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The normal approximation for data

Statistics 224 Fred Boehm 29 January 2014. The normal approximation for data. Course logistics. Homework 1 is due now (on front table) Homework 2 will be posted before Friday due Friday, February 7, in class Future homeworks due on Fridays (not Wednesdays) Office hours:

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The normal approximation for data

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  1. Statistics 224 Fred Boehm 29 January 2014 The normal approximation for data

  2. Course logistics • Homework 1 is due now (on front table) • Homework 2 will be posted before Friday • due Friday, February 7, in class • Future homeworks due on Fridays (not Wednesdays) • Office hours: • Please come with questions • Doing homework (without asking questions) is better suited for the tutorial lab

  3. Course logistics • Please ensure that you have access to Freedman et al.'s text • Homework 2 (and others) rely on it • Readings are primarily from Freedman et al. • Email Fred if you are having difficulty in finding or buying it

  4. Lecture overview • Draw heavily on Freedman et al., Ch 5 • Normal curve • Areas under the normal curve • Normal approximation • Change of scale

  5. Standard normal curve

  6. Standard normal curve properties • Why is it 'standard'? • Consider its equation: • What are three famous irrational numbers?

  7. More properties of standard normal • Symmetric (about zero) • Monotonically decreasing on positives • Monotonically increasing on negatives • Area under the curve? • Calculate the integral

  8. Standard normal: Area under the curve http://allpsych.com/researchmethods/distributions.html

  9. Area under the curve: Integral - Use integral to calculate the area under the curve - For a contiguous area under the curve, use integration limits that correspond to the endpoints of the area

  10. Area under the curve - Table of integrals usually have one endpoint as (negative infinity) - How do we use these tables to determine the area under the curve from a to b, where a is finite?

  11. Area under the curve Think of the differences of two intervals: We know both integrals on the left-hand side:

  12. Normal approximation for data Assume that the population mean height (for men) is 69 inches • Population variance is 9 inches^2 • So, standard deviation is 3 inches • According to the normal approximation, what percentage of men have height between 66 inches and 72 inches?

  13. Normal approximation • Mean height: 69 inches; SD = 3 inches • Translate 66 inches to 1 SD below the mean, since 69-3 = 66 • Translate 72 inches to 1 SD above the mean • Look back at the standard normal curve • See that about 68% of the area under the curve is within 1 sd of the mean

  14. More on Normal approximation • How many SD's from the mean is an observed value? • Z value • For mean mu & SD sigma, and observed value x, • z is the number of standard deviations from the mean to the observation • Sign of z gives direction from mean to observation

  15. Change of scale • What is the average of the numbers: 1,3,4,5,7 • What is their standard deviation? • For this calculation, use the equation: • Transform each original data point to a z-value: • What is the mean of the five z-values? • What is the standard deviation of the five z-values?

  16. Lecture overview • Draw heavily on Freedman et al., Ch 5 • Normal curve • Areas under the normal curve • Normal approximation • Change of scale

  17. Before Friday's lecture - Read carefully all of Chapter 5 in Freedman et al. - Review today's lecture materials - Freedman et al. has many exercises to practice these concepts. Do them. - Check your wisc.edu email - Homework 2 availability

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