Download MA105 Reading Notes

Reading Notes
Math 105 — Fall 2015
Consult these notes as you read from our course textbook. For many textbook sections, these notes
will advise you on what details you may safely skip in your reading. Everything else in each assigned
section should be read carefully and thoughtfully.
§1-1 Review and Preview
Start reading at the Chapter 1 Problem, on page 3, before continuing.
Note that in statistics, the word “population” does not necessarily refer to a group of people. The
“individuals” mentioned in the definition could be animals, manufactured products, or some other
objects being studied.
We will almost never use data from a census in this course. The main long-term purpose of the
course is to introduce tools that let us draw conclusions about a whole population based on only a
sample.
§1-2 Statistical and Critical Thinking
The idea of statistical significance, which is introduced in this section, is extremely important and
will come up throughout the course.
If we are working under a certain assumption, but then observe something that would be very
unlikely if our assumption were correct, that observation is called “statistically significant.” A
statistically significant observation can be considered evidence that our assumption is actually
incorrect.
The discussion of “correlation” and “causation” in this chapter also touches on an important topic
that will be revisited, particularly in Chapters 2, 9, and 10.
The warning about “small samples” on page 9 is somewhat vague for now, but later we will learn
very exact ways to determine whether a sample is large enough to draw meaningful conclusions.
There are some details on page 10 about improper ways to gather or report information. In this
course, we will usually assume that the data we work with has been obtained correctly, although it
is important to have a basic understanding of what can “go wrong.”
You will hopefully find the notes about percentages (page 11) unnecessary, since you should already
§1-3 Types of Data
Page 2
be familiar with how to work with percentages from everyday life. You will need to understand
percentages throughout this course.
§1-3 Types of Data
It is important to understand the distinction between a parameter and a statistic (page 16) because
we will often use a statistic to estimate a parameter.
You do not need to memorize the names or definitions of the “four levels of measurement” (starting
near the bottom of page 18), but do understand that certain kinds of calculations do not make
sense with certain kinds of data.
§1-4 Collecting Sample Data
Read pages 23 and 24, then skip ahead to “Designs of Experiments,” starting on page 28 and
continuing to the end of the section.
Make sure you know the difference between gathering data through observations and through
experiments. You must understand why an experimental study can give us information about
cause-and-effect that an observational study never could.
§1-5 Introduction to the TI-83/84 Plus Calculator
Statistics in the real world often deals with very large lists of data. The statistical methods that
help to understand such data are usually applied using a computer with special statistical software
(for example, Excel, S, SPSS, Minitab, MATLAB, or their free counterparts like OpenOffice Calc,
R, GeoGebra, and Octave.). In this course, we use a particular kind of graphing calculator instead,
because it is portable, less expensive, and has many of the capabilities of statistical software.
However, you should be aware that using a graphing calculator to create statistical graphics and
deal with data is only an approximation of what can be done with more advanced statistics programs
on a computer.
At this point in the course, it will be impossible for you to proceed without an acceptable calculator.
All TI-83 and TI-84 calculators are nearly identical for our purposes. But no other type of calculator,
including the TI-89, or any other electronic device, is allowed in this course.
Instead of reading about “setting the calculator mode” on page 35, the easiest way to make sure
that your calculator is in the ideal condition to be used in this course is to completely reset the
§2-1 Review and Preview
Page 3
calculator, as described below.
The most important parts of this section are “Scientific Notation” (page 36) and “Working with
Data Lists” (page 37). Make sure you know how to store lists of data – see “Creating a New List”
on page 38. You will find that we work with lists of data so often in this course that this process
will soon become very familiar to you.
You can safely ignore the discussion of “APPs” at the bottom of page 37. We will not use APPs
during class, and they will not be allowed during exams. “TRIOLA11” provides a simple way to
load lists of data from Appendix B of the textbook, so you might find it useful as you study some
textbook examples, or you could simply open the book to Appendix B and enter the data by hand
as needed.
Resetting Your Calculator: Before taking any exam in this course, you will be asked to
completely reset your calculator’s memory. It would also be a good idea to reset your calculator’s
memory from time to time throughout the semester. This process will permanently erase all data
and any programs that aren’t built-in.
The following instructions will be displayed on the classroom projector screen before each exam.
