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Normal Distributions

Normal Distributions

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Normal Distributions

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  1. Normal Distributions Wikipedia

  2. Standard Normal Distribution • = 0 • s=1 • To convert a normal distribution to standard normal: • z = (x - m)/s

  3. Normal Approximation of Binomial When there are a large number of trials calculation of a probability using the binomial formula may be very difficult. The normal probability distribution provides an easy-to-use approximation of binomial probabilities where: n > 20, np> 5, and n(1 - p) > 5

  4. Normal Approximation of Binomial • The mean and standard deviation of a binomial distribution are: • = np

  5. Normal Approximation of Binomial The continuity correction factor is used because a continuous distribution is being used to approximate a discrete distribution. Examples: P(x = 3) is approximated by P(2.5 < x < 3.5) P(x > 3) is approximated by P(x > 3.5) P(x > 3) is approximated by P(x > 2.5)

  6. Normal Approximation of Binomial When there are a large number of trials calculation of a probability using the binomial formula may be very difficult. The normal probability distribution provides an easy-to-use approximation of binomial probabilities where: n > 20, np> 5, and n(1 - p) > 5

  7. Normal Approximation of Binomial Applet http://onlinestatbook.com/stat_sim/normal_approx/index.html

  8. Normal Approximation of Binomial A software firm’s technical support staff employs 200 people. On any randomly chosen day 5 percent of the staff are not at work. What is the probability that less than 15 people will not be at work?

  9. Normal Approximation of Binomial • Step 1: Find mean and standard deviation • = np = (.05)(200) = 10 • Step 2: Find z score (remember continuity correction factor) • z = (14.5 – 10)/3.08 = 1.46 • Step 3: Find corresponding value in standard normal table • P(z < 1.46) = .9279

  10. Normal Approximation of Binomial What is the probability that at least 8 people will not be at work?

  11. Normal Approximation of Binomial • Step 1: Find mean and standard deviation • = np = (.05)(200) = 10 • Step 2: Find z score (remember continuity correction factor) • z = (7.5 – 10)/3.08 = -.81 • Step 3: Find corresponding value in standard normal table • P(z > -.81) = 1 - .2090 = .7910

  12. Chapter 7 Sampling and Sampling Distributions

  13. Population The set of all the elements of interest in a study Sample A subset of the population

  14. Parameter A numerical characteristic of the population Statistic A numerical characteristic of a sample

  15. Point Estimator The value of the sample statistic when it is used to estimate the value of the population parameter

  16. Deviations from the Population Value • The statistic calculated from a sample may different from the parameter because of: • Systemic error – The sample was collected in a way that increased the probability that observations with certain characteristics would be selected • Random error – Differences between the characteristics of the sample and the population due to the “luck of the draw”

  17. Simple Random Sample from a Finite Population A sample selected such that each observation in the population has an equal probability of being chosen • Simple Random Sample from an Infinite Population • A sample selected such that: • Each element selected comes from the population • Each element is selected independently

  18. Interval Estimate An interval around a point estimate designed to have a given likelihood of including the population parameter

  19. Sampling Distribution The distribution of a statistic for all possible samples of a given size.

  20. Sampling Distribution, Example Assume we have a population consisting of five observations with the following values of x: Obsx 2 3 3 3 4

  21. Sampling Distribution, Example Assume we are going to draw a sample of 3 observations to estimate the population mean. How many samples could we draw? (5!)/[3!(5-3)!] = [(5)(4)]/[(2)(1)] = 10

  22. Sampling Distribution, Example Possible samples: Obs.X valuesSample Mean 1,2,3 2,3,3 2 2/3 1,2,4 2,3,3 2 2/3 1,2,5 2,3,4 3 1,3,4 2,3,3 2 2/3 1,3,5 2,3,4 3 1,4,5 2,3,4 3 2,3,4 3,3,3 3 2,3,5 3,3,4 3 1/3 2,4,5 3,3,4 3 1/3 3,4,5 3,3,4 3 1/3

  23. Sampling Distribution, Example The resulting sampling distribution would be: 2 2/3 3/10 3 4/10 4 3/10 One way to think about a sampling distribution is that is the proportion of the time we would draw some value (or range of values) for the sample statistic.

  24. Sampling Distribution of the Mean

  25. Sampling Distribution of the Mean It is not necessary to use the finite population correction factor even when the population is finite if the sample size is less than or equal to 5% of the population (n/N < .05)

  26. Properties of the Standard Error • The greater the dispersion in the variable the larger the standard error • The larger the sample size the small the standard error • The size of the standard error decreases at a decreasing rate as observations are added • Assume s = 100: • nStandard errorD • 500 4.5 • 1,000 3.2 -1.3 • 1,500 2.6 -0.6 • 2,000 2.2 -0.4

  27. Shape of the Sampling Distribution of the Mean If the variable has a normal distribution the sampling distribution will have a normal distribution

  28. Shape of the Sampling Distribution of the Mean If the variable does not have a normal distribution the Central Limit Theorem applies. Central Limit Theorem In selecting simple random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large (typically n > 30).

  29. Central Limit Theorem http://onlinestatbook.com/stat_sim/sampling_dist/index.html See page 273