Chapter 11

Normal distributions

1. Sketch the basic shape of the density curve for a Normal random variable.
2. Identify the two parameters that specify the center and spread of the density curve for a Normal random variable.
3. Recognize the convention in statistics of using Roman letters for sample statistics (\(\bar{x}\), \(s\), etc.) and Greek letters for population parameters (\(\mu\), \(\sigma\), etc.).
4. Recognize, explain, and use the notation \(X \sim N(\mu, \sigma)\).
5. Sketch a normal density curve for any \(N(\mu, \sigma)\) random variable, including its: mean, and mean \(\pm\) 1, 2, and 3 standard deviations.

The 68–95–99.7 rule

1. Use the 68–95–99.7 rule to reason about probabilities in the body of the density curve of a Normal random variable.

The standard Normal distribution

1. State the mean and standard deviation of a standard Normal random variable.
2. Explain why, for a Normal random variable with mean \(\mu\) and standard deviation \(\sigma\), subtracting its mean and then dividing by its standard deviation transforms the random variable into a standard Normal random variable.
3. Compute a \(Z\)-score (equivalently, \(Z\)-statistic or standardized score) given a data value from a \(N(\mu, \sigma)\) random variable.

Finding Normal probabilities

1. Convert probability queries of the form
   
   • \(P(X \leq a)\)
     
   • \(P(X \geq a)\)
     
   • \(P(X \leq a \text{ or } X \geq b)\)
     
   • \(P(a \leq X \leq b)\)
     
   into questions about the probability in the left, right, both tails, or body of a Normal density curve, or vice versa.
   
2. Sketch the area corresponding to probabilities in the:
   
   • Left tail
     
   • Right tail
     
   • Both tails
     
   • Body
     
   of a Normal random variable with a given mean and standard deviation.
   
3. Identify which tail pnorm returns the probability for, by default.
   
4. Find the corresponding probabilities using pnorm in R.
   

Finding percentiles

1. Define the percentile of a distribution.
2. Define the quantile of a distribution.
3. Convert a question asking for a quantile to a question asking for a percentile or vice versa.
4. Compute percentiles or quantiles using qnorm.
5. Describe what the first letter in the functions pnorm and qnorm are indicating about the function.
6. Given a probability \(p\), find the value of \(a\) for the equalities \(p = P(X \leq a)\) and \(p = P(X \geq a)\).