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<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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