Open the memory menu by pressing 2nd and + . Then select Reset. Next, select All Memory (this
is located in the ALL menu on a TI-83 Plus or TI-84 Plus). Finally, select Reset.
The results for a TI-83 Plus are shown above. On any calculator, after resetting the memory,
the screen should display the message Mem cleared. I will check your calculator’s screen for this
message before giving you a copy of each exam.
§2-1 Review and Preview
As with every chapter, you should read the Chapter Problem (page 49) before continuing.
For now, we will hardly deal with the “changing characteristics of the data over time.” Also, do not
worry about memorizing the five characteristics given in this section. Ignore any future references
to the silly “CVDOT” mnemonic. You can (and should) be aware of the important characteristics
of data, which are addressed in Chapters 2-3, without memorizing a list of their names.
§2-2 Frequency Distributions
Page 4
§2-2 Frequency Distributions
We will not use the idea of “class boundaries” (page 51). The “procedure for constructing a
frequency distribution” on pages 51-52 makes a big deal out of something that is actually fairly
simple to do by hand, and can also be done with the help of your calculator.
It might be best to skip the calculator instructions on page 53 until after reading §2-3, at which
point the instructions will make more sense. In general, §2-2 and §2-3 are so very closely related
that most of the ideas in §2-2 will be easier to understand once you have seen all of §2-3.
§2-3 Histograms
If you know what a bar graph is, make sure you understand the ways in which a histogram is
different from a bar graph.
For now, it is best if you skip “Assessing Normality: Normal Quantile Plot” on pages 63-64.
Once you read the instructions for constructing a histogram using your calculator (bottom of
page 64), go back and take a look at the instructions on page 53 for constructing a frequency
distribution. The trick for automatically constructing a frequency distribution from a list of data
is to create a histogram with the class width and boundaries you want, then use the TRACE button
and “left” and “right” arrow keys to read off the frequency distribution. See also the following
addition example.
Additional Example: Suppose we want a histogram and/or frequency distribution of the following
data, using lowest class limit 80 and class width 8:
95 104 105 101 107 103 120 93 111 96 102 109 85 109 98 103 121 103 101 91
86 102 96 104 101 84 97 91 99 96 106 104 123 106 102 109 91 108 108 102
The first step is to enter the data as a calculator list, for example L3. Then select Plot1 in the STAT
PLOT menu and set up the plot as shown here. Press ZOOM , then 9 , to show a histogram. This
histogram will not have the desired class limits and widths, so press WINDOW and choose a lowest
class limit (“Xmin”) of 80 and class width (“Xscl”) of 8. Since we have increased the class width,
let’s also increase the height of the graph window (“Ymax”) to, say, 20, in case the histogram bars
become a bit taller. Press GRAPH to show the histogram.
§2-4 Graphs that Enlighten and Graphs that Deceive
Page 5
The desired frequency distribution could easily be created without a calculator, using only the
data given. But since we have the corresponding histogram, we can read the distribution from it.
Pressing TRACE and using the / and . keys, we can read the following distribution:
Value
80-87
88-95
96-103
104-111
112-119
120-127
Freq.
3
5
16
13
0
3
§2-4 Graphs that Enlighten and Graphs that Deceive
The most important kind of plot mentioned in Chapter 2 is the histogram, but all of the others
explained in this section are also useful. Particularly, make sure to learn how to create a scatterplot
with your calculator (page 68), and also understand what scatterplots are used for.
Do not spend too much time on dotplots, frequency polygons, or ogives. These types of visuals
have their uses, but we will not use them much.
§3-1 Review and Preview
Pay attention to the distinction between descriptive statistics and inferential statistics.
§3-2 Measures of Center
Note that we have separate symbols for a sample mean x̄ (pronounced “x bar”) and a population
mean µ (pronounced “mew”) because x̄ is only an estimate of µ.
Although we will usually calculate means using a simple calculator function (described on page 91),
you must understand how means are calculated, according to Formula 3-1 (page 92).
The “Caution” on page 94 is only partially correct. Statisticians sometimes use the word “average”
when reporting their findings to the public, because it is less grammatically awkward than the word
“mean.” The word “average” will sometimes be used (referring to a mean) in later parts of this
course.
You may stop reading before “Part 2” begins on page 98.
