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Math 3307 Lecture NotesPerkowsky text May’13

Chapters 4 – 6

Homework Assignments10 points each problem ( 140 points total)

Chapter 4 2, 4, 6

Chapter 5 2, 4, 6, 8, 12, 16

Chapter 6 2, 4, 8, 10, 12

Homework style sheet and rules:

Work on one side only; pdf it and upload it before the deadline on the calendar.

Work that is poorly scanned or illegible will be given a zero. This includes sideways or upside down scans!

Do NOT crowd the work, leave at least 3” between problems. Label the answers carefully so the grader can grade efficiently.

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Chapter 4 – Data with Two Variables

Pages 85 – 117

We dealt with “univariate” data in Chapter 2…these are data sets with only one variable. Often, there are 2 variables to deal with – we’ve seen this in College Algebra. When there are 2 variables we call the data set “bivariate”. This happens because the two variables are often related to one another and if they are related we call them “correlated”. Think of time and technology prices…as time increases, the price for a particular laptop typically decreases. This is not easy to see if you study time separately from price!

Initially we will focus on linear relationships between two variables.

4.1 Scatter Plots and Correlation

With a scatter plot, we graph the data points just like we’re in College Algebra on the Cartesian plane. Now two differences show up right away! Sometimes in Statistics, there is no “origin”…the intersection of the axes is at some other point values than (0,0) – check the numbers at the SW intersection (always) so you don’t get caught by this fact. And the fact that real data is almost never perfect is the second difference – the points will be “off” the line that we use as a model. This is due to error, variation in experiments, the very “real” nature of what we’re measuring.

Correlation is a number that tells how “perfect” your real data is. The formula is complex and we’ll just use technology to come up with it.

As we are only working with linear data, we’ll use terminology from lines.

We can have a perfect positive correlation

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We can have a strong positive correlation

We can have a weak positive correlation

We can have NO correlation

We can have each of these with NEGATIVE correlations, too: perfect, strong, and weak . Let’s look at pictures of these, too.

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Think of the numbers as grades! 95% is an excellent grade and an excellent correlation, too. A small caution: you want n = 30 or bigger.

Now, one big caution:

There is a strong correlation between time and a laptop price…but time passing doesn’t CAUSE the price decrease…people just stop paying so much because it’s old technology now and they’re looking forward to the next big thing. A good correlation coefficient doesn’t imply that one thing is causing the other…just that there is a relationship.

Let’s look at the diagrams on page 88 at the bottom and assign correlation coefficients to them along with describing them.

A

B

C

D

E

F

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Also from the internet – more “regression lines”

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Strong or weak? Negative or positive?

What is that line called? What does it mean and what is it used for?

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Strong or weak? Negative or positive?

What is that line called? What is it used for?

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To show that not all regression curves are lines! Note the exponential formula !

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Reviewing the “cause and effect” notion, let’s look at page 89 and check out the graph of manatee deaths and the stock market.

From an internet listing we find:

What is the difference between correlation and cause and effect?

Correlation means that two or more sets of data move in some consistent pattern. Perhaps during a 10-year period the number of cars sold in the U.S. moved in the same direction as the country’s rate of inflation. Even with a 10-year correlation between the two sets of data, it is unlikely that more inflation caused an increase in the number of cars sold. In other words, correlation does not assure that there is a cause and effect relationship.On the other hand, if there is a cause and effect relationship, there will have to be correlation.

Exploration:

Given the four quadrants and a data set that goes through the origin,

Where will you find the points with a positive correlation?

Negative?

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4.2 Pearson’s Correlation Coefficient

Now let’s look at the formula for the correlation coefficient:

this is the square root of the number used above

Let’s take this apart:

These formulas are all relatively similar!

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Now, depending on WHERE the points are the product in is positive (Q1 or Q3) or negative (Q2 or Q4). And note that this product is the AREA between the data point and the two mean lines…even though it is sometimes negative. See the picture on page 91! Let’s sketch this here: (2, 3) mean x = 5, mean y = 4

Let’s check out the properties on page 93 for Pearson’s Correlation Coefficient and let’s remember that we will, indeed, use technology to get that number!

1. It’s always:

the endpoints are perfect; 0 is no correlation

The size of the sample matters…the more data points the better. Try to be over 30.

2. Changing the units has no effect on it. Feet to inches; ounces to pounds;Dollars to renminbi…no problem. It is what it is.

