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Random Variables and Normal Models
Random Variables and Normal Models
A random variable assigns a numerical value to each outcome of a chance experiment.
Expected Value
For a discrete random variable, $$E(X)=\sum xP(x).$$
Normal Distribution
The normal model is bell-shaped, symmetric, and described by mean $\mu$ and standard deviation $\sigma$.
68-95-99.7 Rule
About 68% of observations lie within $1\sigma$, 95% within $2\sigma$, and 99.7% within $3\sigma$ of the mean.
Example 1
If test scores are normal with mean 70 and standard deviation 10, what interval captures about 95% of scores?
$70 \pm 20$, so about $50$ to $90$.
Practice Problems
1. What does a random variable do?
2. State the 68-95-99.7 rule.
3. Mean $=100$, SD $=15$. Give the 68% interval.
4. Mean $=100$, SD $=15$. Give the 95% interval.
5. Is the normal distribution symmetric or skewed?
Show Answer Key
1. It assigns a number to each outcome
2. 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD
3. $85$ to $115$
4. $70$ to $130$
5. Symmetric