MDM4U More Z-Scores, etc. 2016-05-13

[sorry, looks like I missed the “Publish” button again. I’m just posting this on Sunday instead of Friday :(]

We practised a bunch with the z-score formula and relating small questions to the graph of the normal curve. Here are the problems to solve for homework:

1. For a normal distribution with $\mu=100, \sigma=10, x=112$ find $P(X.
2. For a normal distribution with $\mu=250, \sigma=9$ find $x$ so that $P(X>x)=10%$.
3. For a normal distribution with $\mu=20, \sigma=4$ find $P(X<18) + P(X>25)$.

Spoiler Alert: Solutions

1. Use the z-score formula to find $z = \frac{112-100}{10}=1.2$. In the table look up the z-score 1.20 to find $P(X<112)=.8849$ or 88.49%. [Some interpretations: Graphically this means that 88.49% of the area under the normal curve is to the left of 112. In a survey this means that if you select a participant at random there is an 88.49% chance their “value” will be less than 112.]
2. Use the Standard Normal Probabilities table to find the z-score closest to 90%, or .9000 (we use 90% because we want the value that gives us $P(X>x)$, not less than $x$. The probabilities in our table are .8997 and .9015, so .8997 is closest to 90%. That z-score is 1.28. Now, use that z-score with the z-score formula $1.28=\frac{x-250}{9}$ to solve for the missing value $x=261.52$.
3. $P(X<18)=0.3085$ and $P(X>25)=0.1056$, so $P(X<18)+P(X>25)=0.4141$. [Note: this is the same as $P(X<18 \text{ or } X>25)$. We’ll talk about this more on Monday.]
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