3. The value is not affected by which variable is called x and which is called y.This is because there is NO cause and effect relationship involved.

***Pull out the TI’s page 95 has the instructions for this!

Let’s discuss the data: air temp and bat behavior. Page 95 and 96

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4.3 Slopes and Equations of Fitted Lines

“Fitting” a line to the data is actually quite scientific these days and easy, too, with all the technology we have available to us.

For our uses, let’s review two forms of the equation of a line:

The Point-Slope Equation

This equation works very well with data sets….you can get the slope and use a point quite efficiently.

The Slope-Intercept Equation

Once we’ve used the preceding equation, we’ll convert to this form for ease in reading!

Let’s look at the data on page 96 – and at the lines for the two girls work.NOTE that the data starts at (700, 50) and not the origin. See the squiggle on the graph that helps you see this?

Now, which girl is right?

What about the difference in the slopes?Let’s count out the two slopes on the graph….rise and run

Let’s find the equation for EACH picture:

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Now to uses: we will want to predict the number of stories from the height…whether it is a higher skyscraper (extrapolation – going beyond the data range above or below) or one in between the shortest and the tallest for which there is no data point (this is called interpolation – finding a point between the extremes).

What if we extrapolate back to zero? Or up to 1800?

What about a building that is 1200 feet? We are interpolating here.

How can we find these equations, these models, with a minimum of fuss – we’ve got two ways and we’ll look at each in turn!

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4.4 The Least Squares Line4.5 The Median-Median Line

First the Least Squares Line Method.

Let’s define the perpendicular distance from a data point to the regression line as the “residual”. And our goal is to make the sum of the residuals as small as possible. Residuals can be positive (point above the line) or negative (point below the line).

Let’s look at some residuals:

This is the “Tent and Sleeper” data on page 101.I also used Geogebra to create these visuals.

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Note the regression line, the , and the residual plot on this presentation.

Now let’s check out the analysis in the textbook on pages 102 and 103.

Note the area boxes on page 103. Note the discussion about the coefficients on page 103 and 104!

Now let’s use the calculator to get the 3 S’s (see page 104 and make sure you get the SAME number!

Let’s look at our formulas for a linear regression line:

The instructions are on page 105 to do this AUTOMATICALLY! Let’s do it the automatic way!

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RCurve Problem 1

Varnish Drying Time

Amt -gmHours to dry

1 7.2

2 6.7

3 4.7

4 3.7

5 4.7

6 4.2

7 5.2

8 5.7

Sometimes it is useful to watch paint dry. Let’s see what we can find out about this data

Graph the data.

Find the best fit using technology. Be sure to support your conclusion by getting the Regression Coefficient

Extension: How long will it take 10 grams of varnish to dry?

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RCurve Problem 2

Let’s look at a made up set of data and discuss it.

x 1 2 3 4 10 10y 1 3 3 5 1 11

Make the scatter plot and find the regression line. Calculate the correlation coefficient.

What has happened to the data here? Is this a good experiment? Why or why not?

Would you consider any of the points to be anomalies?

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RCurveProblem 3 – an extension of what’s in the book

Given the following data for the stretch of a spring, select the best model from among those below:

What do x and y represent?

x 5 10 15 20 25 30 35 40 45

y 0 19 57 94 134 173 216 256 343

a.

b.

c.

How do you know you are right?

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Looking at the Median-Median Line page 106

To look at this we first need to group the data. We’ll still use the sleeper tent data for this illustration:

Sort the data is ascending order by floorsize, the x-values. Divvy the list into reasonable groups, trying to get the same number of points in each group (this is NOT always possible when you work with the natural breaks in the data). If it can’t be even, put the larger number of points in an extreme group.

In each group, find the median point (x, y). SEE THE CAUTION p. 103

Find the line through the medians of the extreme values. Find the line that is parallel to this line through the middle point. Move the first line 1/3 of the way (up or down) toward the second, middle point point line (same slope, y-intercept one third between the other two).

This also can be done in your calculator. See page 110 for the instructions!

Problem MM 1

Given the following data find the median, median line.

(−8, 1.7), (−3, 2.9), (−1, 4), (3, 6.7), (7, 8), (10, 10.5), (11, 11)

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Chapter 4 Summary

Sample questions: 4

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Chapter 5 – Probability

5.1 What is Probability?

A probability is a number between 0 and 1 that characterizes the likelihood that something will happen. 0 means it will never happen, 1 means it will definitely happen and proper fractions are in between these two number giving other likelihoods. Sometimes we convert to decimals to describe probability.

