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Lecture 11 The Normal Distribution Math 1107 Introduction to Statistics

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Math 1107 – The Normal Distribution Inferential Statistics Sampling & Central Limit Theorem Confidence Intervals, Hypothesis Testing, Regression, ANOVA, etc. Descriptive Statistics Visualization, Summarization, Outliers Categorical Data Analysis Analyzing Specific Data Looking for Outliers Looking for Relationships Drawing Conclusions from Representative Data Making Decisions Looking for Relationships

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There are many types of distributions: Binomial – 2 outcomes (success or failure…H or T); Poisson – Infinite possibilities, with discrete occurrences; Normal – Bell Shaped continuous distribution Math 1107 – The Normal Distribution

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A family of continuous random variables whose outcomes range from minus infinity to plus infinity. Bell shaped and symmetric about the mean μ. Mean = μ, Median = μ, Mode = μ. The standard deviation is σ. The area under the normal curve below μ is.5. The area above μ is also.5. Probability that a Normal Random Variable Outcome: Lies within +/- 1 std dev of the mean is.6826 Lies within +/- 2 std dev of the mean is.9544 Lies within +/- 3 std dev of the mean is.9974 Math 1107 – The Normal Distribution

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68%95%99% 0123-2-3

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Math 1107 – The Normal Distribution The Standard Normal Distribution looks like a Normal Distribution, but has important statistical properties: mean = 0 std dev = 1 Remember from earlier in the semester that: The Std Normal Distribution enables the calculation of Z- scores Z-Scores can be compared against ANY populations using any scale

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Remember from earlier in the semester that: The Std Normal Distribution enables the calculation of Z-scores; Z-Scores can be compared against ANY populations using any scale; Z-scores are stated in units of standard deviations; So, typical Z-scores will range from 0 (the mean) to 3 and can be negative or positive. And…most importantly…we can use Z-scores to determine the associated probability of an outcome. Math 1107 – The Normal Distribution

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How do we use a z-score to find a probability? Z=(x-mu)/std dev Where, X is a value of interest from the distribution; Mu = the average of the distribution; Std dev = the std dev of the distribution.

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Math 1107 – The Normal Distribution Prior to solving any Normal Distribution problem using Z-scores, ALWAYS draw a sketch of what you are doing. This will provide you with a guide for what is a “reasonable” answer.

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Math 1107 – The Normal Distribution Example: Watts Corporation makes lightbulbs with an average life of 1000 hours and a std dev of 200 hours. Assuming the life of the bulbs is normally distributed, what is the probability of buying a bulb at random that lasts for up to 1400 hours? X=1400 Mu = 1000 Std dev = 200 So, Z=(1400-1000)/200 = 2. A z-score of 2 equals.4772. We add.5 to this and get a probability of.9772.

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Example: Unlucky Larry bought a Watts Corporation bulb and it only lasted 800 hours. What is the probability that a bulb selected at random would last between 800 and 1000 hours? X=800 Mu = 1000 Std dev = 200 So, Z=(800-1000)/200 = -1. A z-score of -1 equals.3413. So, there is a 34.13% chance of selecting a bulb at random that generates between 800 and 1000 hours of light. Math 1107 – The Normal Distribution

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Example: What is the probability of selecting a bulb at random that generates less than 800 hours? The total area under the curve less than the average is.50 or 50%. So, if we know the area between 800 and 1000 is.3413, then the area less than 800 is.5-.3413 or.1587. Math 1107 – The Normal Distribution What is the probability of selecting a bulb at random that generates more than 800 hours? The total area under the curve more than the average is.50 or 50%. So, if we know the area between 800 and 1000 is.3413, then the area less than 800 is.5+.3413 or.8413.

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Math 1107 – The Normal Distribution Example: Coca Cola Bottlers produce millions of cans of coke a year. The average can holds 12 ounces with a std dev of.2 ounces. What is the probability of getting a coke with between 11.8 and 12 ounces? X=11.8 ounces Mu = 12 Std dev =.2 So, Z=(11.8-12)/.2 = -1. A z-score of -1 equals.3413.

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Example: Coca Cola Bottlers produce millions of cans of coke a year. The average can holds 12 ounces with a std dev of.8 ounces. What is the probability of getting a coke with between 11.8 and 12 ounces? X=11.8 ounces Mu = 12 Std dev =.8 So, Z=(11.8-12)/.8 = -.25. A z-score of -.25 equals.0987, or 9.87% Math 1107 – The Normal Distribution

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Example from Page 243: Airlines have designed their seats to accommodate the hip width of 98% of all males. Men have hip widths that are normally distributed with a mean of 14.4 inches and a standard deviation of 1.0. What is the minimum hip width that airlines cannot accommodate? This is the 98 th percentile.

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Z=(x-mu)/std dev Math 1107 – The Normal Distribution In this example, we are working “backward”. We know the Probability (98%) and we want to know the value that generates this probability. Given the Z formula, we now solve for x. 2.05=(x-14.4)/1 2.05 = x-14.4 2.05+14.4 = x-14.4+14.4 16.45 = x

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