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Confidence interval of difference of means | Probability and Statistics | Khan Academy

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Confidence Interval of Difference of Means
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{'title': 'Confidence interval of difference of means | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 807.515, 'start': 729.016, 'weight': 0.887}], 'summary': 'Compares weight loss of low-fat diet and control group over four months, showing mean weight loss of 9.31 pounds for low-fat group and 7.40 pounds for control group, aiming to establish a 95% confidence interval around the incremental weight loss, and discusses the process of creating a 95% confidence interval using z-scores with a critical z-value of 1.96.', 'chapters': [{'end': 166.529, 'segs': [{'end': 56.191, 'src': 'embed', 'start': 25.097, 'weight': 0, 'content': [{'end': 27.479, 'text': 'Group 1 is the low-fat group, to see if it actually works.', 'start': 25.097, 'duration': 2.382}, {'end': 31.01, 'text': 'After four months, the mean weight loss was 9.31 pounds for group 1.', 'start': 28.027, 'duration': 2.983}, {'end': 31.911, 'text': 'So let me write this down.', 'start': 31.01, 'duration': 0.901}, {'end': 32.832, 'text': 'Let me make it very clear.', 'start': 31.931, 'duration': 0.901}, {'end': 35.594, 'text': 'So the low-fat group, the mean weight loss was 9.31 pounds.', 'start': 32.872, 'duration': 2.722}, {'end': 56.191, 'text': 'So our sample mean for group 1 is 9.31 pounds, with a sample standard deviation of 4.67.', 'start': 35.614, 'duration': 20.577}], 'summary': 'Low-fat group lost 9.31 pounds in 4 months.', 'duration': 31.094, 'max_score': 25.097, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk25097.jpg'}, {'end': 176.548, 'src': 'embed', 'start': 147.857, 'weight': 2, 'content': [{'end': 151.319, 'text': 'Or is there a chance that we can actually go the other way with the low-fat diet?', 'start': 147.857, 'duration': 3.462}, {'end': 153.841, 'text': 'So really just this video 95% confidence interval.', 'start': 151.34, 'duration': 2.501}, {'end': 158.204, 'text': "In the next video, we'll actually do a hypothesis test using this same data.", 'start': 153.861, 'duration': 4.343}, {'end': 164.388, 'text': "And now to do a 95% confidence interval, let's think about the distribution that we're thinking about.", 'start': 158.764, 'duration': 5.624}, {'end': 166.529, 'text': "So let's look at the distribution.", 'start': 165.128, 'duration': 1.401}, {'end': 169.961, 'text': "Of course, we're going to think about the distribution that we're thinking about.", 'start': 167.438, 'duration': 2.523}, {'end': 176.548, 'text': 'We want to think about the distribution of the difference of the means.', 'start': 171.022, 'duration': 5.526}], 'summary': 'Analyzing low-fat diet impact using 95% confidence interval.', 'duration': 28.691, 'max_score': 147.857, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk147857.jpg'}], 'start': 0.779, 'title': 'Comparing low-fat diet weight loss', 'summary': 'Discusses a study comparing weight loss of low-fat diet and control group over four months, showing mean weight loss of 9.31 pounds for the low-fat group and 7.40 pounds for control group, aiming to establish a 95% confidence interval around the incremental weight loss.', 'chapters': [{'end': 166.529, 'start': 0.779, 'title': 'Low-fat diet weight loss study', 'summary': 'Discusses a study comparing the weight loss of a low-fat diet group and a control group over four months, revealing a mean weight loss of 9.31 pounds for the low-fat group and 7.40 pounds for the control group, and aims to establish a 95% confidence interval around the incremental weight loss.', 'duration': 165.75, 'highlights': ['The mean weight loss for the low-fat diet group was 9.31 pounds, with a sample standard deviation of 4.67, while the control group had a mean weight loss of 7.40 pounds with a sample standard deviation of 4.04, indicating a potential difference in weight loss between the two groups.', 'The difference in weight loss between the low-fat diet group and the control group was calculated to be 1.91 pounds, suggesting a potential incremental weight loss of 1.91 pounds every four months for those on the low-fat diet.', 'The chapter aims to establish a 95% confidence interval around the incremental weight loss to determine if there is consistent weight loss with the low-fat diet or if there is a chance of weight gain, setting the stage for a hypothesis test in the subsequent video.']