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Youth Forward Scholarship Winter 2018 – My Service To My Community

Name: Lesslie Hasman
From: El Cajon, California
School: Santana High School
Votes: 0 2.1A Measures of Relative Standing and Density; Z-scores

• _________________________measure a score as related to the other
scores in the data set (Measures the relative standing).

• _____________________________________________________(Z-score)

• Puts everything on the same scale in terms of standard deviation.

• Standardized values (z-score):

• z =

• Tells us how many ___________________________________________ the
original observation is from the mean, and in which direction:

• Observations larger than the mean are _________________________.

• Observations smaller than the mean are
__________________________.

• My husband and I argue about who is smarter….

• My SAT score: 1350

• N(1000, 200)

• His ACT score: 28

• N(18, 6)

• Standardizing Women’s Heights: N(64.5, 2.5)

1. A women 68 inches tall has a standardized height of what?

1. A women 58 inches tall has a standardized height of what?

2.1B Measures of Relative Standing and Density Curves; Percentiles

• We can standardize data by using z-scores, but we can also use
____________________

____________________________________________________________________________________________.

• We say that the pth percentile of a distribution is the value
where _________________ of the observations are less than or
equal to it.

• If I scored a 78 on my math test, which was better than 25 of my 28
class mates, then I am in the 25/28=89 percentile for the class.

• Example: Here are the scores of 25 math students in my class

 79 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 83 89 84 82 77 72

• If your score was 80 on the test, what percentile would you be in on
this test?

• Have you wondered if there is an easy way to convert Z-scores to
percentiles?

• The percent of observations falling at or below a particular Z-score
depends on the________________________ of the distribution.

• An observation that has a score equal to the mean has a Z-score of
0. In a heavily left skewed distribution where the mean is less
than the median, this observation will be below the 50th
percentile (the median).

2.1C Measures of Relative
Standing and Density Curves; Density Curves

Density Curves

• Sometimes the overall pattern of a distribution is such that we can
describe it with a __________________________________. It is
remarkable how many natural phenomena appear to be related to
a_________________________________________________

__________________________________________________. When appropriate,
using a normal distribution model to represent distributions that
occur in real-life situations can be extremely useful in statistical
analysis.

• __________________________________________ – a mathematical model
for the distribution

• Displays the overall pattern (____________________) of a
distribution, but is an idealized description, ignoring minor
____________________________________________.

• Has an area of exactly
_________________________________ underneath it.

• Is on or above the
______________________________________.

• The area under the curve and above any range of values is the
proportion of all observations that fall in that range.

• Types

• Normal curve (symmetric normal density curve)

• Right-skewed curve

• Left-skewed curve

The median and mean of a density curve

• ____________________________________________________________

• The point that divides the area under the curve into
_______________________.

• _________________________________________________: half the area
under the curve to its left and the remaining half of the area to
its right.

• ____________________________________________________________

• The “balance point” of the curve (think of a
teeter-totter).

• The point at which the curve would balance if made of solid
material.

• ____________________________________________________________________________________________

• The median and mean are the same for a symmetric density curve.
They both lie at the ____________________________ of the curve.

• The mean of a skewed curve is pulled _________________________ from
the median in the direction of the
_________________________________________.

Normal Distributions

• ________________________________________

• Density curves that are _________________________,
______________________________, and
________________________________________.

• Describe normal distributions.

• Described by its ___________________
and ________________________________________
.

• is
located at the ____________________ of the symmetric curve and is
the _____________________ as median.

• controls
the __________________________ of a normal curve.

• A curve with a larger standard deviation is
____________________________________________.

• Locating
by eye:

1. Imagine you are skiing down a mountain that has the shape of a
______________________________________.

2. At first the descent is _______________________________.

3. Then the slope begins to grow ________________________ as you go out
and down.

4. You can feel the change as you run a pencil down a normal curve.

• The points at which this change of curvature takes place are
called
__________________ _________________________
and are located at distance

on either side of the mean .

• Why are the normal distributions important in statistics?

1. Normal distributions are good descriptions for some distributions of
________________________________.

(i.e. SAT exam scores, psychological test scores, lengths of
cockroaches).

1. Normal distributions are good ____________________________ to the
results of many kinds of __________________________________________
(i.e. tossing a coin many times).

2. Many _______________________________________ procedures based
on normal distributions work well for other
__________________________________ distributions.

