Title: The Normal Distributions
1Chapter 3
2Density Curves
- Here is a histogram of vocabulary scores of n
947 seventh graders - The smooth curve drawn over the histogram is a
mathematical model which represents the density
function of the distribution
3Density Curves
- The shaded bars on this histogram corresponds to
the scores that are less than 6.0 - This area represents is 30.3 of the total area
of the histogram and is equal to the percentage
in that range
4Area Under the Curve (AUC)
- This figure shades area under the curve (AUC)
corresponding to scores less than 6 - This also corresponds to the proportion in that
range AUC proportion in that range
5Density Curves
6Mean and Median of Density Curve
7Normal Density Curves
- Normal density curves are a family of bell-shaped
curves - The mean of the density is denoted µ (mu)
- The standard deviation is denoted s (sigma)
8The Normal Distribution
- Mean µ defines the center of the curve
- Standard deviation s defines the spread
- Notation is N(µ,?).
9Practice Drawing Curves!
- The Normal curve is symmetrical around µ
- It has infections (blue arrows) at µ s
10The 68-95-99.7 Rule
- 68 of AUC within µ 1s
- 95 fall within µ 2s
- 99.7 within µ 3s
- Memorize!
This rule applies only to Normal curves
11Application of 68-95-99.7 rule
- Male height has a Normal distribution with µ
70.0 inches and s 2.8 inches - Notation Let X male height X N(µ 70, s
2.8) - 68-95-99.7 rule
- 68 in µ ? ? 70.0 ? 2.8 67.2 to 72.8
- 95 in µ ? 2? 70.0 ? 2(2.8) 64.4 to 75.6
- 99.7 in µ ? 3? 70.0 ? 3(2.8) 61.6 to 78.4
12Application 68-95-99.7 Rule
- What proportion of men are less than 72.8 inches
tall? - µ s 70 2.8 72.8 (i.e., 72.8 is one s
above µ)
Therefore, 84 of men are less than 72.8 tall.
13Finding Normal proportions
- What proportion of men are less than 68 tall?
This is equal to the AUC to the left of 68 on
XN(70,2.8)
To answer this question, first determine the
z-score for a value of 68 from XN(70,2.8)
14Z score
- The z-score tells you how many standard deviation
the value falls below (negative z score) or above
(positive z score) mean µ - The z-score of 68 when XN(70,2.8) is
-
Thus, 68 is 0.71 standard deviations below µ.
15Example z score and associate value
-0.71 0 (z values)
16Standard Normal Table
Use Table A to determine the cumulative
proportion associated with the z score
See pp. 79 83 in your text!
17Normal Cumulative Proportions (Table A)
z .00 .02
?0.8 .2119 .2090 .2061
.2420 .2358
?0.6 .2743 .2709 .2676
.01
?0.7
.2389
Thus, a z score of -0.71 has a cumulative
proportion of .2389
18Normal proportions
The proportion of mean less than 68 tall
(z-score -0.71 is .2389
.2389
19Area to the right (greater than)
Since the total AUC 1 AUC to the right 1
AUC to left Example What of men are greater
than 68 tall?
1?.2389 .7611
.2389
20Normal proportions
The key to calculating Normal proportions is to
match the area you want with the areas that
represent cumulative proportions. If you make a
sketch of the area you want, you will almost
never go wrong. Find areas for cumulative
proportions from Table A (p. 79) Follow the
method in the picture (see pp. 79 80) to
determine areas in right tails and between two
points
21Finding Normal values
- We just covered finding proportions for Normal
variables. At other times, we may know the
proportion and need to find the Normal value. - Method for finding a Normal value
- 1. State the problem
- 2. Sketch the curve
- 3. Use Table A to look up the proportion
z-score - 4. Unstandardize the z-score with this formula
22State the Problem Sketch Curve
Problem How tall must a man be to be taller than
10 of men in the population? (This is the same
as asking how tall he has to be to be shorter
than 90 of men.) Recall XN(70, 2.8)
23Table AFind z score for cumulative proportion
.10
z .07 .09
?1.3 .0853 .0838 .0823
.1020 .0985
?1.1 .1210 .1190 .1170
.08
?1.2
.1003
zcum_proportion z.1003 -1.28
24Visual Relationship Between Cumulative proportion
and z-score
-1.28 0 (Z value)
25Unstandardize
- x µ zs 70 (-1.28 )(2.8) 70
(?3.58) 66.42 - Conclude A man would have to be less than 66.42
inches tall to place him in the lowest 10 of
heights