2.11.E—Calculate the associated intervals and areas of a normal distribution

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.