§3-3 Measures of Variation
Page 6
§3-3 Measures of Variation
Just like with means, the standard deviation of a sample, written as s, is an estimate of the standard
deviation of the entire population, written σ (pronounced “SIGG-muh”). But unlike the sample
mean and population mean, s and σ are calculated slightly differently from each other. You do
not need to know the formulas for s or σ by heart, but you must remember that they are different.
Above all else, focus on what the standard deviation tells us about a sample or a population.
We will not really use the variance (s2 or σ 2 ) in this course, although it is sometimes very important
in more advanced areas of statistics and probability.
For now, the only part of the Empirical Rule on page 115 that you should know by heart is the
“about 95%” part (see Example 6). We will expand upon this idea in Chapter 6. We will not use
Chebyshev’s Theorem (page 116) at all.
Make sure to learn how to use your calculator to find the range and standard deviation of sample
data. Note that this is done with the same 1-Var Stats function mentioned in §3-2.
§3-4 Measures of Relative Standing and Boxplots
There is very little unimportant information in this section. At the bottom of page 131, we will not
use the “10–90 percentile range,” the “midquartile,” or the “semi-interquartile range.” However, the
“interquartile range” is sometimes useful as a measure of variation that is not sensitive to extreme
values.
The concept of a z score is incredibly important, especially later in Chapters 6 and 7. You will need
to understand that idea as thoroughly and deeply as possible.
You will not need to remember the formulas for determining whether a data value is an outlier
(page 135). Instead, just use your calculator to create a modified boxplot (see page 136) and check
it for outliers. The values of the outliers can be seen by pressing TRACE and using the arrow keys.
— End of Exam 1 material —
§4-1 Review and Preview
Page 7
§4-1 Review and Preview
The “Rare Event Rule for Inferential Statistics” is basically a restatement of the idea of statistical
significance, from back in Chapter 1. This concept is based on probability, which is the main focus
of Chapters 4 through 6. Most of the rest of this course is built upon a good understanding of
probability.
§4-2 Basic Concepts of Probability
On page 154, it is worth pointing out that if you read other resources (websites or books, for
example) about probability, the complement of an event A is also sometimes written as AC or A0 .
Do not read about “odds” (pages 157-158), unless you are interested.
§4-3 Addition Rule
As mentioned on page 165, you will not do well in Chapter 4 if you see it as a series of rules or
formulas to be memorized and used. You must understand the reasoning behind each formal rule
– why the formulas “work” – or you will inevitably try to apply them in situations where they do
not fit.
§4-4 Multiplication Rule: Basics
Again, make sure you understand the reasoning behind the formal rules.
The “5% Guideline” (page 172) should only be used if it would be extremely tedious to avoid it.
§4-5 Multiplication Rule:
Probability
Complements and Conditional
Read the whole section. Also carefully consider the following additional example:
If we draw five cards from a well-shuffled deck, the probability that at least one of them is an ace is
48 47 46 45 44
·
·
·
·
≈ 0.3412.
P (at least one ace) = 1 − P (no aces) = 1 −
52 51 50 49 48
§4-6 Counting
Page 8
§4-6 Counting
You do not need to memorize the formulas for n Pr or n Cr , since they are available in your calculator
(see page 191), but you will probably find it helpful to read and think about where the formulas
come from. In fact, students who understand how to calculate n Pr and n Cr the “long way” are
usually much better prepared to use these numbers properly. In any case, you need to understand
what the numbers n Pr and n Cr represent. Note: n Pr is usually pronounced “n permute r,” and n Cr
is usually pronounced “n choose r.”
We will not use the “Permutations Rule (When Some Items Are Identical to Others)” on page 190.
Therefore, skip Example 4.
Instead of thinking about the “factorial rule” and the “permutations rule” separately, you might
just notice that n Pn always means the same as n!.
§5-1 Review and Preview
As with any chapter, read both the Chapter Problem and the first section.
§5-2 Probability Distributions
You do not need to know the formula for the standard deviation (or variance) of a probability
distribution (page 214), since a calculator can find it automatically (see pages 216-217). However,
you will find it useful to know and understand Formula 5-1 for the mean of a probability distribution.
You must understand what the mean and standard deviation of a probability distribution measure.
You will not be required to construct a probability histogram, as described on page 216, though you
should understand what a probability histogram represents (see the one on page 156, for example).
