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Download stat226_2-3-16 - Iowa State University
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Session packet Statistics 226 Supplemental Instruction Iowa State University Leader: Course: Instructor: Date: Luyun Stat 226 Anna Peterson 2/3/16 Recap: 1. The normal distribution Every normally distributed random variable follows a probability distribution that is characterized by two distinct parameters ______: describe the center of the distribution ______: provides information about how much variability we can expect to see in our data. Notation: to denote the normal distribution we use 2. The 68-95-99.7 rule (empirical rule) The 68-95-99.7 rule applies to all normal distribution (i.e. for any choice of μ and σ. Approx. 68% of the data fall within ____ standard deviation of the mean. i.e. within the range of _______________________ Approx. 68% of the data fall within ____ standard deviation of the mean. i.e. within the range of _______________________ Approx. 68% of the data fall within ____ standard deviation of the mean. i.e. within the range of _______________________ 3. The standard normal distribution Is a special normal distribution Has a mean ________ and a variance ________. Is denoted by _______ When we want to transform any normal distribution with man and variance of a standard normal distribution, i.e. the values from any _________ are transformed to the corresponding values from a __________. This is called standardizing. 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. Question… Can you tell the difference between a normal distribution and a standard normal distribution? 1060 Hixson-Lied Student Success Center 515-294-6624 [email protected] http://www.si.iastate.edu 1. A new line of cars has gas mileage represented by random variable X that is normally distributed with a mean of 32 mpg with a standard deviation of 4mpg. Use the empirical rule to answer questions below. a. What is the notation for this distribution? b. The middle 68% of cars get between how many mpg? c. The middle 95% of cars get between how many mpg? d. 2.5% of all cars get no more than how many mpg? e. 0.15% of all cars get no more than how many mpg? f. Only 16% of all cars get more than how many mpg? g. Only 0.15% of all cars get more than how many mpg? h. What is the approximate probability of obtaining gas mileage between 24 and 32 mpg? i. What is the approximate probability of obtaining gas mileage between 32 and 36 mpg? j. What is the approximate probability of obtaining gas mileage between 24 and 44 mpg? 2. Let X be a normal N (8, 4²) random variable, that is X~N (8, 4²). a. Sketch a picture of the normal distribution, N (8, 4²), and shade the area corresponding to P (X< 6). b. What z-score corresponds to x=6? c. Compute P (X< 6). d. What z-score corresponds to the 77th percentile of the standard normal distribution, N (0, 1)? You still need to draw an appropriate standard normal distribution graph for this. e. What value of corresponds to the 77th percentile of X. the normal distribution, N (8, 4²) 3.Since the length of a downhill ski is related to the height of the individuals renting them, it is fair to assume that a normal distribution would describe the length of women’s skis at rental outlets in Colorado. The mean of the distribution is 150 cm and the standard deviation is 10 cm. a. What is the proportion of women’s ski lengths that are less than 130 cm? b. What is the proportion of women’s ski lengths that are greater than 125 cm? c. What is the proportion of women’s ski lengths that are between 125 and 155? d. Very long skies are expensive and there are not many people who rent them. What is the longest women’s ski a rental shop should carry so that only 2 percent of the costumers will ask to rent a longer ski?
