critical values - need help understanding solution (tendon thickness in athletes)

Herondaleheir

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Apr 1, 2019
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I have the solution to a question (both shown in the pic below) and I was wondering if someone could explain where 1.645 came from? Is there a formula to get that value?

11652
 
The 1.645 is the value of z for which the probability of being less than z is 0.95 in a normal distribution, so that the probability of being greater (the tail that is shaded in the graph) is 0.05. It is commonly found from a table. Your textbook may have used this number many times with full explanations, and at this point your teacher may expect you to have practically memorized the number (or at least know where to look it up).
 
The 1.645 in this problem is one of the "magic" numbers in statistics that, after you work enough problems, you'll simply have memorized. As part of the problem, the author needs to know how many standard deviations above the mean the 95th percentile is. That is to say, 5% of the data will fall to the right of this value. In nearly every case involving normal distributions ("bell curve"), these numbers are found by looking them up on a z-table. The table gives an approximation that's generally been agreed upon by statisticians as being "close enough." These values from the z-table are in turn calculated based upon this kind of ugly integral:

\(\displaystyle \int\limits_{Z}^{\infty} \frac{1}{\sqrt{2\pi}} \cdot \text{exp} \left( \frac{-x^2}{2} \right) \: dx = 0.05\)

Where \(exp(x)\) is equivalent notation to \(e^x\). Having seen this, I'm sure you can agree that solving this integral is anything but trivial. In fact, I don't think it's actually possible to directly solve it to get an exact value, only to use numerical methods to get an approximation out to any desired degree of accuracy.
 
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