![]() The final example of this section explains the origin of the proportions given in the Empirical Rule. ![]() A z-score tells you where exactly the score lies on a normal distribution curve. Therefore P ( − 5.22 < Z < 2.15 ) = 0.9842 − 0.0000 = 0.9842 A score of 1.8 is 1.8 and the standard deviation is present above the mean. We can see from the first line of the table that the area to the left of −5.22 must be so close to 0 that to four decimal places it rounds to 0.0000. Similarly, here we can read directly from the table that the area under the density curve and to the left of 2.15 is 0.9842, but −5.22 is too far to the left on the number line to be in the table. Therefore P ( 1.13 < Z < 4.16 ) = 1.0000 − 0.8708 = 0.1292 The Standard Normal Distribution: The Standard Normal Distribution is a special case of the normal distribution that takes on a specific shape with a given mean and standard deviation: mean 0 standard deviation 1 We also refer to the Standard Normal Distribution as the Z-Distribution because we often speak about this distribution in terms. We can see from the last row of numbers in the table that the area to the left of 4.16 must be so close to 1 that to four decimal places it rounds to 1.0000. We obtain the value 0.8708 for the area of the region under the density curve to left of 1.13 without any problem, but when we go to look up the number 4.16 in the table, it is not there. We attempt to compute the probability exactly as in Note 5.20 "Example 6" by looking up the numbers 1.13 and 4.16 in the table.
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