}], 'duration': 165.75, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk779.jpg', 'highlights': ['The mean weight loss for the low-fat diet group was 9.31 pounds, with a sample standard deviation of 4.67, while the control group had a mean weight loss of 7.40 pounds with a sample standard deviation of 4.04, indicating a potential difference in weight loss between the two groups.', 'The difference in weight loss between the low-fat diet group and the control group was calculated to be 1.91 pounds, suggesting a potential incremental weight loss of 1.91 pounds every four months for those on the low-fat diet.', 'The chapter aims to establish a 95% confidence interval around the incremental weight loss to determine if there is consistent weight loss with the low-fat diet or if there is a chance of weight gain, setting the stage for a hypothesis test in the subsequent video.']}, {'end': 527.49, 'segs': [{'end': 195.334, 'src': 'embed', 'start': 167.438, 'weight': 2, 'content': [{'end': 169.961, 'text': "Of course, we're going to think about the distribution that we're thinking about.", 'start': 167.438, 'duration': 2.523}, {'end': 176.548, 'text': 'We want to think about the distribution of the difference of the means.', 'start': 171.022, 'duration': 5.526}, {'end': 192.171, 'text': "So it's going to have some true mean here, which is It's going to have some true mean, which is the mean of the difference of the sample means.", 'start': 177.489, 'duration': 14.682}, {'end': 193.072, 'text': 'Actually, let me write that.', 'start': 192.191, 'duration': 0.881}, {'end': 194.293, 'text': "It's not a y.", 'start': 193.532, 'duration': 0.761}, {'end': 195.334, 'text': "It's an x1 and x2.", 'start': 194.293, 'duration': 1.041}], 'summary': 'Analyzing the distribution of the difference of means x1 and x2.', 'duration': 27.896, 'max_score': 167.438, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk167438.jpg'}, {'end': 379.324, 'src': 'embed', 'start': 337.575, 'weight': 0, 'content': [{'end': 342.051, 'text': "So let's look it up for 97.5%.", 'start': 337.575, 'duration': 4.476}, {'end': 342.391, 'text': '97.5% on our z table.', 'start': 342.051, 'duration': 0.34}, {'end': 342.771, 'text': "Let's see.", 'start': 342.431, 'duration': 0.34}, {'end': 343.672, 'text': 'We have 97 right here.', 'start': 342.811, 'duration': 0.861}, {'end': 344.172, 'text': 'This is 0.975, or 97.5.', 'start': 343.712, 'duration': 0.46}, {'end': 345.593, 'text': 'And this gives us a z value of 1.96.', 'start': 344.172, 'duration': 1.421}, {'end': 363.453, 'text': 'So this is z is equal to 1.96.', 'start': 345.593, 'duration': 17.86}, {'end': 374.04, 'text': 'Only 2.5% of the results or of the samples from this population are going to be more than 1.96 standard deviations away from the mean.', 'start': 363.453, 'duration': 10.587}, {'end': 379.324, 'text': 'So this critical z value right here is 1.96 standard deviations.', 'start': 374.06, 'duration': 5.264}], 'summary': '97.5% corresponds to a z value of 1.96, indicating 2.5% beyond 1.96 standard deviations from the mean.', 'duration': 41.749, 'max_score': 337.575, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk337575.jpg'}, {'end': 508.862, 'src': 'embed', 'start': 459.116, 'weight': 1, 'content': [{'end': 463.658, 'text': 'Or we can say that there is a 95% chance that 1.91, which is the sample statistic or the statistic that we got,', 'start': 459.116, 'duration': 4.542}, {'end': 465.738, 'text': 'is within 1.96 times the standard deviation of this distribution.', 'start': 463.658, 'duration': 2.08}, {'end': 479.687, 'text': 'of the true mean of the distribution.', 'start': 475.806, 'duration': 3.881}, {'end': 480.647, 'text': 'Or we could say it the other way around.', 'start': 479.707, 'duration': 0.94}, {'end': 485.128, 'text': "There's a 95% chance that the true mean of the distribution is within 1.96 