§5-3 Binomial Probability Distributions
We will not use Formula 5-5 directly (as done in Example 2), nor will we ever need Table A-1
(mentioned at the top of page 230). As a general note, ignore future references in the textbook to
anything in Appendix A (probability tables). Your calculator makes those tables unnecessary.
It is not absolutely necessary for you to understand why Formula 5-5 is correct, but knowing how
and why it works would probably help you to understand when binomial probability is applicable.
§5-4 Parameters for Binomial Distributions
Page 9
Be sure to read carefully about your calculator’s binompdf and especially binomcdf functions. In
the long run, the binomcdf function is by far the more useful and important of the two.
Here is a further illustration of how the binompdf and binomcdf functions can be used. About 18%
of American adults are regular smokers. This means that if we select an American adult at random,
the probability that he or she smokes regularly is about 0.18. If we survey a random sample of
1400 adult Americans, asking each one if he or she smokes regularly, we can think of the survey as
a binomial experiment with n = 1400 and p = 0.18.
(a) What is the probability that exactly 18% of our sample will be regular smokers? 18% of 1400
is 252, so the probability is given by the calculator command binompdf(1400,0.18,252).
The probability is about 0.0277. It is fairly unlikely that exactly 18% of our sample will be
regular smokers.
(b) So how far off might our survey result be? For example, how likely would it be that
210 or fewer of the people in our survey will be regular smokers? This probability can
be written as P (x ≤ 210), or as P (x < 211). To calculate it, we can use the command
binomcdf(1400,0.18,210), and the result is only about 0.00161. It’s very unlikely that
our survey will find so few regular smokers.
(c) Similarly, the probability that 310 or more of the people in our survey will be regular
smokers can be written as P (x ≥ 310), or as P (x > 309). The simplest way to calculate
this probability is to think of it as 1 − P (x ≤ 309), which can be found using the command
1-binomcdf(1400,0.18,309). The result is about 0.0000482. Again, it’s very unlikely that
the proportion of smokers in our survey will differ much from the expected 252 people.
§5-4 Parameters for Binomial Distributions
This section may seem like a minor side note to §5-3 right now, but you will see that it turns out to
be very important information later. You must memorize the formulas for the mean and standard
deviation of a binomial distribution (Formulas 5-6 and 5-8). We will need these formulas beyond
Chapter 5.
— End of Exam 2 material —
§6-1 Review and Preview
Page 10
§6-1 Review and Preview
The normal distribution is the basis of a lot of the inferential statistics we will be doing for the rest
of the course.
§6-2 and §6-3: Normal Distributions
As you read, note that we will not deal with “uniform distributions” much in this course, so it is
fairly safe to skip some of the early parts of §6-2, including Examples 1 and 2. Uniform distributions
are only mentioned in the beginning as a simple example of a continuous distribution.
As you read the rest of §6-2 and §6-3, focus on the details of how to do calculations with your TI-83
or TI-84. Note carefully that any mention of Table A-2 is irrelevant to us! With that in mind, here
is an overview of the main contents of §6-2 and §6-3.
The command∗ normalcdf(a, b) calculates the probability that a randomly selected value taken
from a normal distribution has a z score between a and b.
normal distribution
of z scores
normalcdf(a,b)
a
b
The command normalcdf(a, b, µ, σ) calculates the probability that a randomly selected value
taken from a normal distribution with mean µ and standard deviation σ is between a and b.
normal distribution with
mean µ and standard
deviation σ
normalcdf(a,b,µ,σ)
a
∗
b
As with some of the calculator features in Chapter 5, the calculator input shown in these notes would be slightly
different with the newest versions of Texas Instruments calculators. Consult the textbook for details.
§6-4 Sampling Distributions and Estimators
Page 11
The command invNorm(p, µ, σ) calculates a data value x with the property that the proportion of
all values that are less than x is p.
normal distribution with
mean µ and standard
deviation σ
p
invNorm(p,µ,σ)
The simpler command invNorm(p) calculates a z score with the property that the proportion of all
z scores that are less than that score is p.
normal distribution
of z scores
p
invNorm(p)
Everything in the overview given above is covered in §6-2 and §6-3, with enough examples to get
you started.
As a final note, §6-2 focuses a lot on “bone density tests,” with results that happen to follow the
standard normal distribution. This is silly. The purpose of the standard normal distribution is to
describe the distribution of z scores, not some kind of raw data.
