Chapter Two Highlights
1.
Density Curve a.
Area Under Curve = 1 b.
Curve is always above the horizontal axis
2.
Normal Distribution a.
Symmetric b.
Single Peaked c.
Bell-shaped d.
Empirical Rule (68%-95%-99.7%) e.
z-score: number of standard deviations from the mean f.
z-score: also called standardized score 𝑥−𝜇 g.
𝑧 = 𝜎
3.
Determining Approximate Normality a.
Plot histogram or boxplot b.
Normal Probability Plot i.
Last choice on statplot menu ii.
Data Axis: x iii.
Zoom Nine iv.
If the the Normal Probability Plot appears approximately linear →~Normal
4.
Calculator Functions a.
To Find probability / percentage/ area i.
ncdf(low, high, 𝜇, 𝜎 ) ii.
-1E99 used for -∞ iii.
1E99 used for +∞ b.
If Given probability / percentage / area: to Find z-score i.
invnorm(% as a decimal)

Density curve:
 Displays the overall pattern (shape) of a distribution.


Has an area of exactly 1 sq. unit underneath it.
 Is on or above the horizontal axis.
Looks Like:
Example Problems:
Or

Density Curves and Relative Position of Mean and Median:
Median: Equal area point (half area to left / half area to the right
Mean: Balancing Point (think teeter-totter)
Rescue Run Results: Male 45 to 49
Place Name City
1
Over Chip Gun
Bib No Age all Time Time Pace
Saugatuck MI 1693 49 20 20:33 20:35 6:38/M
2
3
Muskegon MI 1686 47 23 21:05 21:08 6:48/M
Holland MI 795 48 25 21:24 21:29 6:54/M
4 Brian
5 Chris
Zeeland MI 1697 48 42 23:46 25:50 7:40/M
Hudsonville MI 1651 48 56 24:36 24:43 7:56/M
6 Scott
7 Phil
Holland MI 1700 45 58 24:49 25:02 8:00/M
Holland MI 1604 45 71 25:53 26:10 8:21/M
8 Dan
9 Curt
10 Kurt
11 Dean
Zeeland MI 1649 46 74 26:20 26:26 8:30/M
Holland MI 1645 47 89 27:42 27:57 8:56/M
Holland MI 766 46 117 29:40 29:54 9:34/M
Holland MI 1632 45 119 29:47 30:20 9:36/M
12 Michael Holland MI 1619 49 127 30:27 30:52 9:49/M
13 Tim Kindred Zeeland MI 763 47 148 34:16 34:38 11:03/M
14 Ken 724 49 197 50:00 50:24 16:08/M a.
Make a histogram of the data b.
Calculate Mean and Median c.
How is the shape of the distribution of times related to the mean and median you calculated?

The Normal Distribution: a.
Picture: b.
Notation: N (µ , σ ) c.
Features i.
Symmetric ii.
Bell-Shaped iii.
Area Under Curve = 1 d. Real-Life Data is often Approximately Normal
Examples: _______ , ________, _________ , __________
Empirical Rule / 68-95-99.7 Rule:
Fill-In: The middle ____ % of data are within 1 standard deviation of the _____
Helpful Diagram:
The middle ____ % of data are within___ standard deviation of the _____
The middle ____ % of data are within ____ standard deviation of the _____

Empirical Rule Examples:
What if our data does lie right on a 1, 2, or 3 standard deviations from the mean?
THE FAMOUS z-SCORE: 1. z =
−
or in words z =
−
Example Using z-scores to rank data:
2. Converts a _______ value into the number of ____________ from the ______
3. Also called ___________ value

Normal Curve Calculation Examples:

Determining Approximate Normality: The Normal Probability Plot a.
Plot histogram or boxplot b.
Normal Probability Plot i.
Last choice on statplot menu ii.
Data Axis: x iii.
Zoom Nine iv.
If the the Normal Probability Plot appears approximately linear →~Normal
Unemployment Data (2009, USA):
4.1% = N. Dakota
14.7% =
Determine if the following data is ~Normal:
What does a NPP of skewed data look like:
# of Children 1
Frequency 7
2
15
3
5
4
6
5
2
Entering the data for this set is a little tricky. Methods:
6
1
Construct a DOTPLOT
Construct NPP:
1
Compare dotplot with the NPP.
2 3 4 5
7
1
1.
2.
6
# of children in a family
8
1
7
9
1
8
10
1
9 10

Review & Challenges:
2.
Acme Co. makes fasteners designed to keep an AP stats student’s cell phone in her pocket during class. The strength (lbs.) of the fasteners is modeled by the following: N( 80, 5 ) a.
Draw the normal distribution modeling this situation. Identify the mean and the location of the standard deviation. b.
Approximately what percent of the fasteners would fail below 88 lbs.? c.
What Percent of the fasteners would fail less than 75 lbs.?

3.
The height of AP stats students can be considered approximately normal with the following distribution:
N( 67”,2.75”). a.
What proportion of AP staters should be less than 65 inches tall? b.
c.
At least how tall are the top 15% of AP staters?
What are the heights of the middle 90% of AP staters?
4.
Learning to use the TI-84 calculator is crucial for AP stat students. Unfortunately only 25% of staters are able to use the calculator after 3 months, but 90% are proficient after 8 months (WHEW). Assuming calculator proficiency is normally distributed; determine the mean and standard deviation for this model.
5.
Scores on the ACT are normally distributed with a mean of 18 and a standard deviation of 6. Determine the IQR. z-score for 25% = _____ (invnorm!!!!!) z-score for 75%=_________
IQR = ____ - _____ = ______
6.
The test grades for math majors at Euclid Univ. have an approximately normal distribution with a mean of 50.
What is the standard deviation so that 80% of the students are within 12 points (above and below) the mean. z-score for lower 10% = _______ z-score for highest ____ % = _____

7.
The average yearly snowfall in Kindredia is normally distributed with a mean of 75”. If the snowfall of Iceberg exceeds 85” in 5% of the years, what is the standard deviation?
8.
The upper and lower deciles of any distribution are the points that mark off the lowest 10% and highest 10%. a.
What are the lower and upper deciles of the standard normal distribution? (Like asking for the z-scores) b.
The length of human pregnancies is approximately normal with a mean of 266 days and standard deviation of 16 days. What pregnancy lengths are at the lower and upper deciles of this distribution?
9.
The amount of time an HHS student spends on homework each evening is approximately normally distributed.
If 22% of 1200 students spend at least 47 minutes doing homework and Conbo, who spends 65 minutes has a standardized (z) score of 1.932, determine: a.
Mean b.
Standard Deviation c.
Approximately what numbers of HHS students do less than 15 minutes of homework each evening?
10.
Mrs. Kindred teaches the same stats course, once in the morning and once in the afternoon. Over a period of time, the average and standard deviation in the final exam are computed. Assume normality.
Class Time Mean Standard Deviation
Morning
Afternoon
81.23
74.97
4.04
5.68 a.
Carinna took the morning course and got a 91, while Carielle took the afternoon course and got an
89. Whose grade was higher in relation to everyone in the same time slot? Show work. b.
If Dunkie got a 75 in the morning course’s final, determine what grade he would need in the afternoon course to have the same relative grade. Show work.

11.
The length of time Ty’s Iphone lasts between charges is described by a normal distribution 3.6 hours of continuous use with standard deviation of .45 hours. a.
What percentage of time would the iphone last more than 4.75 hours? b.
c.
d.
e.
f.
g.
What percentage of time would the iphone last less than 3 hours?
What percentage of time would the phone last between 2 to 4 hours?
How many hours of use would be in the top 10 percentile?
Q1 = _________
Q3 = _________
Outliers Discussion? Q1-1.5IQR???? Q3+1.5IQR???????