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140 CHAPTER 2 • Modeling One-Variable Quantitative Data

To get the corresponding ITBS vocabulary score x, use the z-score formula to
TEACHING TIP: “unstandardize”:
Differentiate z = x µ−
σ
For students who have difficulty with x − 6.84
algebra, the process of “unstandardizing” 1.28 = 1.55
to get an ITBS score doesn’t need to 1.28(1.55)6.84 = x
+
involve algebra. Once z =1.28 has 8.824 = x
been obtained, students can find the So the 90th percentile of the distribution of ITBS vocabulary scores for Gary, Indiana,
ITBS score if they can interpret the seventh-graders is 8.824.
It is also possible to find the 90th percentile of either distribution in Figure 2.18
z-score. In this case, the ITBS score is with technology.
1.28 standard deviations greater than
SD = 1-- for review purposes only.
the mean, so we can find our answer by
+
computing 6.841.28(1.55). Of course TECH Finding values from areas in any normal distribution
this is algebraically equivalent to the CORNER
“unstandardizing” method, but it avoids You can use technology to find values from areas in any normal distribution. Let’s use an applet and the TI-83/84 to
the formal algebra that is difficult for confirm our earlier result about the 90th percentile of ITBS vocabulary scores for Gary, Indiana, seventh-graders.
With technology, we can do the calculation using the standard normal distribution or the “unstandardized”
some students. normal distribution with mean 6.84 and standard deviation 1.55.
(i) Using the standard normal distribution: TI-83/84
What is the 90th percentile of the standard 1. Press 2nd VARS (Distr) and choose invNorm(.
normal distribution?
Applet OS 2.55 or later: In the dialog box, enter these values:
area: 0.90, µ:0, σ :1, choose Paste, and then press
1. Launch the Normal Distributions applet at ENTER .
bfwpub.com/spa4e. Older OS: Complete the command invNorm(0.90,0,1)
2. Select the option to “Calculate a value corre- and press ENTER .
sponding to an area.”
3. Enter 0 for the mean and 1 for the standard
deviation. Then click “Plot distribution” to see the
graph of the standard normal distribution.
4. To find the 90th percentile, select “Calculate
boundary value(s) for a left-tail area” from the
pull-down menu, type 0.90 for the area, and click
“Calculate value(s).”
Normal Distributions Note: The most recent TI-84 Plus CE OS has added
(C) 2021 BFW Publishers
Operation: Calculate a value corresponding to an area an option for specifying area in the LEFT, CENTER,
Mean = 0 Plot distribution or RIGHT of the distribution. Choose LEFT in this
case.
(ii) Using the unstandardized normal distribution:

What is the 90th percentile of a normal
distribution with mean µ = 6.84 and standard
deviation σ =1.55?
–3 –2 –1 0 1 2 3
Calculate boundary value(s) for a left-tail area of .90
Calculate value(s) Value = 1.282

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140 CHAPTER 2 • Modeling One-Variable Quantitative Data

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