Loading...

PPT – Statistical Analysis PowerPoint presentation | free to download - id: 6fbdab-NDk4O

The Adobe Flash plugin is needed to view this content

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 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

- 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

- 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)

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

- 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

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

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.

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

- 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

- 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 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

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

- 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

- 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 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.

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

- 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

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.

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

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 (gt), then you must subtract a

half-unit - Probability of getting a value equal to or lesser

than (lt), 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 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?

- 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

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

Question 4.16 answers

- Expected value np 500(.03) 15
- 32.93
- 25.46