§6-4 Sampling Distributions and Estimators
In this section, skip “Sampling Distribution of the Sample Variance” on pages 289-290.
Section 6-5 gets deeper into the specifics of the distribution of sample means.
The symbol p̂ for a sample proportion (page 290) is pronounced “p hat.” Because of §5-4, we already
know quite a bit about the distribution of sample proportions.
§6-5 The Central Limit Theorem
Page 12
§6-5 The Central Limit Theorem
Read pages 297-305 carefully, especially the examples, but do not read about the “Correction for a
Finite Population,” starting on on page 305.
√
In Example 4 (on page 304), rather than calculate 0.11/ 36 ≈ 0.018333 and then type this
result into the command normalcdf(12.19,999999,12,0.018333),
it is easier to simply type
√
normalcdf(12.19,999999,12,0.11/ (36)). This same idea could be used in the other examples,
and it is generally a better way to do the calculations, because it helps to avoid errors.
§6-6 Assessing Normality
You do not need to read about “Data Transformations” at the end of this section.
§6-7 Normal as Approximation to Binomial
You only need to know the most basic idea of this section, which is given in the first bullet point at
the top of page 319. To illustrate the idea, let’s return to the topic of a previous example. Suppose
we take a sample of 100 randomly selected adult Americans and find out how many of them smoke
regularly. Since about 18% of all adult Americans smoke regularly, this is a binomial experiment
with n = 100 and p = 0.18. The probability distribution for the possible number of regular smokers
in the sample is shown in the histogram below. (It is possible to find more than 40 smokers, but
extremely unlikely, so that part of the histogram is not shown.)
0.10
binomial probabilities
for n = 100 and p = 0.18
0.08
0.06
0.04
0.02
3
6
9
12
15
18
21
24
27
30
33
36
39
number of successes
Notice that the histogram is strikingly bell-shaped. As it turns out, a binomial probability
distribution can always be approximated by a normal distribution if the number of trials, n, is
large enough. The rule of thumb that we will use in this course is that if np ≥ 5 and nq ≥ 5, then it
§7-1 Review and Preview
Page 13
is suitable to say that the binomial distribution is approximately the same as a normal distribution
√
with mean µ = np and standard deviation σ = npq . This is the most basic idea of §6-7.
The relationship between a binomial distribution with large n and a normal distribution was once an
important tool for easily approximating cumulative binomial probabilities, which are very tedious
without an advanced calculator. Now, that relationship is used for other purposes, as we will see
in §7-2.
You do not really need to read this section. The overview above gives the one important idea. We
will not need to use the “continuity correction” that is used throughout §6-7.
§7-1 Review and Preview
This section reintroduces the idea of “critical values.” Critical values were defined back in §6-2
(page 269), but now we will see why they are very useful.
The most commonly used critical value is z0.025 (also written z0.05/2 ). The important thing about
z0.025 is that the middle 95% of values in a normal distribution are up to z0.025 standard deviations
away from the mean. In other words, 95% of all values in a normal distribution are within z0.025
standard deviations away from the mean. The Empirical Rule (see §3-3) says z0.025 is about 2. With
a calculator, we can find that it is closer to 1.95996. Make sure you can do this small calculation
using the invNorm function. Critical values like this one are used in some very useful formulas in
this chapter. The symbol zα is pronounced “z sub alpha.” Similarly, z0.025 is “z sub point oh two
five.”
As another example, 88% of all values in a normal distribution are within z0.12/2 (or z0.06 ) standard
deviations away from the mean. The critical value z0.06 is about 1.55477. In other words, the middle
88% of values in a normal distribution are no more than 1.55477 standard deviations away from
the mean. See if you can find this critical value using the invNorm function.
§7-2 Estimating a Population Proportion
Almost all of this section is important to understand. However, do not worry too much about “Using
Confidence Intervals for Hypothesis Tests” (pages 341-342). Also, your calculator can construct
confidence intervals automatically, so steps 2 and 3 of the “procedure” on page 344 are not necessary.
Skip all of page 350.
This section is broken into two different days of class time. The main purpose of the second day
is to make sure you know, understand, and are able to use Formulas 7-2 and 7-3 on page 348, and
the “round-off rule” on page 347. When using these formulas, do not round zα/2 , do not memorize
it, and do not look it up in a table.