The _________________________________ rule

• In a normal distribution with mean
and standard deviation
:

• _____________
of the observations fall within __________ of
.

• _____________ of the observations fall within __________ of
.

• _____________ of the observations fall within ___________
of
.

• A short notation for the normal distribution
with mean

and standard deviation
:

• N(,)

• For example: N(64.5, 2.5)

2.2B Normal Distribution; Standard Normal Distributions

Standard
Normal Distribution:

The
standard normal distribution is the normal distribution with
______________ where ________________and ________________.

• z = Standardized
Normal Distribution Normal Distribution Calculations:

• An area under a density curve is
a_____________________________________________________

________________________________________________________________.
Any question about what proportion of observations lie in some range
of values can be answered by finding an area under the curve.

• Table A is a table of areas under the standard normal curve.
The table entry for each value z is the area under the curve
to the left of z.

z-score chart (go to closest #.
vs. <? no difference) – Draw sketches

1. z -2.25

1. z -2.25

1. z > 1.77

1. –2.25 z
1.77

• I wonder what percentile I lie in for my SAT scores:

• 68-95-99.7 rule will only give percentiles for 1, 2, or 3
standard deviations from the mean, but I am 1.15 standard
deviations above the mean.

• Finding normal proportions

Step 1: State
the problem in terms of the observed variable
x.
____________________

_____________________ of the distribution and
_________________ the area of interest under the curve.

Step 2:
______________________________ to restate the problem in terms of a
standard normal variable
z.
Draw a picture to show the area of interest under the standard
normal curve.

Step 3: Find the area under the standard normal curve, using
_____________________ and the fact that the total area under the
curve is __________.

Step 4: Write your conclusion in the context of the problem.

• So, what is the area below 1.75 standard deviations?

• More Practice

1) What percent of people score less than 800 on the SAT if
N(1000, 200)?

2) What percent of people score more than 1400 on the SAT?

3) What percent of people score between 900 & 1350 on the
SAT?

Finding a value given a _________________________

• We may want to find the observed value with a given proportion of
the observations above or below it. To do this,
__________________________________________.

• Find the given proportion in the _________________ of the table.

• Read the corresponding z from the left column and top row.

• “_____________________________________” to get the observed
value.

• Scores on the SAT verbal test in recent years follow approximately
the N(505, 110) distribution. How high must a student score
in order to place in the top 10% of all students taking the SAT?

1. ____________________________________________. We want to find
the SAT score x with are 0.1 to its right under the
normal curve with mean = 505 and
standard deviation = 110. That’s
the same as finding the SAT score x with area 0.9 to its
left.

2. _____________________________________. Look in the body of
Table A for the entry closest to 0.9. It is 0.8997. This is the
entry corresponding to z = 1.28. So z = 1.28 is the
standardized value with area 0.9 to its left.

3. ______________________________________ to transform the
solution from the z back to the original x scale.

Solving this equation for x gives:

This equation should make sense: it says that x lies
___________________ above the mean on this particular normal curve.
That is the “unstandardized” meaning of z
=________________

A student must score at least _________________ on the SAT verbal
test to place in the highest 10% of students who take the test.

• The general formula for unstandardizing a z-score:

• N(1000, 200) What score do I need to be in the top 5% so I
can get into Harvard?

I
would need to score at least ______________________ to place in the
top 5% of students who take the SAT.

______________________________________________

• In the latter part of this course we will want to invoke various
tests of significance to try to answer questions that are important
to us.

• These tests involve sampling people or objects and inspecting them
carefully to gain insights into the populations from which they
come.

• Many of these procedures are based on the assumption that the host
population is approximately normally distributed.

• Consequently, we need to develop methods for assessing normality.

Step
1:
Construct a ________________________________________________
or a ______________________________. See if the graph is

Step
2:
Use the __________________________________ to check for
normality.

Step
3:
Construct a
_________________________________________________________.
A normal probability plot provides a good assessment of the adequacy
of the normal model for a set of data. Use statistics utilities like
Minitab, or a TI-83 or TI-84 calculator to construct normal
probability plots.

If
the data distribution is close to a normal distribution, the plotted
points will lie close to a ___________________________________.
Conversely, nonnormal data will show a
________________________________ trend. Outliers appear as points
that are far away from the overall pattern of the plot.

Any
normal distribution produces a
________________________________________ on the plot because
standardizing is a transformation that can change the slope and
intercept of the line in our plot but cannot change a line into a
curved pattern.