6-1 The Standard Normal Deviation

• The specific normal distribution has the 3 properties:

  • Bell shaped

  • mu = 0 (mean equals zero)

  • sigma = 1 (standard deviation equals 1)

• If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, and it can be described by the equation given as Formula 6-1, it has a normal distribution

• Uniform distributions have 2 very important properties:

  • The area under the graph of a continuous probability distribution is equal to 1

  • There is a correspondence between area and probability, so area = height * width

• A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape

• The graph of any continuous probability distribution is called a density curve, and any density curve must satisfy the requirement that the total area under the curve is exactly 1

• Because the total area under any density curve is equal to 1, there is a correspondence between area and probability

• The standard normal distribution is a normal distribution with the parameters of mu=0 and sigma=1. The total area under its density curve is 1.

• When working with a normal distribution, be careful to avoid confusion between z scores and areas

• The area corresponding to the region between 2 z-scores can be found by finding the difference between the 2 areas

• For the standard normal distribution, a critical value is a z score on the borderline separating those z scores that are significantly low or significantly high

6-2 Real Applications of Normal Distributions

• z = (x-mu) / sigma

• Make sure to choose the correct side of the graph when calculating the z-scores

• A z-score must be negative whenever it is located in the left half of the normal distribution

• Areas (or probabilities) are always between 0 and 1, and they are never negative

6-3 Sampling Distributions and Estimators

• When samples of the same size are taken from the same population, the following two properties apply:

  • Sample proportions tend to be normally distributed

  • The mean of sample proportions is the same as the population mean

• The sampling distribution of a statistic is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.

• The sampling distribution of the sample proportion is the distribution of sample proportions with all samples having the same sample size n taken from the same population

• p is population proportion

• p hat is sample proportion

• The distribution of sample proportion tends to approximate a normal distribution

• Sample proportions target the value of the population proportion in the sense that the mean of all sample proportions p hat is equal to the population proportion p

• The sampling distribution of the sample mean is the distribution of all possible sample means, with all samples having the same sample size n taken from the same population

• The sampling distribution of the sample variance is the distribution of sample variances (s squared), with all samples having the same sample size n taken from the same population

• The distribution of sample variances tends to be a distribution skewed to the right

• An estimator is a statistic used to infer (or estimate) the value of a population parameter

• An unbiased estimator is a statistic that targets the value of the corresponding population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the corresponding population parameter

6-4 The Central Limit Theorem

• For all samples of the same size n with n > 30, the sampling distribution of x bar can be approximated by a normal distribution with mean mu and standard deviation sigma / root n

• Sigma (subscript x bar) is called the standard error of the mean and is sometimes denoted as SEM

6-5 Assessing Normality

• A normal quantile plot is a graph of points (x, y) where each x value is from the original set of sample data, and each y value is the corresponding z score that is expected from the standard normal distribution

• If the distribution of the logarithms of the values is a normal distribution, the distribution of the original values is called a lognormal distribution

6-6 Normal as Approximation to Binomial

• Requirements to use normal distribution as approximation to the binomial distribution:

  • Sample is a simple random sample of size n from a population in which the proportion of successes is p

  • np >= 5, nq >= 5

• A continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by the interval form x-0.5 to x+0.5