254 CHAPTER 5 Normal Probability Distributions Transforming a z-Score to an x-Value Recall that to transform an x@value to a z@score, you can use the formula z = x - m s . This formula gives z in terms of x. When you solve this formula for x, you get a new formula that gives x in terms of z. z = x - m s Formula for z in terms of x zs = x - m Multiply each side by s. m + zs = x Add m to each side. x = m + zs Interchange sides. To transform a standard z@score to an x@value in a given population, use the formula x = m + zs. Transforming a z@Score to an x@Value Finding an x-Value Corresponding to a z-Score A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weight x corresponding to each z@score. Interpret the results. 1. z = 1.96 2. z = -0.44 3. z = 0 SOLUTION The x@value that corresponds to each standard z@score is calculated using the formula x = m + zs. Note that m = 9 and s = 2. 1. For z = 1.96, the corresponding weight x is x = 9 + 1.96122 = 12.92 pounds. 2. For z = -0.44, the corresponding weight x is x = 9 + 1-0.442122 = 8.12 pounds. 3. For z = 0, the corresponding weight x is x = 9 + 0122 = 9 pounds. Interpretation From the figure at the left, you can see that 12.92 pounds is to the right of the mean, 8.12 pounds is to the left of the mean, and 9 pounds is equal to the mean. TRY IT YOURSELF 3 A veterinarian records the weights of dogs treated at a clinic. The weights are normally distributed, with a mean of 52 pounds and a standard deviation of 15 pounds. Find the weight x corresponding to each z@score. Interpret the results. 1. z = -2.33 2. z = 3 3. z = 0.58 Answer: Page A39 EXAMPLE 3 0 12.92 z 8.12 9 x Weight (in pounds) −0.44 1.96
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