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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
    asked for)
  • 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?
  • Answers
  • .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?
  • Answers
  • 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
Question 4.2 Answers
  • Answers to Question 4.2
  • 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
Question 4.7 Answers
  • 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
  • But suppose sometime later, your manager asks you
    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?
  • Answer
  • 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
Question 4.16 answers
  1. Expected value np 500(.03) 15
  2. 32.93
  3. 25.46
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