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Math 1107 Introduction to Statistics

Math 1107 Introduction to Statistics. Lecture 11 The Normal Distribution. Math 1107 – The Normal Distribution. Drawing Conclusions from Representative Data Making Decisions Looking for Relationships. Analyzing Specific Data Looking for Outliers Looking for Relationships.

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Math 1107 Introduction to Statistics

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  1. Math 1107 Introduction to Statistics Lecture 11 The Normal Distribution

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

  3. Math 1107 – The Normal Distribution 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

  4. Math 1107 – The Normal Distribution • 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

  5. Math 1107 – The Normal Distribution

  6. 68% 95% 99% Math 1107 – The Normal Distribution -3 -2 -1 0 1 2 3

  7. 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

  8. Math 1107 – The Normal Distribution • 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.

  9. Math 1107 – The Normal Distribution 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.

  10. 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.

  11. 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.

  12. Math 1107 – The Normal Distribution 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.

  13. Math 1107 – The Normal Distribution 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. 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.

  14. 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.

  15. 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 .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%

  16. Math 1107 – The Normal Distribution 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 98th percentile.

  17. 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. Z=(x-mu)/std dev 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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