense, or why it intuitively makes sense.', 'start': 623.156, 'duration': 5.543}, {'end': 631.141, 'text': "So it's going to be the square root of 100.", 'start': 629.56, 'duration': 1.581}, {'end': 639.46, 'text': "So it's going to be this guy divided by 10.", 'start': 631.141, 'duration': 8.319}], 'summary': 'Standard deviation of sampling distribution is distance divided by sample size square root.', 'duration': 47.167, 'max_score': 592.293, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw592293.jpg'}, {'end': 668.706, 'src': 'embed', 'start': 641.122, 'weight': 3, 'content': [{'end': 647.708, 'text': 'The only way to figure out what that guy is is to actually survey 100 million people, which would have been impossible.', 'start': 641.122, 'duration': 6.586}, {'end': 657.559, 'text': 'So to estimate the standard deviation of this, we will use our sampling standard deviation as our best estimate for the population standard deviation.', 'start': 648.089, 'duration': 9.47}, {'end': 664.805, 'text': 'So we could say, now remember, this is an estimate.', 'start': 663.305, 'duration': 1.5}, {'end': 668.706, 'text': 'We cannot come up with the exact number for this just from a sample.', 'start': 664.845, 'duration': 3.861}], 'summary': 'Estimating standard deviation using sampling for 100 million people survey.', 'duration': 27.584, 'max_score': 641.122, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw641122.jpg'}, {'end': 779.392, 'src': 'embed', 'start': 745.778, 'weight': 1, 'content': [{'end': 757.365, 'text': "I'm really confident that there's a 95% chance that the true mean is within two standard deviations.", 'start': 745.778, 'duration': 11.587}, {'end': 758.366, 'text': 'Or let me put it this way.', 'start': 757.486, 'duration': 0.88}, {'end': 761.368, 'text': "There's a 95% chance that the true mean is in that interval.", 'start': 758.386, 'duration': 2.982}, {'end': 762.369, 'text': 'So let me write this down.', 'start': 761.428, 'duration': 0.941}, {'end': 779.392, 'text': 'I want to find an interval such that such that I am reasonably confident.', 'start': 763.109, 'duration': 16.283}], 'summary': '95% chance true mean within 2 standard deviations', 'duration': 33.614, 'max_score': 745.778, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw745778.jpg'}], 'start': 435.422, 'title': 'Confidence interval for voting percentage', 'summary': 'Discusses estimating the confidence interval for the true mean voting percentage, based on a sample mean of 43% with a standard deviation estimate of 5% and targeting a 95% confidence level in the interval.', 'chapters': [{'end': 900.998, 'start': 435.422, 'title': 'Confidence interval for voting percentage', 'summary': 'Discusses estimating the confidence interval for the true mean voting percentage, using a sample mean of 43% with a standard deviation estimate of 5% and aiming for a 95% confidence level in the interval.', 'duration': 465.576, 'highlights': ['The sample mean is 43%, with a standard deviation estimate of 5%, aiming for a 95% confidence level in the interval.', 'The sampling distribution of the sample mean is discussed, with 100 possible values and an approximation using a normal curve.', 'The estimation of the standard deviation for the sampling distribution is explained, using the sample standard deviation as the best estimator.', 'The aim is to find an interval with a 95% chance of the true mean of the population being within two standard deviations.']}], 'duration': 465.576, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OwPSuHXmiPw/pics/OwPSuHXmiPw435422.jpg', 'highlights': ['The sample mean is 43%, with a standard deviation estimate of 5%, targeting a 95% confidence level.', 'The aim is to find an interval with a 95% chance of the true mean of the population being within two standard deviations.', 'The sampling distribution of the sample mean is discussed, with 100 possible values and an approximation using a normal curve.', 'The estimation of the standard deviation for the sampling distribution is explained, using the sample standard deviation as the best estimator.']}], 'highlights': ['The sample mean is 43%, with a standard deviation estimate of 5%, targeting a 95% confidence level.', 'A random survey of 100 people is conducted, where 57 people are likely to vote for person A and 43 for person B.', 'The distribution of people voting for candidate A or B can be represented using the Bernoulli distribution, where the mean is equal to p', 'The aim is to find an interval with a 95% chance of the true mean of the population being within two standard deviations.', 'In a country of 100 million people, p percent will vote for candidate B and 1 minus p percent will vote for candidate A']}