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Session packet Statistics 226 Supplemental Instruction Iowa State University Leader: Course: Instructor: Date: Luyun Stat 226 Anna Peterson 2/2/16 RECAP: The standard normal distribution Is a “special” normal distribution. Has a mean ________ and a variance ________. Is donated by ________________. Nearly all the area is between ____ and _____. Having knowledge of the mean μ and the standard deviation σ of a normal distribution allows us to determine What proportion of individuals we can expect to fall within a specific range. What percentile a given individual falls at if you know their data value. What data value corresponding to a given percentile. (backward calculation) For the standard normal distribution, the proportion of observations falling into a specified range is tabulated. This is the only normal distribution for which we have readily available tabulated values. We therefore need to transform any normal distribution with mean μ and variance to a standard distribution, i.e. the values from any _________ are transformed to the corresponding values from a _________. This is called s____________. Standardizing, z-score If x is an observation from a normally distributed random variable X, i.e. X follows a normal distribution with mean μ and standard deviation σ, then the standardized value Z is given by Z= A standardized value is often called a z-score. A z-score tells us how many standard deviations the original observation is off the mean and in which direction. Observations _______ than the mean are positive when standardized, and observations________ than the mean are negative when standardized. 1. Let x denote an observation from a normal distribution with a mean of 50 and a standard deviation of 10. For any given x-values, find the corresponding z-score. For any given zscore, find the corresponding x-value from the distribution. Interpret each z-score/xvalue. a. What is the normal distribution notation for this distribution? b. X=33 1060 Hixson-Lied Student Success Center 515-294-6624 [email protected] http://www.si.iastate.edu c. X=8 d. X=49 e. X=63 f. X=72 g. Z=3.4 h. Z=1.2 i. Z=0 j. Z=-2.7 k. Z=-0.4 2. Former ISU basketball player Kelvin Cato is 83 inches tall. Assuming that heights follow approximately a normal distribution with mean 70 and standard deviation σ = 3, a. What is his corresponding z-score? b. What proportion of men are taller than him? 3. For each problem below draw a picture of the normal curve and shade the area you have to find. Let Z represent a variable following a standard normal distribution. a. Find the proportion that is less than z=2.00 b. Find the proportion that is between z = -0.13 and z = 1.75 c. Find the proportion that is greater than z = 1.86 d. Find the z-score for the 64th percentile. e. Find the z-score for the 24th percentile.