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Lesson 2.4
LESSON OVERVIEW VIDEO The Empirical Rule and
Watch the Lesson 2.4–2.6 overview
video for guidance from the authors on Assessing Normality
teaching the content in these lessons.
Find it by clicking on the link in the L E AR N I N G TAR G E T S
TE-book or by logging into the teachers’
resources on our digital platform. • Use the empirical rule to estimate the proportion of values in a specified
interval in a normal distribution.
• Use the empirical rule to estimate the value that corresponds to a given
LEARNING T AR GET KEY percentile in a normal distribution.
• Use graphical and numerical evidence to determine if a distribution of
The problems in the test bank are quantitative data is approximately normal.
(C) 2021 BFW Publishers -- for review purposes only.
keyed to the learning targets using
these numbers:
Why are normal distributions important in statistics? Here are three reasons:
• 2.4.1 1. Normal distributions are good descriptions for some distributions of real data.
• 2.4.2 Distributions that are often close to normal include:
• 2.4.3 • Scores on tests taken by many people (such as SAT exams and IQ tests)
• Repeated careful measurements of the same quantity (like the diameter of a
tennis ball)
• Characteristics of biological populations (such as lengths of crickets and yields
BELL RINGER of corn)
2. Normal distributions are good approximations to the results of many kinds of
According to the Centers for Disease chance outcomes, like the proportion of heads in many tosses of a fair coin.
Control and Prevention, American 3. Many of the inference methods in Chapters 8 – 11 are based on normal
females aged 20 and over have a mean distributions.
height of 64 inches. Suppose that the
standard deviation of their heights is The Empirical Rule
2.5 inches. How tall would a woman be In Lesson 2.3 we saw that the distribution of Iowa Test of Basic Skills (ITBS) vocab-
,
who is exactly 1 standard deviation taller ulary scores for seventh-grade students in Gary, Indiana, is approximately normal
.
than average? 2 standard deviations with mean µ = 6.84 and standard deviation σ = 1.55 How unusual is it for a Gary
seventh-grader to get an ITBS vocabulary score less than 3.74? The figure shows the
taller? 3 standard deviations taller? 1 normal density curve for this distribution with the area of interest shaded. Note that
standard deviation shorter than average? the boundary value, 3.74, is exactly 2 standard deviations below the mean.
2 standard deviations shorter? 3 standard
deviations shorter?
TEACHING TIP 2.19 3.74 5.29 6.84 8.39 9.94 11.49
ITBS vocabulary score
Don’t let students say a distribution
of data is normal. Strictly speaking,
all distributions of real data are only
approximately normal, never perfectly 118
or exactly normal. While this might
seem like a small detail, it can mean
that a student doesn’t understand
the difference between an idealized 03_StarnesSPA4e_24432_ch02_088_153.indd 118 07/09/20 1:55 PM
mathematical model and real-world
observed data.
118 CHAPTER 2 • Modeling One-Variable Quantitative Data
03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 118 10/11/20 7:45 PM
