Math151 Day 8

Math 151 , Spring 2002, W Feb. 13, Day 8 Hit reload to get most current versionFinal version, after class

Homework questions:
Median, mean, percentiles, standard deviation are defined for a density curve in analogy to those for a histogram.
-- Median has half of area below and half above. First quartile has 1/4 of area below, 3/4 above. Etc.
-- Mean is balance point. On the long-tail side of median if distribution is skewed. Same as median if symmetric.
Many densities have tables to describe them, showing area to the left of (below) a given value. Or to the right of (above) a given value. Tables for the simple distributions on the handout are similar to the normal distribution table .

"Normal" distributions:("Gaussian", "Bell-shaped") part 1 (pp. 51-5, 57-8)

What "mechanism"? Thing measured is the result of many small independent influences.
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.
Symmetric. Mean at middle. Curvature changes at 1 standard deviation on either side of mean.
"Standard normal": mean = 0, s.d. = 1
"Middle s. d." A center region one standard deviation wide spans about 38%
68-95-99.7 rule: 68% within 1 s.d. of mean, 95% within 2 s.d. of mean, 99.7% within 3 (roughly).

These are back-of-the-envelope tools. Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ First standard normal table use, then standardizing~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal table use. Our tables give area to the left of a z value.
    Sketch the density, mark the area you're looking for.
    Figure out how to get it using areas to the left of one or more z-values.
        Think cutting up paper bell-curves. (Remember whole area is 1.)
Check with Activstats Normal Tool.

Example: Proportion of observations between 0.5 and 1.4 P(0.5 < z <1.4) =
Proportion of observations below 1.4 minus Proportion of observations below 0.5
P (z < 1.4) - P(z < 0.5) = .9192 - .6915 = .2277

. bell curves. Use 202x515 pixels to print.

Example: Proportion of observations above 0.5,    P( z > 0.5) =
                ONE minus proportion of observations below 0.5,   1 - P( z < 0.5)= 1-.6915 = .3085
.
What is the proportion of observations between -1.25 and + 1.25?
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We'll start here Friday, Day 9
Reading table backward:. What z value has area ..... to the left/right of it? See text p. 61-2
        Sketch roughly.
        Restate (if needed) as "What z value has area A to the LEFT of it."
        Look in body of table for the value closest to A.
        Go to edge(s) of table to find what z that goes with.
Example: What z value has 10% of the observations above it? This is the same z as the one for:
        What z value has 90% of the observations below (to the left of) it.

        Find in the table .8997 and .9015 -- .9000, our number, is between them.
                    .8997 is a little closer to.9000, so use it.
           For .8997, the z value is 1.28.
What z value has area .16 above it?

PreClass assignment Day 8 for Day 9

Review standardizing, (ActivStats 4-4--last activity--"standardizing, " 5-3)

HW assignment, Day 8, Wed. Feb 13
Moore: From David S. Moore, The Basic Practice of Statistics
Reading: Read table use pp.57-8, 61-2. Read ahead, rest of sec. 1.3

Hand in (All problems are from Moore)
p. 64 1.61 eyeball sigma
p. 54 1.53&54 Normal, men's hts--68-95-99.7 rule.
p. 64 1.63 pregnancies--68etc rule
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table use: Always sketch the normal curve first, mark the z-values, shade the area you are looking for!
A)Use the table in the back of the book, and answer the same questions as you did in HWCh5 ACT-3(Normal Densities). Check your answers.
(You can check your answers below using theACT normal tool if you like)
p.61 1.57 z's ..
The following will be part of Friday, Day 9 HW
"Backward"Always sketch a normal curve first, roughly mark the proportion=area you are given.
p. 62, 1.59 (backward z)

Read, to discuss

Optional (more practice)

1.55 wechsler ais, 68etc rule
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p. 65 1.65 z's

"Backward" (with Day 9HW)
p. 65, 1.66 (backward z)

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