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## Statistical Analysis

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### Statistical Analysis Chapter 4 Normal Distribution What is the normal curve? In chapter 2 we talked about histograms and modes A normal distribution is when a set ... – PowerPoint PPT presentation

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Title: Statistical Analysis

1
Statistical Analysis Chapter 4 Normal
Distribution
2
What is the normal curve?
• In chapter 2 we talked about histograms and modes
• A normal distribution is when a set of values for
one variable, when displayed in a histogram (or
line graph) has one peak (mode) and looks like a
bell. Here is an example using height

3
Characteristics of the Normal Curve
1. Bell shaped, fading at the tails. In other
words, more values are in the middle, and odd or
unusual values fall at the tails
2. All (100) of the data fits on the curve, with
50 before the mean and 50 after
3. 68 of the data falls within -1 and 1 standard
deviations of the mean
4. 95 of the data falls between -2 and 2 standard
deviations
5. The percentage of data between any two points is
equal to the probability of randomly selecting a
value between the two points (remember classical
probability from Ch. 3)

4
Standard Deviations and Z-Score
• Z scores the number of standard deviations
away from the mean.
• z-score x - µ
• s
• (x data for which we want to know
the z-score)
• We use the characteristics of the normal curve,
and the z-score, to find out the probability of a
particular event or value occurring (remember
classical probability from Chapter 3)

5
Solving Normal Curve Problems Using Z-Scores
• (steps listed at bottom of p. 111)
• Draw a normal curve, showing values for (-2
through 2)
• Shade the area in question
• Calculate the z scores and cutoffs (percentages
• Use the z-scores and cutoffs to solve the normal
curve problem

6
Find Percentages on the Normal Curve Table
• Lets do these questions as a class
• What is the percentage of data from z 0 to z
0.1?
• What is the percentage of data from z 0 to z
2.16?
• What is the percentage of data from z -1.11 to
z 1.11?
• What is the percentage of data above z 1.24?
• What is the percentage of data below z -0.6?
• .039839.8
• .484648.46
• .3665 .3665 .73373.3
• .50 - .3925 .107510.75
• .50 - .2257 .274327.43

7
Working backwards from percentages
• When working backwards from percentages, we still
use the normal tablebut look for the percentage
to give us the z-score
• What is the z-score associated 10.2 of the data?
• What is the z-score(s) for the middle 30 of the
normal curve?
• What is the z-score of data in the upper 25 of
the normal curve?
• z 0.26
• z -.39 to z .39
• z 0.67

8
Lets do Question 4.2
• Use the normal curve table to determine the
percentage of data in the normal curve
• Between z 0 and z .82
• Above z 1.15
• Between z -1.09 and z .47
• Between z 1.53 and z 2.78
• Work backward in the normal curve table to solve
the following
• 32 of the data in the normal curve data can be
found between z 0 and z ?
• Find the z score associated with the lower 5 of
the data.
• Find the z scores associated with the middle 98
of the data.

9
• 29.39
• 12.51
• 54.29
• 6.03
• Between z 0 and z .92, or between z 0 and z
-.92

10
Question 4.7
• Use the normal curve table to determine the
percentage of data in the normal curve
• Between z 0 and z .38
• Above z -1.45
• Above z 1.45
• Between z .77 and z 1.92
• Between z -.25 and z 2.27
• Between z -1.63 and z -2.89
• Work backward in the normal curve table to solve
the following.
• 15 of the data in the normal curve can be found
between z 0 and z ?
• Find the z score associated with the upper 73.57
of the data.
• Find the z scores associated with the middle 95

11
• 14.80
• 92.65
• 7.35
• 19.32
• 58.71
• 4.97
• z .39 or -.39
• z -.63
• Between z -1.96 and z 1.96

12
Binomial Distributions and Sampling
• Binomial means two categories in a population
• Males and females
• Sports game players vs. Non sports game players
• Incomes over 40,000 vs. incomes under 40,000
• Quick note Rememberfor binomial distributions,
we would visualize this data through a pie
chartbecause we do not have enough categories
for a histogram

13
Sampling from a Two-Category Population
• With two-category populations, we can describe
the population by p the percentage of values in
one category
• This is the same p from the last chapter on
probability (classical probability)
• P(event) s (number of chances for
success)
• n (total equally likely
possibilities)
• We know (actually.statisticians know) that if we
randomly sampled from a population, then
• ps p

