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ANATOMY OF CONVERTING AN X VALUE TO A Z VALUE If x is a normally (or approximately normally) distributed random variable, with mean other than 0 and/or standard deviation other than 1, the value(s) of x must be converted to the Standard Normal Distribution in order to determine probabilities. The formula used to convert x to a z-value will depend upon whether an individual unit or a sample from a population is being considered. Individual unit from a population: When considering an individual unit from population with mean, , and standard deviation, , use the formula below to convert a value of x from a nonstandard normal distribution, to a z-value from the Standard Normal Distribution. z x z EXAMPLE: Given that the scores of students on the ACT college entrance examination in a recent year were normally distributed with mean, = 18.6 and standard deviation = 5.9, What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher? In this case we are considering a single unit from the population. In order to find P(x > 21) we will need to convert the x-value to a z-value from the Standard Normal Distribution as is shown below. Remember to draw a picture of what you are looking for. z x Sample from a population: When considering a sample from a population with mean, , and standard deviation , , we will use the formula below to convert the mean of the sample (x-bar) from a nonstandard normal distribution, to a z-value from the Standard Normal Distribution. __ EXAMPLE: Given that the scores of students on the ACT college entrance examination in a recent year were normally distributed with mean, = 18.6 and standard deviation = 5.9, what is the probability that the mean score obtained from a sample of 50 students who took the test is 21 or higher? In this case we are considering a sample from the population. __ In order to find P( x 21) we will need to convert the value of x to a z-value from the Standard Normal Distribution as is shown below. Remember to draw a picture of what you are looking for. z We can now go to the Standard Normal Table to find P(z > .41) = .3409. ( Obtaining this probability will vary depending on the table format.) P( x 21) P( z .41) .3409 n __ 21 18.6 2.4 .41 5.9 5.9 x x n 21 18.6 2.4 2.4 2.87 5.9 5.9 .835 7.07 50 We can now go to the Standard Normal Table to find P ( z 2.87) .0021. ( Obtaining this probability will vary depending on the table format.) __ P( x 21) P( z 2.87) .0021 Area of interest to right of 2.87 Area of interest to right of .41 0 .41 0 2.87