Page 68 - 2021-bfw-SPA-4e-TE-sample.indd

Chapter 2 Main Points 149

Normal Distributions entry for each z-score is the area under the curve
■ Some distributions of quantitative data can be to the left of z.
modeled by symmetric, single-peaked, bell-shaped ■ You can use Table A or technology to determine
density curves called normal curves. area for given values of the variable or the value
that corresponds to a given percentile in any nor-
■ Any normal distribution is completely specified
by two numbers: its mean µ and standard devia- mal distribution. Main Points
tion σ . The mean is the center of the curve, and σ ■ To find the area in a normal distribution corre-
is the distance from µ to the change-of-curvature sponding to given values:
points on either side. Step 1: Draw a normal distribution with the hori-
zontal axis labeled and scaled using the mean and
standard deviation, the boundary value(s) clearly
identified, and the area of interest shaded.
Step 2: Perform calculations—show your work!
Do one of the following:
(C) 2021 BFW Publishers -- for review purposes only.
σ
(i) Standardize each boundary value and use
Table A or technology to find the desired area
under the standard normal curve; or
(ii) Use technology to find the desired area with-
out standardizing. Label the inputs you used
for the applet or calculator.
■ The empirical rule (also called the 68–95–99.7
rule) describes the approximate percent of values Be sure to answer the question that was asked.
in any normal distribution that fall within 1, 2, ■ To find the value in a normal distribution corre-
and 3 standard deviations of the mean. sponding to a given percentile (area):
■ To assess normality for a given set of quantita- Step 1: Draw a normal distribution with the hori-
tive data, we first observe the shape of a dotplot, zontal axis labeled and scaled using the mean and
stemplot, or histogram. Then we check how well standard deviation, the area of interest shaded
the data fit the empirical rule. and labeled, and unknown boundary value clearly
marked.
■ All normal distributions are the same when values Step 2: Perform calculations—show your work!
are standardized. If a quantitative variable can be Do one of the following:
modeled by a normal distribution with mean µ (i) Use Table A or technology to find the value of
and standard deviation σ, we can standardize the z with the appropriate area under the standard
original distribution of x-values using normal curve, then “unstandardize” to trans-
value − mean x − µ form back to the original distribution; or
z = = (ii) Use technology to find the desired area with-
standard deviation σ out standardizing. Label the inputs you used
The standardized values can be modeled using the for the applet or calculator.
standard normal distribution with mean 0 and Be sure to answer the question that was asked.
standard deviation 1. ■ You can find the mean or standard deviation of a
■ Table A in the back of the book gives percentiles normal distribution using one or more percentiles by
for the standard normal distribution. The table solving for the missing value in the z-score formula.

03_StarnesSPA4e_24432_ch02_088_153.indd 149 07/09/20 1:59 PM

CHAPTER 2 • Main Points 149

03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 149 10/11/20 7:49 PM