Download Chapter 6 Notes

Chapter 6 Notes: The Normal Curve
A Density Curve is a curve that:
*is always on or above the horizontal axis
*has an area exactly 1 underneath it.
*describes the overall pattern of a distribution
**How will you know it is normal?? You will be told if it is normal for now
The area under the curve and above any range of values is the proportion of all
observations that fall in that range.
Density curves are mathematical models. They are an “idealized description of a data set”
not an actual set of data.
The median is the 50th percentile. Exactly 50% of the data lies on each side of the median.
The population mean ( - lower case Greek mu) is the point at which the curve would
balance if made of solid material.
The mean and the median are the same for a symmetric distribution. For non-symmetric
dist. remember that a tail pulls the mean toward it.
The population standard deviation ( - lower case Greek sigma) controls the spread about
.
Normal Curves are important density curves that are symmetric, single-peaked, and bellshaped and they describe normal distributions.
On a normal curve,  can be roughly located by eye – It is where the change in curvature
(inflection points) takes place on either side of .
Notation:
Normal Curve:
(Parameters of the population)
Big Concepts*Symmetry
*Area underneath is 1
The Empirical Rule:
68% of the observations fall within 1 of .
95% of the observations fall within 2 of .
99.7% of the observations fall within 3 of .
** Show example 1…………………………
** The Empirical rule is great unless we want to know something that is not on the curve at
a specific standard deviation. What do we do then??
A Normal Distribution can be standardized so that area at any value (not just the empirical
rule values) can be found easily. The letter z is used for standardized observations on a
normal curve, thus we call them z-scores.
z = x-

Using this method, we standardize any normal distribution so that we can use the areas
from the standard normal distribution where the mean is 0 and the standard deviation is
1, or N(0,1).
Normal distribution notation:
N(,)
so… for a normal distribution whose mean is 500 and standard deviation is 100, we would
note that as
N(500, 100)
** Show example 2…………………………………….
Normal distributions are common for three reasons:
1) Some real data has a normal distribution.
2) They approximate many chance outcomes in probability.
3) Many symmetric distributions that are large can be approximated using normal
dist.
However… Not all data follows a normal distribution, in fact many do not. Don’t assume
data to be normal. See assessing normality later.
Percentiles: The “pth” percentile has p% that lies to the left and 100% - p% that lies to the
right.
**So, to get credit, you must show the following:
Finding Normal Proportions:
1) State the problem (as an inequality)
2) Standardize
3) Draw a picture
4) Use the table.
Examples:
# 1. IQ is normally distributed N(100, 16). Based on this distribution, answer the following
using the Empirical Rule:
A) What proportion of scores falls above 68?
B) What proportion of scores falls below 116?
C) What proportion of scores falls between 84 and 116?
D) What proportion of scores falls between 116 and 132?
E) What proportion of scores fall above 148?
# 2. IQ is normally distributed N(100, 16).
Based on this distribution, answer the following using z-scores:
A) What proportion of scores fall below 110?
B) What proportion of scores fall below 80?
C) What proportion of scores fall above 120?
D) What proportion of scores fall above 90?
E) What proportion of scores fall between 95 and 105?
F) What proportion of scores fall between 90 and 120?
G) What score represents the 80th percentile?
H) What score represents the 95th percentile?
# 3. Jordan and Kelsey are very good friends but highly competitive when academics are
concerned. Jordan took the SAT which is normally distributed where each section has a
mean of 500 and a standard deviation of 100. She scored a 720 on her math section. Kelsey
took the ACT which is normally distributed where each section has a mean of 18 and a
standard deviation of 6. She scored a 32 on her math section. Who made the better score
relative to the test they took?
# 4. Cholesterol levels of adult American women can be described by a Normal model with a
mean of 188 mg/dL and a standard deviation of 24 mg/dL.
A) What percent of adult American women do you expect to have cholesterol levels over 200
mg/dL?
B) What percent of adult American women do you expect to have cholesterol levels between
150 and 170 mg/dL?
C) Estimate the interquartile range of the cholesterol levels.
D) Above what value are the highest 15% of adult American women’s cholesterol levels?
# 5. While only 5% of babies have learned to walk by the age of 10 months, 75% are
walking by 13 months of age. If the age at which babies develop the ability to walk can be
described by a normal model, find the population parameters (the mean and standard
deviation).
# 6. Adult female height is normally distributed with a mean of 65 inches and a standard
deviation of 3.2 inches. What would this distribution be in feet?
# 7. Jim will enter a triathlon next week. His times in each event are normally distributed.
He can do the biking portion with a mean of 35 minutes and a standard deviation of 7
minutes, the running portion with a mean of 52 minutes and a standard deviation of 12
minutes, and the swimming portion with a mean of 22 minutes and a standard deviation of
5 minutes. Assuming the three portions are independent, what is the mean and standard
deviation for the entire triathlon?
# 8. Bicycles arrive at the bike shop in boxes. Before they can be sold, they must be
unpacked, assembled, and tuned (lubricated, adjusted, etc…). Based on past experience,
the shop manager makes the following assumptions about how long this will take:
*the times for each set up phase are independent
*the times for each phase follow a Normal model
*the means and standard deviations of the times (in minutes) are as shown:
Phase
Unpacking
Assembly
Tuning
Mean
3.5
21.8
12.3
SD
0.7
2.4
2.7
A) What are the mean and standard deviation for the total bicycle setup time?
B) A customer decides to buy a bike like one of the display models but wants a different
color. The shop has one, still in the box. The manager says they can have it ready in half an
hour. Do you think the bike will be set up and ready to go as promised? Explain.