Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 1.3 1 Summarizing Data: Theoretical models A density curve is a curve that models the shape of the histogram of a data distribution. The area between the curve, the horizontal axis, and any two vertical lines at x = a and x = b measures the (theoretical) proportion of the data that lies between x = a and x = b. The total area underneath the curve measures 1 = 100%. The mean of a distribution is the point at which the curve would balance if it were a physical solid. The median of a distribution is the point which separates the area under the curve into two equal halves. The mean and median of a symmetric density curve are equal. The mean of a skewed density curve lies further out in the tail of the distribution than the median. The theoretical mean associated with a density curve is labeled m (Greek letter mu). The theoretical standard deviation is denoted s (Greek letter sigma). Section 1.3 2 Summarizing Data: The Normal Distribution The most common and important type of density curve is a normal curve. • It faithfully describes many real data sets. • It approximates important types of chance outcomes. • It is the basis for most statistical inference techniques. A normal curve is symmetric and bell-shaped. Its mean m lies at the position of the central peak. The points on either side of the mean at which the curve changes concavity are a distance s from the mean. We denote the normal distribution with N(m, s). [TI-83: If a quantity x satisfies a normal distribution N(m, s), then the proportion of values of x which lie between two particular values a and b ( a ≤ x ≤ b ) is computed as DISTR normalcdf( a , b , m , s ) If no upper bound b is given, it is understood that b"= ∞; on the calculator, use 1E99 for ∞; if no lower bound is given, it is understood that a"= –∞; on the calculator, use -1E99 for –∞.] Section 1.3 3 The 68–95–99.7 Rule For the normal distribution, • 68% of the data will lie within one standard deviation of the mean (between m – s and m + s); • 95% of the data will lie within two standard deviations of the mean (between m – 2s and m + 2s); • 99.7% of the data will lie within three standard deviations of the mean (between m – 3s and m + 3s). Section 1.3 4 Describing Data: Measuring Relative Standing • The z-score, or standardized score, of an observation x in a normal distribution is its distance from the mean, -m z = xs measured in terms of standard deviations; that is, † x = m + zs . Positive z-scores lie above the mean, negative z-scores below the mean. † • The standard normal distribution N(0, 1) is the one with mean 0 and standard deviation 1. If x is a variable with normal distribution N(m, s), then the corresponding standardized variable z has the standard normal distribution. [TI-83: DISTR normalcdf( a , b )] • extreme z-scores values z* of z which satisfy certain probability conditions of the form P(z ≤ z*) = P for a given proportion value P. [TI-83: DISTR invNorm( P ) ]