6 1 Standard Normal Distribution Part 2 Youtube Step 1: calculate a z score. to compare sleep duration during and before the lockdown, you convert your lockdown sample mean into a z score using the pre lockdown population mean and standard deviation. a z score of 2.24 means that your sample mean is 2.24 standard deviations greater than the population mean. C. z = y − μ σ y − μ σ = 4 − 2 1 4 − 2 1 = 2, where µ = 2 and σ = 1. the z score for y = 4 is z = 2. this means that four is z = 2 standard deviations to the right of the mean. . therefore, x = 17 and y = 4 are both two of their own standard deviations to the right of their respective.
6 1 The Standard Normal Distribution Notes Part 2 Of 3 You Suppose x has a normal distribution with mean 50 and standard deviation 6. about 68% of the x values lie between –1 σ = (–1) (6) = –6 and 1 σ = (1) (6) = 6 of the mean 50. the values 50 – 6 = 44 and 50 6 = 56 are within one standard deviation of the mean 50. the z scores are –1 and 1 for 44 and 56, respectively. The values 50 – 6 = 44 and 50 6 = 56 are within one standard deviation from the mean 50. the z scores are –1 and 1 for 44 and 56, respectively. about 95% of the x values lie within two standard deviations of the mean. therefore, about 95% of the x values lie between –2σ = (–2) (6) = –12 and 2σ = (2) (6) = 12. For example, if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean. the calculation is as follows: x = μ (z)(σ) = 5 (3)(2) = 11. the z score is three. the mean for the standard normal distribution is zero, and the standard deviation is one. Since the mean for the standard normal distribution is zero and the standard deviation is one, then the transformation in equation \ref{zscore} produces the distribution \(z \sim n(0, 1)\). the value \(x\) comes from a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). a z score is measured in units of the standard deviation.
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