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Statistical Analysis – Chapter 4 Normal Distribution . What is the normal curve?. In chapter 2 we talked about histograms and modes

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what is the normal curve
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:
characteristics of the normal curve
Characteristics of the Normal Curve
  • Bell shaped, fading at the tails. In other words, more values are in the middle, and odd or unusual values fall at the tails
  • All (100%) of the data fits on the curve, with 50% before the mean and 50% after
  • 68% of the data falls within -1 and +1 standard deviations of the mean
  • 95% of the data falls between -2 and +2 standard deviations
  • 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)
standard deviations and z score
Standard Deviations and Z-Score
  • Z – scores = the number of standard deviations away from the mean.
  • z-score = x - µ

σ

(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)
solving normal curve problems using z scores
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
find percentages on the normal curve table
Find Percentages on the Normal Curve Table

Let’s 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

  • .0398…39.8%
  • .4846…48.46%
  • .3665 + .3665 = .733…73.3%
  • .50 - .3925 = .1075…10.75%
  • .50 - .2257 = .2743…27.43%
working backwards from percentages
Working backwards from percentages…
  • When working backwards from percentages, we still use the normal table…but 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
let s do question 4 2
Let’s 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.
question 4 2 answers
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
question 4 7
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%
question 4 7 answers
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
binomial distributions and sampling
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: Remember…for binomial distributions, we would visualize this data through a pie chart…because we do not have enough categories for a histogram…

sampling from a two category population
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

sampling distribution
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
sampling from a binomial distribution
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)
sampling from a binomial distribution16
Sampling from a Binomial Distribution
  • But suppose sometime later, your manager asks you to lead another study. But this time, you don’t 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.
sampling from a binomial distribution17
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
binomial distribution and the normal curve
Binomial Distribution and the Normal Curve
  • Now, we need to decide if we can use the normal curve to solve problems…
  • If (np) > 5 and n(1 – p)>5…then the sampling distribution will be normally distributed.
  • So, our sample was 200 students.

Is (np) > 5?

Is n(1 – p)>5?

  • Yes…and yes.

np = (200)(.13) = 26

n(1 – p) = (200)(1 - .13) = (200)(.87) = 174

binomial distribution and the normal curve19
Binomial Distribution and the Normal Curve
  • What do we mean that a sampling distribution is normal?
    • Just like someone’s 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.
how to use the binomial distribution and the normal curve
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 (σ) = √np(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

σ

  • Use the normal curve table to solve the problem
why the continuity correction factor
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.
  • Probability of getting a value equal to or greater than (=>), then you must subtract a half-unit
  • Probability of getting a value equal to or lesser than (=<), you must add a half unit.
  • Probability of getting the exact value, you must get the Z-scores for a half-unit above and a half-unit below
now let s answer a hot topic question
Now let’s 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?
slide23

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 (σ) = √np(1 – p) = √26(1 - .13) = √26(.87) = √22.62 ≈ 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
now let s do question 4 16 as a class
Now let’s 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%
question 4 16 answers
Question 4.16 answers
  • Expected value = np = 500(.03) = 15
  • 32.93%
  • 25.46%