Warm-Up
• Grab a sheet of multiple choice questions and work on those!
Answers to warm-up (C C B D E)
Homework Questions?
Computer Outputs
Finish Section 2.1
Density Curves
1. Always plot your data: dotplot, stemplot, histogram…
2. Look for overall pattern (SOCS)
3. Calculate numerical summary to describe
Add a step…
4. Sometimes the overall pattern is so regular we can describe
it by a smooth curve.
A Density Curve is a curve that:
• Is always on or above the horizontal axis
• Has an area of 1 underneath it
Things to note…
• The mean is the physical balance point of a density curve or a
histogram
• The median is where the areas on both sides of it are equal
Don’t forget…
• The mean and the median are equal for symmetric density
curves!
Start Section 2.2
Normal Curves!
Normal Distributions = Normal Curves
• Any particular Normal distribution is completely specified by
two numbers: its mean (𝜇) and standard deviation (𝜎).
• We abbreviate the Normal distribution with mean 𝜇 and
standard deviation 𝜎 as N(𝝁, 𝝈)
The 68-95-99.7 Rule
Other images to explain the
same thing…in case it helps!
Example
•
A pair of running shoes lasts on average 450 miles, with a
standard deviation of 50 miles. Use the 68-95-99.7 rule to
find the probability that a new pair of running shoes will
have the following lifespans.
Between 400-500 miles
More than 550 miles
Warm-Up (09-26-13)
On the driving range, Tiger Woods practices his golf swing with a
particular club by hitting many, many golf balls. When Tiger hits his
driver, the distance the ball travels follows a Normal distribution with
mean 304 yards and standard deviation 8 yards. What percent of
Tiger’s drives travel at least 288 yards?
What percent of Tiger’s drives travel between 296 and 320 yards?
Example
• The ACT and SAT are both Normally distributed with a mean of 18
and 1500 (respectively) and standard deviation of 6 and 300 (again
respectively). Using this information find the following:
a.
Percentage of scores that are above a 24 on the ACT.
b.
Percentile for a 2100 on the SAT.
c.
Percentage of ACT scores that are between 24 and 30.
d.
Percentage of SAT scores that are between 900 and 1800.
Example
• A vegetable distributor knows that during the month of August, the
weights of its tomatoes were normally distributed with a mean of
0.61 pound and a standard deviation of 0.15 pound.
a. What percent of the tomatoes weighed less than 0.76 pound?
b.
In a shipment of 6000 tomatoes, how many tomatoes can be
expected to weigh more than 0.31 pound?
c.
In a shipment of 4500 tomatoes, how many tomatoes can be
expected to weigh between 0.31 and 0.91 pound?
How does this relate to z-scores?
• If the distribution happens to be normal we can find the area
under the curve and percentiles by using z-scores!
• Standard Normal distribution – mean of 0 and standard
deviation of 1.
•𝒛 =
𝒙−𝝁
𝝈
Table A in your book…
• The table entry for each z-score is the area under the curve to
the left of z.
• If we wanted the area to the right, we would have to subtract
from 1 or 100%
Examples
• Find the proportion of observations that are less than 0.81
• Find the proportion of observations that are greater than -1.78
• Find the proportion of observations that are less than 2.005
• Find the proportion of observations that are greater than 1.53
• Find the proportion of observations that are between -1.25 and 0.81
Homework
• Page 107: 19-38 (Density Curves)
• Page 131: 41-54 (Normal Curves and z-scores)