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Confidence interval example | Inferential statistics | Probability and Statistics | Khan Academy

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Confidence Interval Example
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{'title': 'Confidence interval example | Inferential statistics | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 701.693, 'start': 600.407, 'weight': 0.727}], 'summary': 'Discusses a technology grant available to teachers for installing computers, a survey of 250 teachers, and the calculation of a 99% confidence interval for the proportion of teachers who consider computers essential, along with sampling distribution, sample proportion, and confidence intervals using z-table and standard deviations.', 'chapters': [{'end': 91.289, 'segs': [{'end': 44.462, 'src': 'embed', 'start': 0.836, 'weight': 0, 'content': [{'end': 7.759, 'text': 'In a local teaching district, a technology grant is available to teachers in order to install a cluster of four computers in their classroom.', 'start': 0.836, 'duration': 6.923}, {'end': 13.261, 'text': 'From the 6, 250 teachers in the district,', 'start': 8.259, 'duration': 5.002}, {'end': 20.084, 'text': '250 were randomly selected and asked if they felt that computers were an essential teaching tool for their classroom.', 'start': 13.261, 'duration': 6.823}, {'end': 33.194, 'text': 'Of those selected, 142 teachers 142 teachers felt that the computers were an essential teaching tool.', 'start': 20.704, 'duration': 12.49}, {'end': 42.381, 'text': 'And then they ask us, calculate a 99% confidence interval for the proportion of teachers who felt that the computers are an essential teaching tool.', 'start': 33.694, 'duration': 8.687}, {'end': 44.462, 'text': "So let's just think about the entire population.", 'start': 42.761, 'duration': 1.701}], 'summary': 'Out of 250 teachers, 142 felt computers were essential. calculate 99% confidence interval.', 'duration': 43.626, 'max_score': 0.836, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE836.jpg'}], 'start': 0.836, 'title': "Technology grant and teacher's perception", 'summary': 'Discusses a technology grant available to teachers for installing a cluster of four computers, the survey of 250 teachers regarding the essentiality of computers, and the calculation of a 99% confidence interval for the proportion of teachers who consider computers as an essential teaching tool.', 'chapters': [{'end': 91.289, 'start': 0.836, 'title': "Technology grant and teacher's perception", 'summary': 'Discusses a technology grant available to teachers for installing a cluster of four computers, the survey of 250 teachers regarding the essentiality of computers, and the calculation of a 99% confidence interval for the proportion of teachers who consider computers as an essential teaching tool.', 'duration': 90.453, 'highlights': ['The technology grant offers the installation of a cluster of four computers in classrooms, which is available to teachers in a local teaching district.', 'Out of 250 randomly selected teachers, 142 felt that computers were an essential teaching tool, indicating a proportion of 0.568.', 'The chapter delves into the calculation of a 99% confidence interval for the proportion of teachers who perceive computers as essential in teaching, using the Bernoulli distribution and identifying the mean as the parameter of interest.']}], 'duration': 90.453, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE836.jpg', 'highlights': ['The technology grant offers installation of a cluster of four computers to local teachers.', 'Out of 250 teachers, 142 consider computers essential, indicating a proportion of 0.568.', 'The chapter calculates a 99% confidence interval for the proportion of teachers who perceive computers as essential.']