## Math 151 , Spring '12, Mon. Feb. 13, Day 10.After class. Hit reload..

Reading :  Ch. 3: Normal pp. 73-74, & 68-95-99.7% rule pp.74-77. Use Normal Density Applet curve to check concepts and computation. "Check" problems p. 87: 3.15, 16, 17, 18;... 19, 20. //Standardizing to standard normal pp.77-79 "Check" 3.21. Ahead, rest of chapter.  We WILL learn to use table A. Moore doesn't separate out reading the z-table in the following; focus on just the z-table parts on first reading:  p. 79-83, Cum. proportion and  normal.  "Check" 3.22, 3. 23. "Backward" from prop. to z pp. 83-86.  We'll revisit and learn to deal with x's.

Exams not finished; Sorry!

"Normal" distributions:
("Gaussian", "Bell-shaped") part 1 (pp. 70-74)  Details Day 7   Normal template, optional
Applet: Normal Density Curve
http://bcs.whfreeman.com/bps5e
• What "mechanism"?  Thing measured is the result of many small independent influences.
• a Family:  all the same basic shape, except for scale. Mean gives middle, standard deviation gives spread.
• Knowing mean ("mu") and standard deviation ("sigma") tells you everything.
• "N(mu, sigma)"= "Normal with mean mu, standard deviation sigma"
• Symmetric.  Mean at middle. Curvature changes at 1 standard deviation on either side of mean.
• "Standard normal": mean = 0, s.d. = 1  N(0,1)
• "Middle s. d." A center region one standard deviation wide spans about 38% (not in text)
• 68-95-99.7 rule:  68% within 1 s.d. of mean, 95% within 2 s.d. of mean, 99.7% within 3 (roughly).

• What percent is further than 3 s.d. from the mean?  What percent  is  higher than 1 s.d. from the mean? Etc.
These are back-of-the-envelope tools.  Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.

Example:  "Classic IQ test" scores are approximately N(110, 25).  mean=110, mean +1s.d. = 135, mean + 2s.d.'s = 160,  mean -1s.d. = 95, etc.  See picture
..
Outline:  Details Day 7

Standardizing: (p. 74-5) A "raw value" x is standardized by telling how many standard deviations above the mean it is.
That's the z-score.  Details Day 7
Values in any Normal distribution, after standardizing, become values in a N(0,1) "standard normal" ("Z") distribution.
(Use double ruler lines on Normal curve, one in "raw" units, one in z.    Normal template, optional)

"What proportion"problems in general:  BPS5e pp. 79-86, first pass
Use  Applet: Normal Density Curve   http://bcs.whfreeman.com/bps5e
Work toward answering: Proportion with scores between 100 and 145?   below 100?  Above 145?
Backward: What score is at the 75th percentile?

Use a tableDetails Day 7 Written for Z:  N(0,1). Learn to read and use for Z first, then to use for a different mean and s.d.
Standard Normal table use.  Our tables give area to the left of a z value (Cumulative Proportions)
Using standard normal table:  See text p. 81-83. Table A: p.690-1. Table A (Excel)

Sketch the density, label axis, mark the area you're looking for.
Figure out how to get it using areas to the left of one or more z-values.
.Got to here Monday.
(BPS5e p83-86) Details Day 7
What z value has area ..... to the left/right of it?
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
All of these can be checked using the Applet: Normal Density Curve   http://bcs.whfreeman.com/bps5e

Plan for answering general question:  Proportion with scores between 100 and 145?   below 100?  Above 145?
Sketch, with raw and z ruler line. Standardize, finding z-score(s) for the raw edge-value(s).  Find area from table for Standard Normal.  That's your answer.
Preliminary work.  Sketch, mark area, raw and z ruler lines, standardize. Use Applet to find answer.  http://bcs.whfreeman.com/bps5e

Backward: What score is at the 75th percentile?
Sketch, with raw and z ruler line.  Mark approximately the area.  Find z-score cutting it off .  UNstandardize to get raw value.
Preliminary work.  Sketch, mark area, raw and z ruler lines. Use Applet to find answer, changing mu and sigma to get both raw and z answer.  http://bcs.whfreeman.com/bps5e

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