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Lesson 7 - 4

Lesson 7 - 4. Assessing Normality. Quiz. Homework Problem3: Chapter 7-2 Find the indicated probability of the Z random variable a) P(-1.20 ≤ Z ≤ 2.34) b) P(Z ≥ -0.92) c) Find z such that the area under the curve to the right is 0.89 Reading questions:

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Lesson 7 - 4

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  1. Lesson 7 - 4 Assessing Normality

  2. Quiz • Homework Problem3: Chapter 7-2Find the indicated probability of the Z random variablea) P(-1.20 ≤ Z ≤ 2.34)b) P(Z ≥ -0.92)c) Find z such that the area under the curve to the right is 0.89 • Reading questions: • To access normality of a distribution we look at what type of plot? • To be assessed as normal what must the plot (graph) show?

  3. Objectives • Draw normal probability plots to assess normality

  4. Vocabulary • Normal probability plot – a graph the plots observed data versus normal scores • Normal Scores – is the expected Z-score of the data value if the distribution of the random variable is normal

  5. Normality Plotting Instructions • Arrange the data in ascending order • Compute fi = (i – 0.375) / (n + 0.25), where i in the index (position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is fi. • Find the Z-score corresponding to fi from z-score table (Table IV) • Plot the observed values on the horizontal axis and the corresponding z-scores on the vertical axis. If the sample data are taken from a population that is normally distributed, a normal probability plot of the observed values versus the expected Z-scores will be approximately linear.

  6. Non-Normal Plots • Both of these show that this particular data set is far from having a normal distribution • It is actually considerably skewed right

  7. TI-83 Normality Plots • Enter raw data into L1 • Press 2nd ‘Y=‘ to access STAT PLOTS • Select 1: Plot1 • Turn Plot1 ON by highlighting ON and pressing ENTER • Highlight the last Type: graph (normality) and hit ENTER. Data list should be L1 and the data axis should be x-axis • Press ZOOM and select 9: ZoomStat Does it look pretty linear? (hold a piece of paper up to it)

  8. Example #1 page 396 Cum Prob = normdist(c2, 0, 1, true) Correlation: correl(b2:b7,d2:d7) fi = (i – 0.375) / (n + 0.25) where i is the index value Expected Z-score = norminv(c2, 0, 1) Probability mean standard deviation

  9. Problem 10 Normality Graph

  10. Problem 10 b) x-bar = 20.90 s = 10.52 d) Z (20) = (20 – 20.90) / 10.52 = -0.086 Z (35) = (35 – 20.90) / 10.52 = 1.340 Normcdf (-0.086,1.34) = 0.4441 book got 0.4458 as a result of round-off error using table e) Z (40) = (40 – 20.90) / 10.52 = 1.816 Normcdf(1.816,E99) = 0.3469 Book got 0.0344 as a result of round-off error using table

  11. Summary and Homework • Summary • We can assess whether sample data is approximately normal by using the normal probability plot • If the data is approximately normal, then the normal probability plot (a.k.a. the QQ plot) should be approximately normal also • Homework • pg 399 – 401; 3 – 8, 10 – 12

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