14
Sampling Distribution
• In order to know the odds of getting certain
values from this particular binomial sample, we
have to know the sampling distribution from this
population.
• Under certain conditions, the sampling
distribution for a binomial value is normal (i.e.
the distribution follows the normal curve).
• When the sampling distribution is normal, then we
can make predictions using our table and our
z-scores

15
Sampling from a Binomial Distribution
• Suppose, we defined a population (full time FIT
students who either shop at Hot Topic), and we
have made our measure of interest into a binomial
distribution those who shop at Hot Topic and
those who do not.
• Suppose over the last 10 years, marketers have
surveyed the FIT population hundreds of times and
found that Hot Topic shoppers are p .13. (those
who are non-Hot Topic shoppers is p .87)

16
Sampling from a Binomial Distribution
to lead another study. But this time, you dont
have enough money to survey the whole population,
and you have to get a sample.
• We can assume, because so many studies have been
done in the past that the true value of Hot Topic
shoppers is p .13. Thus, because we know that
ps p, your sample should have approximately the
same value.

17
Sampling from a Binomial Distribution
• For each sample, we can use the number sampled,
and the p value from the population to predict
the total number of Hot Topic shoppers. This is
called the expected value.
• Expected value np
• Thus, if we collected a sample of 200 FIT
students, how many students would we expect to be
Hot Topic shoppers?
• np (200)(.13) 26
• This expected value is the mean of your sample

18
Binomial Distribution and the Normal Curve
• Now, we need to decide if we can use the normal
curve to solve problems
• If (np) gt 5 and n(1 p)gt5then the sampling
distribution will be normally distributed.
• So, our sample was 200 students.
• Is (np) gt 5?
• Is n(1 p)gt5?
• Yesand yes.
• np (200)(.13) 26
• n(1 p) (200)(1 - .13) (200)(.87) 174

19
Binomial Distribution and the Normal Curve
• What do we mean that a sampling distribution is
normal?
• Just like someones age is one value among many
ages that we tally to make a histogram, we can
tally many samples, get the p values of those
sample, and construct histograms from these
means.
• If we took say, 1000 samples, and tallied the p
values for Hot Topic shoppers, then those values,
when turned into a histogram, should form a
normal curve. Just like if we took the heights
of a 1000 women, and tallied those values to get
a normal curve.

20
How to use the Binomial Distribution and the
Normal Curve
• Get the mean (µ)the mean is the expected value
(np)
• Get the standard deviation (s) vnp(1 p)
• Draw a normal curve using mean and standard dev
• Use the continuity correction factor, and add
/- half a unit to the value we want to solve for
• Get the z-scores x - µ
• s
• Use the normal curve table to solve the problem

21
Why the continuity correction factor?
• This is only for discrete values (where values
occupy only distinct points.) For example, in
our study, there is no such thing as a half or
3/4 Hot Topic shopper. Either you are a
shopper or not. Looking at how histograms are
presented, you can see why we have to use the
correction factor.
1. Probability of getting a value equal to or
greater than (gt), then you must subtract a
half-unit
2. Probability of getting a value equal to or lesser
than (lt), you must add a half unit.
3. Probability of getting the exact value, you must
get the Z-scores for a half-unit above and a
half-unit below

22
Now lets answer a Hot Topic Question
• If you collected a sample of 200 FIT students
• What is the probability that 13 will be Hot Topic
shoppers?
• What is the probability that you will have 30 or
more Hot Topic shoppers?
• What is the probability that you will have 25 or
less Hot Topic shoppers?

23
• Question
• What is the probability that 13 will be Hot Topic
shoppers?
• What is the probability that you will have 30 or
more Hot Topic shoppers?
• What is the probability that you will have 25 or
less Hot Topic shoppers?
• Get the mean (µ) expected value np
(200)(.13) 26
• Get the standard deviation (s) vnp(1 p)
v26(1 - .13) v26(.87) v22.62 4.76
• Draw a normal curve using mean and standard dev.
• Use the continuity correction factor to correct
x. (a) 12.5 and 13.5, (b) 29.5, (c) 25.5
• Get the z-scores. (a) -2.83 and -2.62, (b) .735,
(c)-.105
• Solve the problem (a) 4977 - .4956 .002, or
2 (b) .50 - .2704 .23, or 23, (c) .50 - .0596
.4404

24
Now lets do question 4.16 as a class
• In a marketing population of phone calls, 3
produced a sale. If this population proportion
(p 3) can be applied to future phone calls,
then out of 500 randomly monitored phone calls,
• How many would you expect to produce a sale?
• What is the probability of getting 11 to 14
sales?
• What is the probability of getting 12 or less
sales?
• 15
• 32.93
• 25.46

25