Math 247: Continuous Random Variables: The Uniform Distribution (Section 6.1) and The Normal Distribution (Section 6.2)

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1 Math 247: Continuous Random Variables: The Uniform Distribution (Section 6.1) and The Normal Distribution (Section 6.2) The Uniform Distribution Example: If you are asked to pick a number from 1 to 10 and your choice is completely random (this is a questionable assumption!), what is the probability distribution of the number you choose? Construct a graph of the probability distribution of the random variable. What is the probability you will choose a number greater than 7? Use proper notation and illustrate this probability on the graph. Example: Assume the average amount of weight gained by a person over the winter months is uniformly distributed from 0 to 30lbs. Construct a graph of the distribution. What is the probability a person will gain between 10 and 15 lbs during the winter months? Use proper notation and illustrate this probability on the graph.

2 Many physical characteristics, as well as test scores, are normally distributed. The Normal Distribution: The N(, ) curve. = population mean, = population standard deviation The Standard Normal Distribution: N(0,1) curve This curve is the skeleton underneath every normal distribution curve. Example: Assume women s heights are normally distributed with a mean of 65 inches (5 5 ), and a standard deviation of 2.5 inches. Sketch the normal distribution curve, N(65, 2.5), that models these heights. Label the axis! Revisiting the Empirical Rule: What is the area under the entire curve? What does this number represent? What is the probability a randomly chosen women has a height between 62.5 inches and 67.5 inches? Use proper notation and shade this area in.

3 Revisiting z-scores: Continuing with the same heights of women example, label the x-axis and fill in the relevant values for the normal distribution for heights, N(65, 2.5). z-score formula: x z x = data value, = population mean, = population standard deviation Find the z-scores for the following heights: 57.5 in, 60 in, 62.5 in, 65 in, 67.5 in, 70 in, 72.5 in What is the relationship between the z-scores and the standard deviations for the heights? Fill in the z-scores on the Normal Distribution Graph, using a second axis under the x-axis. We can revisit the Empirical Rule questions and think of them in terms of z-scores: What is the probability a randomly chosen women has a height between 62.5 inches and 67.5 inches? is the same question as We can use z-scores to find areas = probabilities = percentages.

4 Finding probabilities using the Standard Normal Curve (z-scores) Example: Use the online calculator to find the following probabilities using z-scores: Use the proper probability notation and illustrate each on the N(0,1) curve. The probability that a z-score will be between -1 and 1 The probability that a z-score will be more than The probability that a z-score will be less than Now we can find probabilities for any normal distribution! Steps to find a percent/probability for a normal distribution using z-scores: 1. PARKING LOT! 2. Write the probability or proportion problem in the proper notation. 3. Sketch the normal curve, label the x-axis in words. 4. Plot the give data (x) value on the axis, and shade in the area described in Step Find the z-score for the x-value using the z-score formula. 6. Find the area using the Normal Distribution Calculator (online) or a table. 7. Answer the question! Examples (do these on another sheet of paper): Males SAT scores. According to data from the College Board, the mean quantitative SAT score for male high school seniors in 2012 was 530. Assume that SAT scores are approximately Normally distributed with a standard deviation of 100. If a male senior is selected at random, what is the probability that his score will be higher than 675? Wide Receivers. The average height of wide receivers in the NFL is 72.7 inches with a standard deviation of.8 inches. The heights are normally distributed. Steve Smith, a former wide receiver with the Carolina Panthers, is 71 inches tall. What percent of wide receivers are as short as or shorter than Mr. Smith?

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