We use probabilities to make decisions. If the probability of rain is 80% (4/5) then we’ll choose to carry an umbrella. If its 20% (1/5) we won’t.

5.2 Outcomes and Events

Vocabulary! is needed to discuss probability.

Experiment

Outcomes

Sample Space

Event (subset of sample space)

Probability:

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See the CAUTION on page 123 at the bottom of the page. Resist the urge to reduce fractions! The denominator shouldn’t be reduced…it means something as it stands.

Law of Large Numbers

Notation:

Mutually Exclusive Events: triple Venn

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Probability Problem 1

Vincent and Joey are playing a game where they each display their right hand with a number of fingers showing simultaneously. (1, 2, 3, or 4 fingers).

What are the outcomes in the sample space?

What is the probability that they each display the SAME number of fingers?

5.3 Basic Probability Rules

Rule Number 1

Probabilities are positive and are between 0 and 1.

Rule Number 2

The sum of the probabilities of all outcomes is 1. (100%)

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Rule Number 3

The probability of an event is one minus the probability of its complement.

Note that this text uses a superscripted bar for complement.

Rule Number 4

Say this in WORDS!

Show this in a VENN diagram for 2 events

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Also note:

Let’s look at page 139 and run through the focus on Understanding together.

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Probability Problem 2

Given the following probabilities for a number on a LOADED die:

Event Probability

1 1/3

2 1/4

3 1/12

4 1/12

5 1/6

6 1/12

Why is it necessary that all the Probabilities add up to one? Do these?

Find the probabilities of the following events:

A. the number observed is a multiple of 3

B. the number is even

C. the number is an even multiple of 3

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Probability Problem 3

Use a tree diagram to analyze the following experiment:

You toss a penny, a quarter, and a tetrahedral die (ie 4 sides!). You record the observable face in that order.

A. what is the sample space?

B. what is the probability that a 3 is on the face of the die

C. what is the probability that 2 heads appear on the coins?

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Probability Problem 4

Suppose the following probabilities are assigned to these events

P({e1}) = 0.3 P({e2}) = P({e3}) and p({e4}) = 4P({e2})

There are only 4 events in the sample space.

A. find the probabilities assigned to each individual event

B. Find P(A) where A = {e1, e3, e4}

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Probability Problem 11

An experiment consists of selecting a random one of the 100 points in the x-y plane with whole number coordinates from 0 – 9 in x and in y. Hint: sketch this array!

Compute the probability for each of the following events.

A The point lies on the line y = x.

B The point lies on the line y = 2x.

C The point lies on the circle .

D The point lies within but not on the circle .

E The point lies within or on the circle .

F The point lies outside the circle .

G The point lies either on the circle but not on the graph of .

H The point lies on the circle or on the line y = x.

I The point lies on the x-axis or the circle y = x.

J. The sum of the x and y-coordinates of the point is 18.

K The sum of the x and y-coordinates of the point is 21.

L The y-coordinate of the point is greater than the square of the x-coordinate.

Put your array on the next page:

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Put your array here:

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5.4 Conditional Probability and Independence

Conditions impose restraints on the Sample Space. The probability of A GIVEN that B happened is different than the probability of A alone.

Probability Rule #5:

( )

Note the change in the Denominator!

There are a couple of worked out problems in this section. Be sure to study them closely.

Now let’s discuss independence – It’s an important concept. Page 137. Let’s look at the formula you use if you KNOW events A and B are independent:

This is quite different from above. Note that you must be able to argue successfully that the events are independent BEFORE you get to use this simpler formula!

Also, not in the book: If two events are independent . In general if SMALL samples are taken from a LARGE group without replacement, then you may assume independence.

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CP Problem 1

Vincent and Joey are playing a game where they each display their right hand with a number of fingers showing simultaneously. (1, 2, 3, or 4 fingers).

What is the probability that they both display 2 fingers given that they display the same number of fingers?

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CP Problem 2

The probability that the stock market goes up on a Monday is 0.6 and the probability that it goes up on a Tuesday given that it went up on Monday is 0.3. Find the probability that the market goes up both days.

