Section 1.1 Commentary
Subsection 1.1.1 Rounding in calculations
In general, do not round off any numbers in a multi-step calculation until the final step. If you must round, keep lots of digits. A rule of thumb is to round to the nearest \(0.001\text{,}\) that is, to the nearest 3rd digit to the right of the decimal point.
Exceptions: When calculating a \(z\) value for normal data, you may round to the nearest 0.01, because that’s the level of accuracy in the \(z\) column on the normal table. When calculating an area under the normal curve, you may usually round the final answer to the nearest whole percent.
Subsection 1.1.2 Percentile (pp.90–91)
The percentile rank of a data value \(A\) is the percent of all data whose value is less than or equal to \(A\text{.}\) If the percent of data less than or equal to \(A\) is \(p\text{,}\) then \(A\) has \(p\)-th percentile. In picture form, the percentile rank of a data value \(A\) is the area under the histogram to the left of \(A\text{.}\) The text introduces percentile and percentile rank in the context of normally distributed data, but the terms percentile and percentile rank apply to all types of data, whether it is normally distributed or not.
Subsection 1.1.3 The term "football-shaped" (p.120)
The text uses the term "football-shaped" or "shaped like a football" very loosely throughout Ch 8–12 to describe a scatter diagram that has linear association. Worse, the term "football-shaped" is sometimes used to imply homoscedasticity. Even worse, "football-shaped" is used to imply that data in thin vertical strips in the scatter diagram is normally distributed (see the box on p.197). Because of this vagueness and ambiguity, we will instead use the terms "shows linear association", "homoscedastic", and "data in thin vertical strips is normally distributed" for these various attributes that may or may not apply to 2-variable data. In particular, in the box on p.197, and also in item 7 of the Ch 11 summary on p.201, replace the first sentence with "Suppose that a scatter diagram shows linear association, is homoscedastic, and data in thin vertical strips is normally distributed."
Subsection 1.1.4 Probability terminology (Ch 13–15)
[Note: The symbols given in this section appear in Friedman’s text in a note on p.227 at the end of Exercise Set B.]
In a box model, an outcome is a sequence of tickets obtained by random draws, either with or without replacement. For example, here are the 12 possible outcomes for 2 draws, without replacement, from the box \(\{A,K,Q,J\}\text{.}\) (This is a box model for "two cards are dealt from a 4-card deck containing an ace, a king, a queen, and a jack".)
\begin{equation*}
\text{set of all outcomes} = \{AK,AQ,AJ,KA,KQ,KJ,QA,QK,QJ,JA,JK,JQ\}
\end{equation*}
In this notation, \(AK\) means the ace is drawn first, and the king second, while \(KA\) means the king is first and the ace is second.
An event is a set of outcomes. For example, the events "get a queen" and "get a king on the first draw" are
\begin{align*}
\text{"get a queen"} \amp = \{QA,QK,QJ,AQ,KQ,JQ\}\\
\text{"get a king on draw 1"} \amp = \{KA,KQ,KJ\}.
\end{align*}
Another example is the event "get the hand ace-king". In a "hand", it does not matter what order the cards are dealt in, so we have
\begin{equation*}
\text{"get the hand ace-king"} = \{AK,KA\}.
\end{equation*}
To say that two events \(E\) and \(F\) "both happen" can be rephrased "\(E \AND F\)". To say that "at least one of the events \(E,F\) happens" can be rephrased "\(E \OR F\)". For example, for the events
\begin{gather*}
E = \text{"get an ace"} = \{AK,AQ,AJ,KA,QA,JA\}\\
F = \text{"get a king"} = \{KA,KQ,KJ,AK,QK,JK\}
\end{gather*}
we have the following.
\begin{align*}
E \AND F \amp = \text{"get an ace and get a king"} = \{AK,KA\}\\
E \OR F \amp = \text{"get an ace or get a king"} = \{AK,AQ,AJ,KA,QA,JA,KQ,KJ,QK,JK\}
\end{align*}
[Comment for students who know about intersection and union of sets: the event "\(E \AND F\) " is the same thing as the intersection of the sets \(E,F\text{,}\) and the event "\(E \OR F\) " is the same thing as the union of the sets \(E,F\text{.}\)]
We will use the correct terms "outcome" and "event" instead of the vague term "thing" used in the text in Ch 13.
We will write \(P(E)\) to denote the probability for an event \(E\text{.}\) For example, in the game of 2 draws, with replacement, from the box \(\{A,K,Q,J\}\text{,}\) the probability of the single outcome \(AK\) is
\begin{equation*}
P(AK) = 1/12 \approx 8.3\%
\end{equation*}
because there are 12 equally likely outcomes for the 2 draws. The probability for getting the hand ace-king is
\begin{equation*}
P(\text{get the hand ace-king}) = 2/12 \approx 16.7\%
\end{equation*}
because there are 2 equally likely outcomes in the event "get the hand ace-king", and there are 12 equally likely outcomes in all for the 2 draws. The probability of getting a queen is
\begin{equation*}
P(\text{get a queen}) = 6/12 = 50\%
\end{equation*}
because there are 6 equally likely outcomes in the event "get a queen". The probability of getting king on the first draw is
\begin{equation*}
P(\text{get a king on draw 1}) = 3/12 = 25\%
\end{equation*}
because there are 3 equally likely outcomes in the event "get a king on draw 1".
The textbook does not give a definition of the term conditional probability; instead, they explain by example in Ch 13 Sec. 2 (p.226). Here is the definition: The conditional probability of event \(E\) given that event \(F\) has happened, denoted \(P(E|F)\text{,}\) is
\begin{equation*}
P(E|F) = \frac{P(E \AND F)}{P(F)}
\end{equation*}
where the event "\(E \AND F\)" is the set of all outcomes that belong to both events \(E\) and \(F\text{.}\) For example, in 2 draws with replacement from the box \(\{A,K,Q,J\}\text{,}\) we have
\begin{equation*}
P(\text{get a queen | get a king on draw 1}) = 1/3 \approx 33.3\%
\end{equation*}
because there is 1 outcome, namely \(KQ\text{,}\) that has a \(Q\text{,}\) among the 3 outcomes that have \(K\) on draw 1. We also have
\begin{equation*}
P(\text{get a king on draw 1 | get a queen}) = 1/6 \approx 16.7\%
\end{equation*}
because there is 1 outcome, namely \(KQ\text{,}\) for which draw 1 is a \(K\text{,}\) among the 6 outcomes that have a \(Q\text{.}\)
The definitions of independent and dependent events (box on p.230) should be replaced by the following: "Two events \(E,F\) are independent if \(P(F|E) = P(F)\text{.}\) Otherwise, the events \(E,F\) are dependent." Or in words, "Two events \(E,F\) are independent if the probability of \(F\) given that \(E\) has happened is equal to the probability of \(F\text{.}\) Otherwise, the events \(E,F\) are dependent."
