2.11 The Normal Distribution

Describe a normal distribution.

• A continuous random variable is a variable that can take on any value within a specified domain. Every interval within the domain has a probability associated with it.
• Many continuous random variables are wellmodeled by a normal distribution.
• A normal distribution can be described as a continuous, unimodal, bell-shaped, and symmetric curve.
• A normal curve can be used to model a distribution of data and a continuous random variable.
• The normal distribution, or the normal curve, is identified by two parameters, the mean, µ, and the standard deviation, σ. The smaller the standard deviation, the taller and more concentrated the normal curve is around its mean. The larger the standard deviation, the shorter and less concentrated the normal curve is around its mean.

Calculate the mean and standard deviation for a normal distribution.

• A standard normal distribution is a normal distribution with mean and standard deviation μ =0 σ =1. Probability, Random Variables, and Probability Distributions UNIT 2

Calculate percentages from a normal distribution using the empirical rule.

• The empirical rule can be used to estimate the area of a region under the graph of the normal distribution curve. For a normal distribution, approximately 68% of observations are within 1 standard deviation of the mean, approximately 95% of observations are within 2 standard deviations of the mean, and approximately 99.7% of observations are within 3 standard deviations of the mean. This is called the empirical rule, or the 68–95–99.7 rule.

Calculate the probability that a particular value lies within a given interval of a normal distribution.

• If the distribution of a random variable is approximately normal, the probability that the random variable takes on values within a particular interval of the random variable is determined by the area under the normal curve within that interval. The total probability or area under the normal curve is 1.

Calculate the associated intervals and areas of a normal distribution.

• The boundaries of an interval associated with a given area in a normal distribution can be determined using technology or using z-scores and a standard normal table.
• Intervals associated with a given area in a normal distribution can be determined by assigning appropriate inequalities to the boundaries of the intervals. To determine the intervals, p is defined as a number between 0 and 100, xa is the lower bound, and xb is the upper bound on a normal distribution.
  • i. P Xx p a() <= 100 means that the lowest p% of the values lie to the left of xa . Probability, Random Variables, and Probability Distributions UNIT 2 76
  • ii. P xX x p ab()<< = 100 means that p% of the values lie between xa and xb.
  • iii. pP()Xx>=b 100 means that the highest p% of the values lie to the right of xb.
  • iv. To determine the most extreme p% of values on both sides requires dividing the area associated with p% into two equal areas on either extreme of the distribution: P Xx p a()<= ( (| ) )| 1 2 100 and P Xx p b()>= ( (| ) )| 1 2 100 mean that half of the p% most extreme values lie to the left of xa and half of the p% most extreme values lie to the right of xb.

Compare measures of relative position for distributions.

• Percentiles and proportions may be used to compare relative positions of individual values within a normal distribution or between normal distributions. Probability, Random Variables, and Probability Distributions UNIT 2