Review "Finding the z-Score for a Specified Area," pp. 277--279 of our textbook. Remember that the notation ?? in a sense represents a function: we input an area, or probability, and the output is a z-score -- i.e., a value of the standard normal random variable Z.Without using a calculating tool, can you find the critical value ?0.5? Explain, using properties of the standard normal distribution. [A helpful reference is Key Fact 6.5, p. 274.]A particular medical test for bone mineral density yields "scores" said to have a standard normal distribution. What score separates the bone mineral density of approximately the uppermost 2.3% of the population from everyone else? I.e., what is the critical value ?0.023? Round the z-score to the nearest tenth.In your answer, include a screenshot or photo showing the calculating tool you used. Some possible tools include the followingThe desmos.com calculator -- I've created a demo for this purpose at https://www.desmos.com/calculator/topzivkbad (Links to an external site.); you will just need to adjust the value of ?.Table II of our textbook.A spreadsheet such as MS Excel, LibreOffice Calc, or Google Sheets.A handheld calculator; or...? 2776.2 Areas under the Standard Normal Curve277Finding the z-Score for a Specified AreaSo far, we have used Table II to find areas. Now we show how to use Table II to findthe z-score(s) corresponding to a specified area under the standard normal curve.EXAMPLE 6.8 Finding the z-Score Having a Specified Area to Its LeftDetermine the z-score having an area of 0.04 to its left under the standard normalcurve, as shown in Fig. 6.19(a).FIGURE 6.19Finding the z-score havingan area of 0.04 to its leftArea = 0.04.Area = 0.04.-3 -2 1-10 12 3-3 -2 1-10 12 3z = ?Z =-1.75(a)(b)Solution Use Table II, a portion of which is given in Table 6.3.TABLE 6.3Areas under the standard normal curveSecond decimal place in z0.090.080.070.060.050.040.030.020.010.000.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 –1.90.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359| –1.80.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446| –1.70.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 || –1.60.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668|| –1.5Micro 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287| –1.90.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359| –1.80.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 –1.70.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548| –1.60.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668 –1.5Youtry it!Search the body of the table for the area 0.04. There is no such area in the table,so use the area closest to 0.04, which is 0.0401. The z-score corresponding to thatExercise 6.73 area is – 1.75. Thus the z-score having area 0.04 to its left under the standard normalon page 280 curve is roughly – 1.75, as shown in Fig. 6.19(b).The previous example shows that, when no area entry in Table II equals the onedesired, we take the z-score corresponding to the closest area entry as an approximationof the required z-score. Two other cases are possible.If an area entry in Table II equals the one desired, we of course use its correspond-ing z-score. If two area entries are equally closest to the one desired, we take the meanFIGURE 6.20The za notationArea = a of the two corresponding z-scores as an approximation of the required z-score. Both ofthese cases are illustrated in the next example.Finding the z-score that has a specified area to its right is often necessary. We haveto make this determination so frequently that we use a special notation, za.O ZaDEFINITION 6.3The za NotationThe symbol za is used to denote the z-score that has an area of a (alpha) toits right under the standard normal curve, as illustrated in Fig. 6.20. Read "za"as "z sub a" or more simply as "z a."

Given: Area having the z-score to its left under the standard normal curve i.e. Pz<Z=0.04

Answered: EXAMPLE 6.8 Finding the z-Score Having…

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Review "Finding the z-Score for a Specified Area," pp. 277--279 of our textbook. Remember that the notation ?? in a sense represents a function: we input an area, or probability, and the output is a z-score -- i.e., a value of the standard normal random variable Z.

Without using a calculating tool, can you find the critical value ?0.5? Explain, using properties of the standard normal distribution. [A helpful reference is Key Fact 6.5, p. 274.]
A particular medical test for bone mineral density yields "scores" said to have a standard normal distribution. What score separates the bone mineral density of approximately the uppermost 2.3% of the population from everyone else? I.e., what is the critical value ?0.023? Round the z-score to the nearest tenth.
In your answer, include a screenshot or photo showing the calculating tool you used. Some possible tools include the following
- The desmos.com calculator -- I've created a demo for this purpose at https://www.desmos.com/calculator/topzivkbad (Links to an external site.); you will just need to adjust the value of ?.
- Table II of our textbook.
- A spreadsheet such as MS Excel, LibreOffice Calc, or Google Sheets.
- A handheld calculator; or...?

277
6.2 Areas under the Standard Normal Curve
277
Finding the z-Score for a Specified Area
So far, we have used Table II to find areas. Now we show how to use Table II to find
the z-score(s) corresponding to a specified area under the standard normal curve.
EXAMPLE 6.8 Finding the z-Score Having a Specified Area to Its Left
Determine the z-score having an area of 0.04 to its left under the standard normal
curve, as shown in Fig. 6.19(a).
FIGURE 6.19
Finding the z-score having
an area of 0.04 to its left
Area = 0.04.
Area = 0.04.
-3 -2 1-1
0 1
2 3
-3 -2 1-1
0 1
2 3
z = ?
Z =-1.75
(a)
(b)
Solution Use Table II, a portion of which is given in Table 6.3.
TABLE 6.3
Areas under the standard normal curve
Second decimal place in z
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 –1.9
0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359| –1.8
0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446| –1.7
0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 || –1.6
0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668|| –1.5
Micro

0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287| –1.9
0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359| –1.8
0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 –1.7
0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548| –1.6
0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668 –1.5
You
try it!
Search the body of the table for the area 0.04. There is no such area in the table,
so use the area closest to 0.04, which is 0.0401. The z-score corresponding to that
Exercise 6.73 area is – 1.75. Thus the z-score having area 0.04 to its left under the standard normal
on page 280 curve is roughly – 1.75, as shown in Fig. 6.19(b).
The previous example shows that, when no area entry in Table II equals the one
desired, we take the z-score corresponding to the closest area entry as an approximation
of the required z-score. Two other cases are possible.
If an area entry in Table II equals the one desired, we of course use its correspond-
ing z-score. If two area entries are equally closest to the one desired, we take the mean
FIGURE 6.20
The za notation
Area = a of the two corresponding z-scores as an approximation of the required z-score. Both of
these cases are illustrated in the next example.
Finding the z-score that has a specified area to its right is often necessary. We have
to make this determination so frequently that we use a special notation, za.
O Za
DEFINITION 6.3
The za Notation
The symbol za is used to denote the z-score that has an area of a (alpha) to
its right under the standard normal curve, as illustrated in Fig. 6.20. Read "za"
as "z sub a" or more simply as "z a."

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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