Chapter 7 The Normal Distribution

1. Section 7.1 Standard Normal Curve
   • probability density curve
     • probability distribution of a continuous variable
     • The area under the curve (and above the x-axis is equal to 1)
     • If `alpha < beta` then the area under the curve between `alpha` and `beta` represents
       1. the proportion of population with values from `alpha` to `beta`
          
       2. the probility that a randomly chosen "individual" (i.e., data point) will have a value between `alpha` and `beta`.
   • The Normal Distribution
   • Using a table to find area under the curve
   • Using the calculator to find area under the curve
   • Finding a `z`-score for a given area
   • Finding a `z`-score for the area to the right of a value
   • The notation `z_alpha`
   • Using the calculator for z-scores (same page as finding area)
2. Section 7.2 Applications of Normal Distribution
   • Converting Normal Values to `z`-scores
   • Finding Areas Under a Normal Curve
   • Finding the Value from a Normal Distribution Corresponding to a Given Proportion
3. Section 7.3 Sampling Distributions and the Central Limit Theorem
   • The probability historgram for the sampling distributon of `bar x`
   • The Central Limit Theorem
     If we start with a population with mean `mu` and standard deviation `sigma`, and we draw random samples each of size `n` where `n > 30` and we compute the sample mean, `bar x`, for each of sample, then the distribution of `bar x` will be approximately normal with the mean of all the `bar x`'s approaching `mu` (i.e., `mu_(bar x) = mu`) and the standard deviation of the `bar x`'s will approach `sigma/sqrt(n)` (i.e., `sigma_(bar x) = sigma/sqrt(n)`).
   • Looking at actual samples.
   • Sample problems and solutions on the calcualtor.
4. Section 7.4 Central Limit Theorem for Proportions
   • The Central Limit Theorem for Proportions:
     If we start with a population that has a characteristic that appears in `p` proportion, and we draw random samples of size `n` where we know that `n*p >= 10` and `n*(1-p) >= 10`, and we look at compute `hat p` for each sample, then the distribution of `hat p` will be approximately normal, with the mean of the `hat p`'s equal to `p`
     (i.e., `mu_(hat p) = p`), and the standard deviation of the `hat p`'s approximately equal to `sqrt( ((p*(1-p))/n )`
     (i.e., `sigma_(hat p) =sqrt( ((p*(1-p))/n )`   )
5. Section 7.5 Normal Approximation to the Binomial Distribution
   • Review:
     • The mean of a binomial random variable `X` with `n` trials and probability of success equal to `p` is given as `mu_X = n*p`
     • The variance of a binomial random variable `X` with `n` trials and probability of success equal to `p` is given as `sigma_X^2 = n*p*(1-p)`
     • The standard deviation of a binomial random variable `X` with `n` trials and probability of success equal to `p` is given as `sigma_X = sqrt(sigma_X^2) = sqrt( n*p*(1-p) )`
   • The Normal Approximation to the Binomial:
     If we start with a binomial distribution that has a success probability of `p`, and we look at a random variable `X` based on `n` trials, then, in those cases where `n*p >= 10` and `n*(1-p) >= 10`, we will find that `X` is approximately normal with mean `mu_X = n*p` and standard devaition `sigma_X = sqrt( (n*p*(1-p) )`.
   • The continuity correction
   • Before looking at any explanation of the differences between proportions and the binomial distribution, we want to look at the changes in mean and standard deviation related to transformations in values.
6. Section 7.6 Assessing Normality
   • What does it take to say that something is not approximately normally distributed.
     • The existence of one or more outliers.
     • Significant (whatever that means) skewness (we want the mean and median to be close)
     • More than one mode (but that depends on how hard we look and how far apart the two supposed modes are)
   • dotplots
   • boxplots
   • histograms
   • stem-and-leaf
   • Normal Quantile Plots (see the web page)