times the standard deviation of the distribution.", 'start': 480.667, 'duration': 4.461}, {'end': 508.862, 'text': 'of 1.91.', 'start': 506.401, 'duration': 2.461}], 'summary': '95% chance that sample statistic is within 1.96 times standard deviation of true mean', 'duration': 49.746, 'max_score': 459.116, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk459116.jpg'}], 'start': 167.438, 'title': 'Statistical inferences', 'summary': 'Discusses the distribution of the difference of means and the process of creating a 95% confidence interval using z-scores, with a critical z-value of 1.96, to ensure a 95% confidence level.', 'chapters': [{'end': 217.786, 'start': 167.438, 'title': 'Distribution of difference of means', 'summary': 'Discusses the distribution of the difference of the means, focusing on the true mean, sample means, and standard deviation of the distribution.', 'duration': 50.348, 'highlights': ['The distribution is centered around the true mean, which is the mean of the difference of the sample means.', 'It involves the sample mean of x1 minus the sample mean of x2.', 'The distribution also has a standard deviation.']}, {'end': 527.49, 'start': 218.086, 'title': 'Creating 95% confidence interval', 'summary': 'Discusses the process of creating a 95% confidence interval using z-scores, with a critical z-value of 1.96, to determine the range within which the true mean of the difference lies, ensuring a 95% confidence level.', 'duration': 309.404, 'highlights': ['The z-table is used to find the critical z-value for a 95% confidence interval, resulting in a z-value of 1.96, indicating that 2.5% of the results or samples from the population are more than 1.96 standard deviations away from the mean.', 'The critical z-value of 1.96 represents the range within which there is a 95% chance that the true mean of the distribution lies, or equivalently, the sample statistic is within 1.96 times the standard deviation of the distribution, forming the basis for constructing the confidence interval.', "The symmetric nature of the z-value's interpretation is highlighted, emphasizing the equivalence of statements regarding the range within which the true mean of the distribution lies and the sample statistic's proximity to the standard deviation of the distribution."]}], 'duration': 360.052, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk167438.jpg', 'highlights': ['The z-table is used to find the critical z-value for a 95% confidence interval, resulting in a z-value of 1.96, indicating that 2.5% of the results or samples from the population are more than 1.96 standard deviations away from the mean.', 'The critical z-value of 1.96 represents the range within which there is a 95% chance that the true mean of the distribution lies, or equivalently, the sample statistic is within 1.96 times the standard deviation of the distribution, forming the basis for constructing the confidence interval.', 'The distribution is centered around the true mean, which is the mean of the difference of the sample means.']}, {'end': 947.625, 'segs': [{'end': 604.046, 'src': 'embed', 'start': 579.908, 'weight': 2, 'content': [{'end': 597.463, 'text': 'And we know that the variance of each of these sampling distributions is equal to the variance of this sampling distribution is equal to the variance of the population distribution,', 'start': 579.908, 'duration': 17.555}, {'end': 604.046, 'text': 'the variance of the population distribution divided by our sample size and our sample size in this case is 100,', 'start': 597.463, 'duration': 6.583}], 'summary': 'Variance of sampling distribution = variance of population distribution / sample size (100)', 'duration': 24.138, 'max_score': 579.908, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk579908.jpg'}, {'end': 822.526, 'src': 'heatmap', 'start': 729.016, 'weight': 0, 'content': [{'end': 750.967, 'text': 'So this is equal to Let me write it right here.', 'start': 729.016, 'duration': 21.951}, {'end': 753.948, 'text': 'This is going to be equal to 0.617.', 'start': 751.447, 'duration': 2.501}, {'end': 762.613, 'text': 'So if we go back up over here, we calculated the standard deviation of this distribution to be 0.617.', 'start': 753.948, 'duration': 