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Class 4

Class 4. Normal Distributions Sampling Distributions Central Limit Theorem. Normal Random Variable. Bell shaped curve. Computing Normal Probabilities. We have computed probabilities for Z, a standard normal. What is E(Z)? Var(Z)?

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Class 4

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  1. Class 4 Normal Distributions Sampling Distributions Central Limit Theorem

  2. Normal Random Variable • Bell shaped curve

  3. Computing Normal Probabilities • We have computed probabilities for Z, a standard normal. What is E(Z)? Var(Z)? • It turns out that if X is any normal random variable with mean  and standard deviation , then (X - )/ is a standard normal random variate. As a result, we write

  4. Computing Normal Probs. (cont.) • Suppose that X has a normal distribution with  = 5 and  = 3. Can you graph the distribution of X? • What is the P{X  11}?

  5. Computing Normal Probs (cont.) • What is P{-1  X  9}?

  6. Using EXCEL • Select the Function Wizard (fx) statistical/normdist. • The syntax of this function looks like normdist(x, , , true or false). • If the fourth argument is true, this will return P{X  x} where X is a normal(, ).

  7. Example • How can you interpret the computation of Z? • The lifetime of a tire has a normal distribution with a mean of 40,000 miles and a standard deviation of 3,000 miles. It is desired to set a warranty on these tires such that 10% of the tires fall under warranty. What is the required value (in miles)?

  8. Example (cont.) • How many standard deviations do we have to go out on a (any) normal distribution to cut off 10%? • Therefore, if w is the warranty limit, we have:

  9. Using EXCEL • The function norminv(prob, , ) will return the value on a normal(, ) distribution that cuts off prob to the left. • Try norminv(.1, 40000, 3000).

  10. Summary (So far) • Describe Data Graphically and Numerically • Populations vs. Samples • Further description of populations--Random Variables • Discrete • Continuous Now we will return to sampling and apply what we have learned.

  11. Sampling • Reasons for sampling as opposed to taking a census • Cost • Speed • Analysis • Feasibility • Types • Nonrandom • Random • Simple Random Sample: A sample where all samples of size n have the same chance of being chosen. • Systematic • Stratified • Cluster Judgment or Convenience

  12. Sampling Distributions • Basic idea: Imagine all simple random samples of size n that can be drawn from a population. Each sample has its own characteristics (such as a sample mean). We might wonder about the likelihood of seeing a particular characteristic in our sample. This is the idea behind a sampling distribution.

  13. Example • Infinite population: • For future reference:  = 1(.2) + 2(.2) + 4(.6) = 3 2 = (1-3)2(.2) + (2-3)2(.2) + (4-3)2(.6) = 1.6

  14. Homework • For the following population: • Compute  and . • Generate the sampling distribution of for a sample of size n = 3.

  15. The Central Limit Theorem • Let be computed by taking a simple random sample of size n from a population with mean  and standard deviation . Then for large n, the distribution of will be approximately normal. Large n means: • n  1 when sampling from a normal distribution, • n  30 when sampling from any distribution. As always when sampling from an infinite population or a population of size N where N>>n,

  16. A Quick Note

  17. Using the CLT • Incomes in a community are normally distributed with a mean of $25,000 and standard deviation of $8,000. If we take a sample of size 4, what is the probability that the average income in the sample is greater than $29,000?

  18. Income Example

  19. Income Example • What is the probability that a single income selected will fall above $29,000? • What proportion of the population will fall above $29,000?

  20. Using the CLT • A company produces lids for tin cans. The lids are supposed to be 4 inches in diameter. The standard deviation of tin can lids is .012 inches. Because a worker suggested that the machine is in need of adjustment, the foreman has taken a sample of 100 lids and found that inches. Should they shut down production to make the adjustment?

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