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CP Problem 3

A cosmetics sales person makes house calls to different neighborhoods. Past experience has shown that the probability that a female resident is home when she calls in the evening is 0.7. Given that the lady of the house is home, the probability of a sale is 0.3. Find the probability that she’s home and she buys a product.

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PIE Problem 1

It is known from old medical records that the probability that a person has cancer is 0.02 and the probability of having heart disease is 0.05. Assume that the two ailments are independent events. What is the probability of having both? Neither? One or the other but NOT both?

PIE Problem 2

A nuclear power plant has a fail-safe mechanism with 6 protective devices that function independently. The respective probabilities are 0.3, 0.2, 0.2, 0.2, 0.1, and 0.1. What is the probability that the reactor will fail?

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5.5 Multiplication Rules

Probability Rule #6

This is a version of Rule #5

And recall:

If two events are independent

Let’s look at page 139: Classroom Connection and the following Focus on Understanding together.

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MR Problem 1

In a survey of 10,000 African Americans, it was determined that 27 had sickle cell anemia.

Suppose we randomly select one person from the sample. What is the probability that this person has sickle cell anemia?

If two individuals are selected randomly what is the probability that both have it?

Hint:

Assuming independence of selection, what is the probability of randomly selecting two people from the sample who have it.

Hint: If two events are independent

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PIE Problem - Enrichment

Past attendance records show that the probability that the chair of the board attends the annual sales meeting is 0.65, that the president of the company attends is 0.9, and that they both attend is 0.6. Would you say that they are acting independently? Show the basis of your reasoning.

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Prob Summary Problem 1

On a TV quiz show a contestant is asked to pick an integer at random from the first 100 natural numbers. Unbeknowst to the contestant, the rule for winning is to pick a number that is divisible by 9 or 12. If a winning number is picked, the contestant will win a trip to the Bahamas.

What is the probability that the contestant will win?

Prob Summary Problem 2

In a certain area, TV Channels 4 and 7 are affiliated with the same national network The probability that channel 4 carries a certain sports program is 0.5, that channel 7 carries it is 0.7 and the probability that they both carry it is 0.3. What is the probability that Mike will be able to watch the program on either channel? (ie Channel 4 OR Channel 7).

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Prob Summary Problem 3

The town of Alvin has 2 ambulance services: the city service and a private company. In an emergency, the probability that the city service responds is 0.6, the probability that the private company responds is 0.8, and the probability that either responds is 0.9. Find the probability that both services will respond to an emergency.

Prob Summary Problem 4

A box contains 3 red, 4 green, and 5 white balls. One ball is picked at random. What is the probability that it is red or white?

What’s different about this situation from the preceding ones? How will you use the formula:

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Prob Summary Problem 5

Two employees, Tom from Plant Operations and Becky from Campus Security, are supposed to check that the University Computing Center doors are locked after 6pm. On any given day, the probability that Tom will check the doors is 0.96 and the probability that Becky will check is 0.98. The probability that they both check is 0.95. What is the probability that neither of them will check the doors?

Prob Summary Problem 6

The probability that a radioactive substance emits at least one particle during any given hour is 0.008. What is the probability that it does not emit any particle at all during any given hour?

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Prob Summary Problem 7

Explain why it is not possible to have two events, A and B, with either of the following probabilities.

A. P(A) = 0.6 and P(A and B) = 0.8

B. P(A) = 0.7 and P(A or B) = 0.6

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Prob Summary Problem 8

If A and B are events with 0.6 and 0.7 as their respective probabilities. Further P(A or B) = 0.9. Find P(not (A and B)).

Prob Summary Problem 9

Past records indicate that the probability that a new customer will open a checking account is 0.7 and the probability that this new customer will open a savings account is 0.4 while the probability of opening both is 0.25.

What is the probability that a new customer will open a savings account or a checking account?

What is the probability that the customer will open neither and opt for a different service entirely?

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5.6 Geometric Probability

Geometric probability comes up in problems with spinners and with rectangular grids. The formula changes just slightly with this type of problem. Before we had

Probability:

Now we’ll have:

Geometric Probability:

Let’s look at the dart problem on page 106 together.

There’s a spinner problem on page 142 – 143, that we can check out, too.

Now let’s look at the game card on page 103 and answer those questions!

And let’s work through the problem in the Focus on Understanding problem on page 164.