}, {'end': 534.409, 'segs': [{'end': 117.009, 'src': 'embed', 'start': 91.389, 'weight': 0, 'content': [{'end': 96.252, 'text': "It's neither 0 or 1, so not an actual value that you could actually get out of a teacher if you were to ask them.", 'start': 91.389, 'duration': 4.863}, {'end': 99.114, 'text': 'They cannot say something in between good and not good.', 'start': 96.532, 'duration': 2.582}, {'end': 102.016, 'text': 'But the actual expected value is something in between.', 'start': 99.414, 'duration': 2.602}, {'end': 104.825, 'text': 'It is P.', 'start': 103.524, 'duration': 1.301}, {'end': 109.726, 'text': "Now what we do is we're taking a sample of those 250 teachers.", 'start': 104.825, 'duration': 4.901}, {'end': 114.788, 'text': 'And we got that 142 felt that the computers were an essential teaching tool.', 'start': 110.066, 'duration': 4.722}, {'end': 117.009, 'text': 'So in our survey, we had 250 sampled.', 'start': 115.208, 'duration': 1.801}], 'summary': 'Out of 250 teachers surveyed, 142 found computers essential for teaching.', 'duration': 25.62, 'max_score': 91.389, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE91389.jpg'}, {'end': 344.795, 'src': 'embed', 'start': 308.837, 'weight': 1, 'content': [{'end': 312.179, 'text': 'And then all of that divided by 250 minus 1 is 249.', 'start': 308.837, 'duration': 3.342}, {'end': 316.102, 'text': "So our sample variance is, well, I'll just say 0.246.", 'start': 312.179, 'duration': 3.923}, {'end': 330.966, 'text': "So our sample variance, I'll write it over here, our sample variance Sample variance is equal to 0.246.", 'start': 316.102, 'duration': 14.864}, {'end': 335.509, 'text': 'If you were to take the square root of that,', 'start': 330.966, 'duration': 4.543}, {'end': 344.795, 'text': "our actual sample standard deviation is going to be let's take the square root of that answer right over there.", 'start': 335.509, 'duration': 9.286}], 'summary': 'Sample variance is 0.246, leading to a standard deviation.', 'duration': 35.958, 'max_score': 308.837, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE308837.jpg'}, {'end': 475.818, 'src': 'embed', 'start': 441.342, 'weight': 2, 'content': [{'end': 444.565, 'text': "But our best estimate of that, and that's why we call it confident.", 'start': 441.342, 'duration': 3.223}, {'end': 452.01, 'text': "We're confident that the real mean or the real population proportion is going to be in this interval.", 'start': 444.585, 'duration': 7.425}, {'end': 457.274, 'text': "But we're confident, but we're not 100% sure because we're going to estimate this over here.", 'start': 452.03, 'duration': 5.244}, {'end': 460.156, 'text': "And if we're estimating this, we're really estimating that over there.", 'start': 457.314, 'duration': 2.842}, {'end': 466.031, 'text': 'can be estimated by this sample standard deviation.', 'start': 462.028, 'duration': 4.003}, {'end': 475.818, 'text': "So then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample,", 'start': 466.491, 'duration': 9.327}], 'summary': 'Confident estimate of population proportion with sample standard deviation.', 'duration': 34.476, 'max_score': 441.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE441342.jpg'}], 'start': 91.389, 'title': 'Sampling distribution and confidence interval', 'summary': 'Explains the process of sampling 250 teachers to determine the proportion of those who consider computers as an essential teaching tool. it involves 142 out of 250 teachers expressing positive views, resulting in a sample proportion of 56.8% and a sample variance of 0.246, leading to the estimation of the standard deviation of the sampling distribution as 0.031.', 'chapters': [{'end': 534.409, 'start': 91.389, 'title': 'Sampling distribution and confidence interval', 'summary': 'Explains the process of sampling 250 teachers to determine the proportion of those who consider computers as an essential teaching tool, with 142 out of 250 teachers expressing positive views, resulting in a sample proportion of 56.8% and a sample variance of 0.246, leading to the estimation of the standard deviation of the sampling distribution as 0.031.', 'duration': 443.02, 'highlights': ['The sample proportion of teachers who thought that computers were a good teaching tool was 56.8% based on the survey of 250 teachers, with 142 expressing positive views and 108 expressing negative views.', 'The sample variance, calculated as 0.246, provides a measure of the variability of the sample data, which is crucial for estimating the true variance of the population.', 'The estimation of the standard deviation of the sampling distribution, derived as 0.031, is key in determining the precision of the sample mean and constructing a confidence interval for the population proportion.']