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If $X$ is a normally distributed random variable and $X$ ~ $\mathrm{N(\mu ,\; \sigma )}$ , then the z-score is:
The z-score tells you how many standard deviations that the value $x$ is above (to the right of) or below (to the left of) the mean, $\mu $ . Values of $x$ that are larger than the mean have positive z-scores and values of $x$ that are smaller than the mean have negative z-scores. If $x$ equals the mean, then $x$ has a z-score of $0$ .
Suppose $X$ ~ $\mathrm{N(5,\; 6)}$ . This says that $X$ is a normally distributed random variable with mean $\mathrm{\mu \; =\; 5}$ and standard deviation $\mathrm{\sigma \; =\; 6}$ . Suppose $\mathrm{x\; =\; 17}$ . Then:
This means that $\mathrm{x\; =\; 17}$ is 2 standard deviations $\mathrm{(2\sigma )}$ above or to the right of the mean $\mathrm{\mu \; =\; 5}$ . The standard deviation is $\mathrm{\sigma \; =\; 6}$ .
Notice that:
Now suppose $\mathrm{x=1}$ . Then:
This means that $\mathrm{x\; =\; 1}$ is 0.67 standard deviations $\mathrm{(-\; 0.67\sigma )}$ below or to the left of the mean $\mathrm{\mu \; =\; 5}$ . Notice that:
$5+\left(-0.67\right)\left(6\right)$ is approximately equal to 1 $\phantom{\rule{20pt}{0ex}}$ (This has the pattern $\mu +\left(-0.67\right)\sigma =1$ )
Summarizing, when $z$ is positive, $x$ is above or to the right of $\mu $ and when $z$ is negative, $x$ is to the left of or below $\mu $ .
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 $X$ = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. $X$ ~ $\mathrm{N(5,\; 2)}$ . Fill in the blanks.
Suppose a person lost 10 pounds in a month. The z-score when $\mathrm{x\; =\; 10}$ pounds is $\mathrm{z\; =\; 2.5}$ (verify). This z-score tells you that $\mathrm{x\; =\; 10}$ is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
This z-score tells you that $\mathrm{x\; =\; 10}$ is 2.5 standard deviations to the right of the mean 5 .
Suppose a person gained 3 pounds (a negative weight loss). Then $z$ = __________. This z-score tells you that $\mathrm{x\; =\; -3}$ is ________ standard deviations to the __________ (right or left) of the mean.
$z$ = -4 . This z-score tells you that $\mathrm{x\; =\; -3}$ is 4 standard deviations to the left of the mean.
Suppose the random variables $X$ and $Y$ have the following normal distributions: $X$ ~ $\mathrm{N(5,\; 6)}$ and $\mathrm{Y\; ~\; N(2,\; 1)}$ . If $\mathrm{x\; =\; 17}$ , then $z=2$ . (This was previously shown.) If $\mathrm{y\; =\; 4}$ , what is $z$ ?
The z-score for $\mathrm{y\; =\; 4}$ is $\mathrm{z\; =\; 2}$ . This means that 4 is $\mathrm{z\; =\; 2}$ standard deviations to the right of the mean. Therefore, $\mathrm{x\; =\; 17}$ and $\mathrm{y\; =\; 4}$ 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 $X$ ~ $\mathrm{N(5,\; 6)}$ represents weight gains for one group of people who are trying to gain weight in a 6 week period and $Y$ ~ $\mathrm{N(2,\; 1)}$ measures the same weight gain for a second group of people. A negative weight gain would be a weight loss.Since $\mathrm{x\; =\; 17}$ and $\mathrm{y\; =\; 4}$ are each 2 standard deviations to the right of their means, they represent the same weight gain relative to their means .
Suppose $X$ has a normal distribution with mean 50 and standard deviation 6.
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