§7-3 Estimating a Population Mean
Page 14
In summary, you must know the following:
ˆ The exact meaning of a confidence interval.
ˆ How to use 1-PropZInt to create a confidence interval for a proportion.
ˆ The requirements for using 1-PropZInt.
ˆ How to write an interval in each of the three formats mentioned on page 344.
ˆ How to use Formulas 7-2 and 7-3 when appropriate.
Optional note: Where did Formulas 7-1 through 7-3 come from?
In a binomial experiment, if n is large enough, the possible proportion
p of “successes” is
approximately normally distributed with mean p and standard deviation pq/n . If we have a
sample proportion p̂, we know that the population proportion p is roughly
p equal to p̂. So the
standard deviation of all possible sample proportions is approximately p̂q̂/n . (Note that q̂,
pronounced “q hat” is simply 1 − p̂.)
When we take a random sample, there is a probability of 1 − α that the sample proportion will
be within zα/2 standard deviations of the mean of allp
possible sample proportions. In other words,
p and p̂ should be no farther apart than about zα/2 p̂q̂/n . This is the margin of error E from
Formula 7-1 on page 344 (which you do not need to memorize).
p
2
If we solve the equation E = zα/2 p̂q̂/n for n, the result is that n = zα/2 p̂q̂ / E 2 . This is
Formula 7-2 (which you must know by heart). If we do not have an estimate p̂, to be sure to use a
large enough sample size, we need to use the largest possible value of p̂q̂. It turns out that p̂q̂ is as
large as it can be when p̂ = q̂ = 0.5. This is why the “worst-case scenario” for p̂q̂ is 0.5 · 0.5 = 0.25.
This gives us Formula 7-3 (which you must also know by heart).
§7-3 Estimating a Population Mean
You can skip everything from “Degrees of Freedom” in the middle of page 358 up to and including
Example 1. Do not worry about how to calculate tα/2 , since we have no practical reason to do
so. As in the previous section, don’t bother with steps 2 and 3 of the “Procedure for Manually
Constructing a Confidence Interval for µ” on page 360. Also like before, do not worry about “Using
Confidence Intervals for Hypothesis Tests” (pages 360-361).
As always, ignore any references to probability tables.
As with §7-2, we spend two days of class time on this section, with the focus of the second day being
to make sure you know, understand, and are able to use Formula 7-4 on page 365. Read page 366
carefully.
§7-3 Estimating a Population Mean
Page 15
Summarizing, you must know the following:
ˆ The exact meaning of a confidence interval.
ˆ How (and when) to use TInterval to create a confidence interval for a mean.
ˆ The requirements for using TInterval.
ˆ How to write an interval in each of the three formats mentioned again on page 360.
ˆ How to use Formula 7-4.
Optional note: Formula 7-4
Remember from §6-5 that if a population is normally distributed, or if the sample size n is large
enough, the √
distribution of all possible sample means will be normal with mean µ and standard
deviation σ/ n. So, when we take a random sample, there is a probability of 1 − α that the sample
mean will be within zα/2 standard deviations
of the population mean µ. In other words, µ and x̄
√
should be no farther apart than zα/2 σ/ n.
h z σ i2
σ
α/2
√
Now, if we solve the equation E = zα/2
. This is Formula 7-4 (which
for n, we get n =
E
n
you must know by heart).
It is helpful to notice that Formula 7-2 and Formula 7-4 are extremely similar (and as mentioned
before, Formula 7-3 is just the “worst-case scenario” of Formula 7-2).
— End of Exam 3 material —
§8-1 Review and Preview
Page 16
§8-1 Review and Preview
The idea of statistical significance, which is what a hypothesis test really judges, has been explained
fairly thoroughly a few times before. Hypothesis testing for a mean µ was introduced in some detail
in §6-5. It is strongly recommended that you re-read “Introduction to Hypothesis Testing” on
page 304. Make sure to review Example 4 on that page, including the “Interpretation” on page 305.
This course does not cover hypothesis tests for σ, which are mentioned in this section and in §8-2.
As a general rule, ignore any mention of hypothesis tests for σ.