8.665}, {'end': 767.795, 'text': 'So now we can actually calculate our interval, because this is going to be 0.617.', 'start': 762.613, 'duration': 5.182}, {'end': 779.962, 'text': "So if you want 1.96 times that, so we get 1.96 times That 0.617, I'll just write the answer we just got.", 'start': 767.795, 'duration': 12.167}, {'end': 780.222, 'text': 'So we get 1.21.', 'start': 779.982, 'duration': 0.24}, {'end': 781.283, 'text': 'So this number right here is 1.21.', 'start': 780.222, 'duration': 1.061}, {'end': 785.785, 'text': 'So the 95% confidence interval is going to be the difference of our means.', 'start': 781.283, 'duration': 4.502}, {'end': 807.515, 'text': '1.91 plus or minus this number, 1.21.', 'start': 801.009, 'duration': 6.506}, {'end': 808.616, 'text': "So what's our confidence interval?", 'start': 807.515, 'duration': 1.101}, {'end': 816.123, 'text': "If we subtract it so the low end of our confidence interval and I'm running out of space the low end 1.91, minus 1.21 is just what is that?", 'start': 808.636, 'duration': 7.487}, {'end': 816.763, 'text': "That's just 0.7..", 'start': 816.143, 'duration': 0.62}, {'end': 822.526, 'text': 'So the low end is 0.7.', 'start': 816.763, 'duration': 5.763}], 'summary': 'The 95% confidence interval is 1.91 plus or minus 1.21, resulting in a low end of 0.7.', 'duration': 68.578, 'max_score': 729.016, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk729016.jpg'}], 'start': 527.85, 'title': 'Confidence interval calculation', 'summary': "Explains how to calculate the 95% confidence interval for the true difference between two populations' means using sample variances, resulting in a confidence interval of 0.7 to 3.12.", 'chapters': [{'end': 947.625, 'start': 527.85, 'title': 'Confidence interval calculation', 'summary': "Explains how to calculate the 95% confidence interval for the true difference between two populations' means using sample variances, resulting in a confidence interval of 0.7 to 3.12.", 'duration': 419.775, 'highlights': ['The standard deviation of the differences of the sample means is equal to the square root of the variances of each of those distributions, which results in a 95% confidence interval of 0.7 to 3.12.', 'The variance of the population distribution divided by the sample size, which in this case is 100, is used to approximate the variance of the sampling distributions, especially because the sample size is greater than 30 for both circumstances.', "The confidence interval gives a 95% chance that the true difference between the populations' means lies within the range of 0.7 to 3.12."]}], 'duration': 419.775, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/hxZ6uooEJOk/pics/hxZ6uooEJOk527850.jpg', 'highlights': ["The confidence interval gives a 95% chance that the true difference between the populations' means lies within the range of 0.7 to 3.12.", 'The standard deviation of the differences of the sample means is equal to the square root of the variances of each of those distributions, resulting in a 95% confidence interval of 0.7 to 3.12.', 'The variance of the population distribution divided by the sample size, which in this case is 100, is used to approximate the variance of the sampling distributions, especially because the sample size is greater than 30 for both circumstances.']}], 'highlights': ['The mean weight loss for the low-fat diet group was 9.31 pounds, with a sample standard deviation of 4.67, while the control group had a mean weight loss of 7.40 pounds with a sample standard deviation of 4.04, indicating a potential difference in weight loss between the two groups.', "The confidence interval gives a 95% chance that the true difference between the populations' means lies within the range of 0.7 to 3.12.", 'The difference in weight loss between the low-fat diet group and the control group was calculated to be 1.91 pounds, suggesting a potential incremental weight loss of 1.91 pounds every four months for those on the low-fat diet.', 'The z-table is used to find the critical z-value for a 95% confidence interval, resulting in a z-value of 1.96, indicating that 2.5% of the results or samples from the population are more than 1.96 standard deviations away from the mean.']}