Download Chapter 2 AP Stats notes Density Curves Always plot data (graph

Chapter 2 AP Stats notes
Density Curves
Always plot data (graph, histogram, stem plot, etc)
Look at SOCS (shape, outliers, center, spread)
Calculate summary (5 number summary)
If overall patter in regular (normal or symmetric) it can be described as a smooth curve.
Smooth curve is a mathematical model
Density curve is any curve where the area under the curve = 1.
(Normal Density Curve)
Area under the curve is = 1
Area under curve = 1
This is a density curve as long as the area under the curve = 1
Median of a density curve is the equal-areas point. (50% on each side)
Median
Mean
Mean of a density curve is the balance point
X = mean computed from actual data
S = standard deviation computed from actual data
Idealized distribution (smooth curve)
μ (mu) = mean
σ (sigma) =standard deviation
Homework
Read pages 78-83 do problems 1 – 4
Median Mean
Normal distributions
Normal curves are called normal distributions
Bigger spread, bigger σ
σ
μ
Smaller spread, smaller σ
σ
μ
inflection point (where the curvature of the graph changes)
σ
σ
μ
Inflection points – 1 standard deviation away from the mean
Empirical rule 68-95-99.7 rule
**Need to know this all year**
(put percents in each area)
N(30,5) we can draw a picture for this. N represents normal, 30 represents μ and 5 is σ
15
20
25
30
35
40
45
What percent is the right of 35?
What percent is between 20 and 35?
What percent is to the left of 20?
What percent is below 40?
Homework read pages 85-90 do 6-9, 11-15
Standard Normal calculations
To compare distributions, we must have them all on the same scale.
X = an observation
μ = mean
σ = standard deviation
z-score tells how many standard deviations the original observation falls from the mean
ex. Height of young women in US
μ = 64.5 inches σ = 2.5 inches
z=
I am 69 inches tall, find how many standard deviations I am from the mean
Z=
z=
z = 1.8
I am 1.8 standard deviations above the mean for height of women in the US
Standard normal distribution
N(0,1)
Mean = 0 Standard deviation = 1
Normal distribution calculations
Standard normal table (Table A FRONT OF BOOK)
The table gives the area under the curve to the left of the line (or our standard deviation)
What percent fall below my 69 inches?
.
.9641
0
1.8
96% of the women in the US are shorter than me?
What percent is taller than me?
.9641
1-.9641 = .0359
Area under the curve is = 1 this means that area to left of line and area to right of line add
together to equal 1.
3.59% of women in the US are taller than me.
Finding normal proportions
1. State problem **Draw a picture and shade area of interest**
2. Standardize (Find z-score)
3. Find required area
4. Write conclusion in context of problem. ***This is extremely important***
Using Table A backwards
N(505, 110)
How high must a student score to be in the top 10% on the SAT?
.10
.90
Find z from the table that corresponds with .9
(find .9 inside table)
z-score is 1.28
Now work equation backwards
Z=
1.28(110) = x – 505
1.28 =
140.8 = x – 505
+505
+ 505
X = 645.8
A student must get higher than 645.8 on the SAT to be in the top 10%.
Homework read pages 93-101 do problems 19-24
Tuesday
Assessing Normality
Many statistical procedures are based on the assumption that the data is approximately normal
Checking for normality
1. Construct histogram or stem plot to see if approximately bell-shaped
2. Check empirical rule (68-95-99.7)
Find mean and standard deviation
Check to see if approximately 68% fall between 1 standard deviation from the mean
Check to see if approximately 95% fall between 2 standard deviations from the mean
Check to see if approximately 99.7% fall between 3 standard deviations from the mean
3. Normal probability plot
Do this on the calculator (page 105)
When we graph the normal probability plot – if points are close to a straight line than data is
distributed approximately normal
If not linear (straight line) than distribution is not approximately normal
Outliers will be points far away from the overall pattern
Homework read page 112 do 40-42, 44-48