§8-2 Basics of Hypothesis Testing
The ideas and vocabulary introduced in this section will become clearer as you see them in the
context of more specific examples in the sections that follow. You may find it useful to refer back to
this section of the textbook from time to time throughout the rest of the chapter. Again, remember
that hypothesis testing was partly introduced in §6-5. Using the language of §8-2, Example 4 on
page 304 was a “right-tailed” test.
You do not need to read about the “Confidence Interval Method for Hypothesis Testing” at
the bottom of page 407. Note carefully that in this course, we will always use the “P -value
Method,” rather than the old-fashioned “Critical Value Method” or “Confidence Interval Method.”
Throughout Chapter 8, ignore any mention of the critical value or confidence interval methods of
hypothesis testing. Ideally, those methods generally have the same result as the “P -value Method,”
but using P -values is the far more modern, straightforward, and popular approach.
§8-3 Testing a Claim About a Proportion
As mentioned before, ignore any references to the “Critical Value Method” or the “Confidence
Interval Method.” We will simply use P -values. Unfortunately, some homework problems will ask
you to give the “test statistic,” which your calculator calls “ z,” although this number is not very
important when we are using the P -value method.
In this course, for the sake of simplicity, we will avoid using the “Exact Method” outlined in Part 2,
whenever possible. But in the professional world, because of advances in technology, there is little
reason to settle for the approximate P -values used in Part 1.
§8-4 Testing a Claim About a Mean
Page 17
§8-4 Testing a Claim About a Mean
The bottom line of this section is that there are simple calculator functions (either T-Test or Z-Test,
depending on the circumstances) which will produce the P -value for any required hypothesis test
about a population mean, as long as certain requirements are met. As before, some homework
problems will ask you to give the “test statistic,” which is called “ t” when using the T-Test
function.
Ignore Examples 2 and 4, and note that Example 3 can easily be completed with a calculator.
§9-1 Review and Preview
As always, remember to take a look at the Chapter Problem on page 455 before starting the rest
of the chapter.
§9-2 Two Proportions
Like 1-PropZTest and 1-PropZInt, the new calculator functions introduced in this section
(2-PropZTest and 2-PropZInt) use normal probability as a kind of approximation, so the
requirements for using the new functions (see page 457) are quite similar to those before.
As in Chapter 8, ignore any references to “critical values.” However, this section does introduce an
entirely new type of confidence interval, which should not be overlooked.
Do not read the “Rationale” that begins at the bottom of page 462.
§9-3 Two Means: Independent Samples
Stop reading before the “Rationale” at the bottom of page 473. We will not use the “Alternative
Methods” described in Part 2 (starting on page 474). In other words, we will never assume that the
two populations’ standard deviations are known (so we will not use the 2-SampZTest or 2-SampZInt
functions), nor will we assume that the standard deviations are equal (so we will never “pool the
variances,” as discussed on page 475).
§9-4 Two Dependent Samples (Matched Pairs)
Page 18
§9-4 Two Dependent Samples (Matched Pairs)
Read everything in this section very carefully (except, of course, for any reference to “critical
values”).
One way to think about these two closely-related sections (§9-3 and §9-4) is that §9-3 deals with
a difference between means, and §9-4 deals with a mean of differences. Note that §9-4 is really
just about using a t test or a t interval when the sample happens to be made up of differences.
These ideas are actually introduced in §7-3 (see Exercises 14 and 15 on page 371) and in §8-4 (see
Exercises 15 and 17 on page 435).
§10-1 Review and Preview
We will only delve into the basics of this chapter, in Sections 10-1 through 10-3, which are very
strongly related to scatterplots and their main purpose (see page 67). Again, the Chapter Problem
(page 509) is the best place to start reading the chapter.
§10-2 Correlation
You can safely skim over the big box of mathematical details that runs from the middle of page 512
through most of page 513. Also skip Examples 2 and 3, for obvious reasons. Besides these
exceptions, read everything until the bottom of page 520 very carefully.
Beyond page 520, simply note that there is a kind of hypothesis test that can judge whether there is
enough evidence to indicate a correlation between two variables. As with other recent procedures,
this can be done in a fairly automatic way using your calculator. See Example 8 and the brief note
about “One-Tailed Tests” that follows it on page 523.
§10-3 Regression
Like before, skim over the big box of details on page 532. Skip Example 2, and stop reading before
Part 2 (which starts on page 537).
— End of Exam 4 material —