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Chapter 5 Summary

Problem 7, 11, 15 are very typical

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Chapter 6 – Counting Techniques

Let’s read the Classroom Exploration on page 152…how would you handle this question?

6.1 The Multiplication Principle or the Fundamental Counting Principle

Let’s look at Robert’s clothes on page 154 and see what occurs on the above exploration?

Remember tree diagrams? How does mutually exclusive fit into tree diagrams?

Let’s read the definition on page 157 for the Fundamental Counting Principle and think about ways to teach it so it makes sense to your students.

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Now let’s look again at the intro problem with the questions from page 159 in mind. See the comments at the top of page 60!

Now let’s look at Venn Diagrams:

Two set Venn diagrams:

If you need a review of union and intersection NOW is the time to speak up!

The usual Venn diagrams might be helpful here:

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The circle on the left is A and the one on the right is B; the box containing the circles is the Sample Space or the Universe.

Find A∩B, AUB, only A, neither A nor B…

Translate A∩B into words…what is the key word?

Translate AUB into words…what is the key word?

VD Problem 1

From a survey of 100 college students, a marketing research company found that 75 students owned Ipads, 45 owned cars, and 35 owned Ipads and cars.

How many students owned either a car or an Ipad?

How many student owned neither?

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VD Problem 2

A survey of 64 informed voters revealed the following information:

45 believe that Elvis is alive49 believe that they have been abducted by aliens42 believe both of these things

How many people believe neither?

How many believe Elvis is alive but don’t believe that they’ve been abducted by aliens?

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Three set Venn diagrams:

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VD Problem 3

A survey of Math faculty at UH revealed the following:

51 admire Moe49 admire Larry60 admire Curley34 admire Moe and Larry32 admire Larry and Curley24 admire all three Stooges1 dislikes all 3

How many people are there in the Math department?How many admire Curly but not Larry and not Moe?How many admire Larry or Curley?How many admire just one Stooge?How many admire exactly 2 Stooges?

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6.2 Permutations

Who remembers the permutations formula? (see page 165 for a reminder!)

What is important about permutations?

Let’s see how we do this with our calculators: page 162 and let’s spend some time on their discussion of scientific notation.

With replacement and without replacement – BOY does it matter! Page 163The first half of this section is without replacement. With replacement starts on page 167 (we’ll get there next).

Check out with replacement.

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6.3 Combinations page 168

Formula: page 171

What’s important about combinations?

Calculator: page 171 and 172

Let’s look at the Focus on Understanding on page 172 and talk our way through the questions

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Now let’s answer the Dr. Math question on page 173…as a group…Focus on Understanding questions 1 and 2.

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6.4 Mixed Counting Problems

Let’s go through each question and discuss WHY the choice of how to solve it is correct and how we would know we were right if the book hadn’t spelled out the answer!

Permutations and Combinations – a quick review + problems

Permutations are the number of possible arrangements when order matters.

A permutation of n objects taken r at a time is computed using

So your answer is a number of outcomes. Note that you still have to calculate the other part of the probability ratio once you’ve done ONE permutation!

A permutation of 11 objects taken 4 at a time is calculated:

which is 7,920.

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Combinations are the number of possible arrangements when order does NOT matter.

A combination of n objects taken r at a time is computed using

Again, your answer is a number of outcomes that you will use in a probability ratio.

A combination of 11 objects taken 4 at a time is calculated

which is 330.

Note that there are far fewer combinations than permutations?

Why is this always true?

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P&C Problem 1

How many 8-letter permutations can be formed from the letters in the word “hesitate”. In how many of these do the e’s occur side by side?

P&C Problem 2

Find the number of numbers greater than 4000 which can be written using the digits 0, 3, 5, 6, 8.

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P&C Problem 3

Find the number of different lines determined by 11 points, no three of which lie on the same line.

P&C Problem 4

From a group of 12 students, in how many ways can a 6 person team be formed if a specific person MUST be included?

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P&C Problem 5

Find the number of ways of arranging the letters in the word “object”. Find the number of ways of arranging the letters if the first letter must be “j”.

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P&C Problem 5

Would you use a permutation or a combination to solve the following problems:

A. the number of ways a 5 card hand can be dealt from a 52 card deck

B. making 3 digit numbers from the digits {1, 2, 3, 4, 5}

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Chapter 6 Summary

Sample Question:

Page 173, Focus on Understanding, questions 1 or 2

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