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?

2. 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?

• What about if your score was 72?

• What about if your score was 89?

• 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 50^th
  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
    _________________________________________.

2.2A
Normal Distributions

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).

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

3. 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

2. z  -2.25

3. z > 1.77

4. –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?

2.2C
Normal Distributions; Standard Normal Curve and Assessing Normality

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
approximately normal about the mean.

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.