}], 'duration': 443.02, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE91389.jpg', 'highlights': ['The sample proportion of teachers who thought that computers were a good teaching tool was 56.8% based on the survey of 250 teachers, with 142 expressing positive views and 108 expressing negative views.', 'The sample variance, calculated as 0.246, provides a measure of the variability of the sample data, which is crucial for estimating the true variance of the population.', 'The estimation of the standard deviation of the sampling distribution, derived as 0.031, is key in determining the precision of the sample mean and constructing a confidence interval for the population proportion.']}, {'end': 1114.057, 'segs': [{'end': 709.277, 'src': 'heatmap', 'start': 589.865, 'weight': 0, 'content': [{'end': 590.805, 'text': "So that's what we want.", 'start': 589.865, 'duration': 0.94}, {'end': 596.989, 'text': 'We want a 99% chance that if we pick a sample from the sampling distribution of the sample mean,', 'start': 590.845, 'duration': 6.144}, {'end': 599.851, 'text': 'it will be within this many standard deviations of the actual mean.', 'start': 596.989, 'duration': 2.862}, {'end': 603.189, 'text': "And to figure that out, let's look at an actual z-table.", 'start': 600.407, 'duration': 2.782}, {'end': 606.012, 'text': 'So we want 99% confidence.', 'start': 603.99, 'duration': 2.022}, {'end': 610.115, 'text': 'So another way to think about it if we want 99% confidence.', 'start': 606.412, 'duration': 3.703}, {'end': 620.288, 'text': 'if we just look at the upper half right over here, that orange area should be 0.475, because if this is 0.475,', 'start': 610.115, 'duration': 10.173}, {'end': 622.27, 'text': "then this other part's going to be 0.475..", 'start': 620.288, 'duration': 1.982}, {'end': 626.034, 'text': "And we will get to our, oh sorry, we want to get to 99%, so it's not going to be 0.475.", 'start': 622.27, 'duration': 3.764}, {'end': 636.434, 'text': "We're going to have to go to 0.495.", 'start': 626.034, 'duration': 10.4}, {'end': 638.935, 'text': 'if we want 99% confidence.', 'start': 636.434, 'duration': 2.501}, {'end': 644.739, 'text': 'So this area has to be 0.495 over here, because if that is, that over here will also be.', 'start': 639.055, 'duration': 5.684}, {'end': 648.24, 'text': 'So that their sum will be 99% of the area.', 'start': 645.079, 'duration': 3.161}, {'end': 657.285, 'text': 'Now, if this is 0.495,, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this,', 'start': 648.701, 'duration': 8.584}, {'end': 657.786, 'text': "it's going to be 0.5..", 'start': 657.285, 'duration': 0.501}, {'end': 659.006, 'text': "So it's going to be 0.5 plus 0.495.", 'start': 657.786, 'duration': 1.22}, {'end': 666.937, 'text': "It's going to be 0.9.", 'start': 659.006, 'duration': 7.931}, {'end': 669.058, 'text': 'Let me make sure I got that right.', 'start': 666.937, 'duration': 2.121}, {'end': 671.38, 'text': "0.995 And so let's look at our z-table.", 'start': 669.078, 'duration': 2.302}, {'end': 676.623, 'text': 'So where do we get 0.995 on our z-table? 0.995 is pretty close, just to have a little error.', 'start': 671.4, 'duration': 5.223}, {'end': 677.423, 'text': 'It will be right over here.', 'start': 676.643, 'duration': 0.78}, {'end': 678.644, 'text': 'This is 0.9951.', 'start': 677.443, 'duration': 1.201}, {'end': 701.693, 'text': 'So another way to think about it is, So this value right here gives us the whole cumulative area up to that, up to our mean.', 'start': 678.644, 'duration': 23.049}, {'end': 709.277, 'text': 'So if you look at the entire distribution like this, if you look at the entire distribution, this is the mean right over here.', 'start': 702.333, 'duration': 6.944}], 'summary': 'Want 99% confidence, need z-table value of 0.995 for entire distribution.', 'duration': 119.412, 'max_score': 589.865, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE589865.jpg'}, {'end': 1099.053, 'src': 'embed', 'start': 1046.49, 'weight': 2, 'content': [{'end': 1047.55, 'text': 'if we subtract 8 from this we get.', 'start': 1046.49, 'duration': 1.06}, {'end': 1058.444, 'text': 'So we are 99% confident that the true population proportion is between these two numbers.', 'start': 1054.181, 'duration': 4.263}, {'end': 1064.827, 'text': 'Or another way, that the true percentage of teachers who think those computers are good ideas is between.', 'start': 1058.524, 'duration': 6.303}, {'end': 1066.288, 'text': "we're 99% confident.", 'start': 1064.827, 'duration': 1.461}, {'end': 1072.772, 'text': "we're confident that there's a 99% chance that the true percentage of teachers that like the computers is between 48.8% and 