Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download AP STATS 2.2 STANDARD NORMAL CALCULATIONS All normal
Transcript
AP STATS 2.2 STANDARD NORMAL CALCULATIONS All normal distributions are the same if we measure in units of size σ about the mean, μ, as center. Changing to these units is called STANDARDIZING. STANDARDIZED OBSERVATIONS If x is an observation from a distribution that has a mean, μ and standard deviation σ , the standardized value of x is z x The letter z is commonly used for a standardized observation. The standardized score is called the z score. A standardized score tells how many standard deviations a particular observation is away from the mean. Standardizing makes all normal distributions into a single distribution with a common scale. The STANDARD NORMAL DISTRIBUTION is the normal distribution with a mean = 0 and a standard deviation = 1, or N (0,1) An area under a density curve is a proportion of the observations in the distribution. Because all normal distributions are the same when we standardize, we can find areas under a curve from a single table that gives the areas under the curve for the standard normal distribution. Table A in the front and back of your text gives the areas under the curve to the LEFT of z. FINDING NORMAL PROPORTIONS State the problem in terms of the observed variable x. Standardize x to restate the problem in terms of a standard normal variable, z. Draw a picture to show the area under the curve. Find the required area under the standard normal curve using Table A and the fact that the total area under the curve is 1. EXAMPLE 1 The heights of young women (18-24) are approximated by the normal distribution N(64.5, 2.5). What is the standardized height of a woman 5 ft. 8 in tall? What proportion of all young women are less than 68 inches tall? EXAMPLE 2 Using the same distribution of young women’s heights, Karen is 4’ 11’’ tall. What percent of all young women are taller than Karen? EXAMPLE 3 The distribution of blood cholesterol levels in a large population of people of the same age and sex is relatively normal. For 14 yr old boys, N(170 mg/dl, 30 mg/dl). What percent of 14 yr old boys have a cholesterol level above 240 mg/dl? (240 mg/dl is considered the danger level that may require medical attention) EXAMPLE 4 – Reverse look-up Scores on the SAT Verbal in recent years approximate the N ( 505, 110 ) distribution. How high must a student score to be in the top 10% of all students taking the SAT?
