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If is a normally distributed random variable and ~ , then the z-score is:
The z-score tells you how many standard deviations that the value is above (to the right of) or below (to the left of) the mean, . Values of that are larger than the mean have positive z-scores and values of that are smaller than the mean have negative z-scores. If equals the mean, then has a z-score of .
Suppose ~ . This says that is a normally distributed random variable with mean and standard deviation . Suppose . Then:
This means that is 2 standard deviations above or to the right of the mean . The standard deviation is .
Notice that:
Now suppose . Then:
This means that is 0.67 standard deviations below or to the left of the mean . Notice that:
is approximately equal to 1 (This has the pattern )
Summarizing, when is positive, is above or to the right of and when is negative, is to the left of or below .
Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has anormal distribution. Let = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. ~ . Fill in the blanks.
Suppose a person lost 10 pounds in a month. The z-score when pounds is (verify). This z-score tells you that is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
This z-score tells you that is 2.5 standard deviations to the right of the mean 5 .
Suppose a person gained 3 pounds (a negative weight loss). Then = __________. This z-score tells you that is ________ standard deviations to the __________ (right or left) of the mean.
= -4 . This z-score tells you that is 4 standard deviations to the left of the mean.
Suppose the random variables and have the following normal distributions: ~ and . If , then . (This was previously shown.) If , what is ?
The z-score for is . This means that 4 is standard deviations to the right of the mean. Therefore, and are both 2 (of their ) standard deviations to the right of their respective means.
The z-score allows us to compare data that are scaled differently. To understand the concept, suppose ~ represents weight gains for one group of people who are trying to gain weight in a 6 week period and ~ measures the same weight gain for a second group of people. A negative weight gain would be a weight loss.Since and are each 2 standard deviations to the right of their means, they represent the same weight gain relative to their means .
Suppose has a normal distribution with mean 50 and standard deviation 6.
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