64.8%.", 'start': 1066.288, 'duration': 6.484}, {'end': 1074.833, 'text': 'Now, that we answered the first part of the question.', 'start': 1072.772, 'duration': 2.061}, {'end': 1086.91, 'text': 'The second part how could the survey be changed to narrow the confidence interval but to maintain the 99% confidence interval??', 'start': 1079.949, 'duration': 6.961}, {'end': 1089.691, 'text': 'Well, you could just take more samples.', 'start': 1087.351, 'duration': 2.34}, {'end': 1099.053, 'text': 'If you take more samples, then our estimate of the standard deviation of this distribution will go down, because this denominator will be higher.', 'start': 1089.831, 'duration': 9.222}], 'summary': '99% confident that 48.8-64.8% favor computers; take more samples to narrow interval', 'duration': 52.563, 'max_score': 1046.49, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE1046490.jpg'}], 'start': 534.989, 'title': 'Confidence intervals', 'summary': 'Covers the calculation of a 99% confidence interval using z-table, standard deviations, and achieving 99% confidence level. it also discusses the 99% confidence interval for population proportion, demonstrating calculations and maintaining 99% confidence.', 'chapters': [{'end': 791.949, 'start': 534.989, 'title': '99% confidence interval calculation', 'summary': 'Discusses the calculation of a 99% confidence interval using z-table, explaining the concept of standard deviations and how to achieve 99% confidence level.', 'duration': 256.96, 'highlights': ['The z-table value for 99% confidence level is 2.58, indicating that a sample will be within 2.58 standard deviations of the mean with a 99% chance.', 'To achieve 99% confidence, the area under the curve on the z-table needs to be 0.495, resulting in a total area of 99.2% within 2.58 standard deviations from the mean.', 'Understanding the z-table and its values is crucial for determining the standard deviations needed to achieve a 99% confidence level.']}, {'end': 1114.057, 'start': 791.969, 'title': 'Confidence interval for population proportion', 'summary': 'Discusses the concept of a 99% confidence interval for the population proportion, demonstrating how to calculate the interval and how to narrow it by taking more samples, maintaining 99% confidence.', 'duration': 322.088, 'highlights': ['The chapter explains that there is a little over 99% chance that any sample mean falls within 2.58 standard deviations of the sampling mean, allowing for a 99% confidence interval for the population proportion.', 'The calculation shows that a 99% confidence interval for the true population proportion is between 48.8% and 64.8%.', 'Taking more samples can narrow the confidence interval by reducing the standard deviation of the distribution, leading to a more precise estimate while maintaining 99% confidence.']}], 'duration': 579.068, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SeQeYVJZ2gE/pics/SeQeYVJZ2gE534989.jpg', 'highlights': ['The z-table value for 99% confidence level is 2.58, indicating a sample within 2.58 standard deviations with 99% chance.', 'To achieve 99% confidence, the area under the z-table needs to be 0.495, resulting in a total area of 99.2% within 2.58 standard deviations.', 'The calculation shows a 99% confidence interval for the true population proportion between 48.8% and 64.8%.', 'Understanding the z-table and its values is crucial for determining the standard deviations needed to achieve a 99% confidence level.', 'Taking more samples can narrow the confidence interval by reducing the standard deviation, leading to a more precise estimate while maintaining 99% confidence.']}], 'highlights': ['The z-table value for 99% confidence level is 2.58, indicating a sample within 2.58 standard deviations with 99% chance.', 'The calculation shows a 99% confidence interval for the true population proportion between 48.8% and 64.8%.', 'The technology grant offers installation of a cluster of four computers to local teachers.', 'Out of 250 teachers, 142 consider computers essential, indicating a proportion of 0.568.', 'The sample proportion of teachers who thought that computers were a good teaching tool was 56.8% based on the survey of 250 teachers, with 142 expressing positive views and 108 expressing negative views.', 'Taking more samples can narrow the confidence interval by reducing the standard deviation, leading to a more precise estimate while maintaining 99% confidence.']}