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Review 3

Review 3. Chapter 6 Normal Distributions Chapter 7 Introduction to Sampling Distributions. Chapter 6 Normal Distributions. Graphs of Normal Probability Distributions Standard Units and Areas Under the Standard Normal Distribution Areas Under Any Normal Curve

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Review 3

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  1. Review 3 • Chapter 6 Normal Distributions • Chapter 7 Introduction to Sampling Distributions

  2. Chapter 6 Normal Distributions • Graphs of Normal Probability Distributions • Standard Units and Areas Under the Standard Normal Distribution • Areas Under Any Normal Curve • Normal Approximation to the Binomial Distribution

  3. The Normal Curve

  4. The Empirical Rule

  5. The Empirical Rule

  6. Control Charts A graph to examine data over equally spaced time intervals. Used to determine if a variable is in statistical control. Statistical Control: A variable x is in statistical control if it can be described by the same probability distribution over time.

  7. Control Chart Example

  8. Determining if a Variableis Out of Control One point falls beyond the 3σ level. A run of nine consecutive points on one side of the center line. At least two of three consecutive points lie beyond the 2σ level on the same side of the center line.

  9. 6.2 Standard Units and Areas Under the Standard Normal Distribution • Z-Scores and Raw Scores • Standard Normal Distribution • Areas Under the Standard Normal Curve • Using a Standard Normal Distribution Table

  10. Computing z Scores

  11. Work With General Normal Distributions Or equivalently,

  12. The Standard Normal Distribution Z scores also have a normal distribution µ = 0 σ = 1

  13. Using the Standard Normal Distribution There are extensive tables for the Standard Normal Distribution. We can determine probabilities for normal distributions: Transform the measurement to a z Score. Utilize Table 5 of Appendix II.

  14. Procedure HOW TO USE A LEFT TAIL STYLE STANDARD NORMALDISTRIBUTION TABLE 1. For areas to the left of a specified z-value, use the table entry directly. 2. For areas to the right of a specified z-value, look up the table entry for z and subtract the area from 1. Note: Another way to find the same area is to use the symmetry of the normal curve and look up the table entry for z. 3. For areas between two z-values, z1 and z2 (where z2 >z1), Subtract the table area for z1 from the table area for z2

  15. 6.3 Areas under Any Normal Curve

  16. Inverse Normal Distribution Sometimes we need to find an x or z that corresponds to a given area under the normal curve. In Table 5, we look up an area and find the corresponding z.

  17. 6.4 Normal Approximation to the Binomial

  18. Continuity Correction

  19. Chapter 7 Introduction to Sampling Distributions • Sampling Distributions • The Central Limit Theorem • Sampling Distributions for Proportions

  20. 7.1 Sampling Distributions • Statistic A statistic is a numerical descriptive measure of a sample. • Parameter A parameter is a numerical descriptive measure of a population.

  21. 7.2 The Central Limit Theorem If x is a random variable with a normal distribution, mean = µ, and standard deviation = σ, then the following holds for any sample size:

  22. The Central Limit Theorem(Any Distribution) If a random variable has any distribution with mean = µ and standard deviation = σ, the sampling distribution of will approach a normal distribution with mean = µ and standard deviation = as n increases without limit.

  23. Sample Size Considerations For the Central Limit Theorem (CLT) to be applicable: If the x distribution is symmetric or reasonably symmetric, n≥ 30 should suffice. If the x distribution is highly skewed or unusual, even larger sample sizes will be required. If possible, make a graph to visualize how the sampling distribution is behaving.

  24. If np > 5 and nq > 5, then can be approximated by a normal variable with mean and standard deviation and 7.3 Sampling Distributions for Proportions

  25. Continuity Corrections Since is discrete, but x is continuous, we have to make a continuity correction. For small n, the correction is meaningful.

  26. Control Charts for Proportions Used to examine an attribute or quality of an observation (rather than a measurement). We select a fixed sample size, n, at fixed time intervals, and determine the sample proportions at each interval. We then use the normal approximation of the sample proportion to determine the control limits.

  27. Procedure

  28. (c) Signal III: at least two out of three consecutive points beyond a control limit (on the same side). If no out-of-control signals occur, we say that the process is “in control,” while keeping a watchful eye on what occurs next.

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