I have some questions that I really need help with.
The mean mark for an IQ test in the population is 100, with a standard deviation of 16.5. The IQ is normally distributed. Your IQ is 113.
a. What percentage of people will have an IQ higher than your own IQ?
b. What proportion of people will have an IQ between your IQ and the mean IQ mark in the population?
c. What decile of the population does your IQ fall into?
How do I even begin? Help me :(
a) $z=\frac{\bar{x}-\mu}{\sigma}=\frac{113-100}{16.5}\approx 0.7879$. To get what percentage of people scored more than you, take the area from that z-score to infinity (use
normalcdf()in your calculator). The area under the curve gives you a percentage (that's how the normal curve works - the entire area is 1 or 100%). The area turns out to be $\approx 0.215384$ and therefore the answer is $\approx 21.5384\%$. You need to do better :).b) The z-score in the middle is zero. So now take the area from a z-score of $0$ to $0.7879$ (same thing as before). The answer is $\approx 28.4616\%$.
c) This is like asking what percentile you're at. If you are at the 99th percentile, let's say, it means that you are doing better than 99% of the population. In other words, the percentage of people doing worse than you is 99. With relation to the normal curve, this would be the area from $-\infty$ to your z-score. Same exact thing - find the area from $-\infty$ to $0.7879$. This turns out to be $\approx 78.4616\%$. Now the question asks for your decile, which you may guess from the name is like a percentile except for tens. You're in the 7th decile since you're doing better than about 70% of the population (but not more than 80% of the population).
And just a tip: if you have TI-83/84,
normalcdf()has two overloaded methods:normalcdf(z_lower, z_upper)(meaning the minimum z-index and the maximum one), or easiest to use isnormalcdf(lower, upper, mean, std_dev)(Lower value, upper value, mean, and standard deviation). If you want to put $\infty$, you can use $_\text{E}99$.