Question

The score of adults on an IQ test are approximately Normal with mean 100 and standart deviation 15. Alysha scores 135 on such a test. She scores highter than what percent of all adults? a. About 5% b. About 95% c. About 99%

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asked 2020-11-12
The score of adults on an IQ test are approximately Normal with mean 100 and standart deviation 15. Alysha scores 135 on such a test. She scores highter than what percent of all adults?
a. About 5%
b. About 95%
c. About 99%

Answers (1)

2020-11-13
Let X be the random variable of IQ test. Then \(\displaystyle{X}\sim{N}{\left({100},{15}^{{2}}\right)}\)
which means that \(\displaystyle\frac{{{X}-{100}}}{{15}}\sim{N}{\left({0},{1}\right)}\)
We need to find \(\displaystyle{P}{\left({X}\le{135}\right)}\)
We write this is a way to get the standard normal variable: \(\displaystyle{P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{{135}-{100}}}{{15}}\right)}={P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{7}}{{3}}\right)}\)
Let Ф denote the standard normal distribution, so \(\displaystyle{P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{7}}{{3}}\right)}=Ф{\left(\frac{{7}}{{3}}\right)}\simФ{\left({2.33}\right)}\sim{0.9901}\)
Thus, (c) is correct.
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