mba 522 Converting_X_value_to_Z_value

mba 522 Converting_X_value_to_Z_value - We have converted...

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Converting X value to Z value If you have a variable that is normally distributed, you must first convert it to standard normal so you can use the Z table to calculate probabilities related to it. You convert the X value from the normal distribution into its equivalent Z value. The standard normal distribution has a mean of zero and its standard deviation is always 1.0. The Z table reports the areas under the curve for this standard normal distribution. But, once we convert our X value into a z value, we can use the normal table to find probability values. Let us say that the average height of females in a class is 65 inches with the standard deviation being 1.5 inches. Susan is 67 inches tall. What is the probability that someone is taller than Susan? We will compute the Z value for Susan's height, and we get: Z=(67-65)/1.5=1.33. That means 67 is 1.33 standard deviations to the right of the mean. We don't have a table that reports probabilities for height when mean is 65 and standard deviation is 1.5. But we have the table for Z-distribution (the standard normal).
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Unformatted text preview: We have converted our normal distribution for height into the Z distribution. On the Z-axis (Z-scale): The mean of 65 is now 0. The standard deviation of 1.5 is now 1. The X of 67 is now 1.33. We can use this data to answer the question in this problem. The table tells shows 0.9082, which means 90.82% of the area under the curve is to the left of Z=1.33. That means, approximately 91% of the females are shorter than 67 inches. Automatically, we can also conclude that approximately 9% of the area under the curve is to the right of Z=1.33. Therefore, the probability that a woman is taller than 67 inches is approximately 0.09. Also, since the curve is symmetrical, 50% of it is to the right of the mean and 50% of it is to the left of the mean. Therefore, since approximately 9% is above 67 inches, then approximately 41% (50% - 9%) is between 65 and 67 inches (i.e., 41% is between Z values of 0 and 1.33)....
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