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5-2
The Standard Normal Distribution
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FIGURE 5-11 Finding z
Scores
Area 0. 025
Area 0. 025
z 1. 96
0
z 1. 96
To find this z score,
locate the cumulative
area to the left in Table A–2 . Locate 0. 975 in the body of Table A–2 .
The examples in this section were contrived so that the mean of 0 and the
standard deviation of 1 coincided exactly with the parameters of the standard normal distribution. In reality, it is unusual to find such convenient parameters, because typical normal distributions involve means different from 0 and standard
deviations different from 1. In the next section we introduce methods for working
with such normal distributions, which are much more realistic.
Using Technology
STATDISK Select Analysis, Probability Distributions,
Normal Distribution. Enter the z score to find area, or enter the
cumulative area from the left to find the z score.
Minitab
● To find the cumulative area to the left of a z score (as in
Table A-2), select Calc, Probability Distributions,
Normal, Cumulative probabilities, enter the mean of 0
and standard deviation of 1, then click on the Input Constant button and enter the z score.
●
To find a z score corresponding to a known probability,
select Calc, Probability Distributions, Normal, then
select Inverse cumulative probabilities and the option
Input constant. For the input constant, enter the total
area to the left of the given value.
Excel
● To find the cumulative area to the left of a z score
(as in Table A-2), click on fx, then select Statistical,
NORMSDIST, and enter the z score.
●
To find a z score corresponding to a known probability,
select fx, Statistical, NORMSINV, and enter the total
area to the left of the given value.
TI-83 Plus
● To find the area between two z scores, press 2nd, VARS,
2 (for normalcdf), then proceed to enter the two z scores
separated by a comma, as in (left z score, right z score).
●
To find a z score corresponding to a known probability,
press 2nd, VARS, 3 (for invNorm), and proceed to enter
the total area to the left of the value, the mean, and the
standard deviation in the format of (total area to the left,
mean, standard deviation) with the commas included.
5-2 Basic Skills and Concepts
Using a Continuous Uniform Distribution. In Exercises 1–4, refer to the continuous
uniform distribution depicted in Figure 5-2, assume that a class length between 50.0 min
and 52.0 min is randomly selected, and find the probability that the given time is selected.
1. Less than 50.3 min
2. Greater than 51.0 min
3. Between 50.5 min and 50.8 min
4. Between 50.5 min and 51.8 min
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CHAPTER 5
Normal Probability Distributions
Using a Continuous Uniform Distribution. In Exercises 5–8, assume that voltages in a
circuit vary between 6 volts and 12 volts, and voltages are spread evenly over the range
of possibilities, so that there is a uniform distribution. Find the probability of the given
range of voltage levels.
5. Greater than 10 volts
6. Less than 11 volts
7. Between 7 volts and 10 volts
8. Between 6.5 volts and 8 volts
Using the Standard Normal Distribution. In Exercises 9–28, assume that the readings
on the thermometers are normally distributed with a mean of 0° and a standard deviation
of 1.00°C. A thermometer is randomly selected and tested. In each case, draw a sketch,
and find the probability of each reading in degrees.
9. Less than 20.25
10. Less than 22.75
11. Less than 0.25
12. Less than 2.75
13. Greater than 2.33
14. Greater than 1.96
15. Greater than 22.33
16. Greater than 21.96
17. Between 0.50 and 1.50
18. Between 1.50 and 2.50
19. Between 22.00 and 21.00
20. Between 2.00 and 2.34
21. Between 22.67 and 1.28
22. Between 21.18 and 2.15
23. Between 20.52 and 3.75
24. Between 23.88 and 1.07
25. Greater than 3.57
26. Less than 23.61
27. Greater than 0
28. Less than 0
Basis for Empirical Rule. In Exercises 29–32, find the indicated area under the curve of
the standard normal distribution, then convert it to a percentage and fill in the blank. The
results form the basis for the empirical rule introduced in Section 2-5.
29. About _____% of the area is between z 5 21 and z 5 1 (or within 1 standard deviation of the mean).
30. About _____% of the area is between z 5 22 and z 5 2 (or within 2 standard deviations of the mean).
31. About _____% of the area is between z 5 23 and z 5 3 (or within 3 standard deviations of the mean).
32. About _____% of the area is between z 5 23.5 and z 5 3.5 (or within 3.5 standard
deviations of the mean).
Finding Probability. In Exercises 33–36, assume that the readings on the thermometers
are normally distributed with a mean of 0° and a standard deviation of 1.00°. Find the
indicated probability, where z is the reading in degrees.
33. P(21.96 , z , 1.96)
34. P(z , 1.645)
35. P(z . 22.575)
36. P(1.96 , z , 2.33)
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5-3
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Applications of Normal Distributions
Finding Temperature Values. In Exercises 37–40, assume that the readings on the thermometers are normally distributed with a mean of 0° and a standard deviation of 1.00°C.
A thermometer is randomly selected and tested. In each case, draw a sketch, and find the
temperature reading corresponding to the given information.
37. Find P90, the 90th percentile. This is the temperature reading separating the bottom
90% from the top 10%.
38. Find P20, the 20th percentile.
39. If 5% of the thermometers are rejected because they have readings that are too low,
but all other thermometers are acceptable, find the reading that separates the rejected
thermometers from the others.
40. If 3.0% of the thermometers are rejected because they have readings that are too high
and another 3.0% are rejected because they have readings that are too low, find the two
readings that are cutoff values separating the rejected thermometers from the others.
5-2 Beyond the Basics
41. For a standard normal distribution, find the percentage of data that are
a. within 1 standard deviation of the mean.
b. within 1.96 standard deviations of the mean.
c. between m 2 3s and m 1 3s.
d. between 1 standard deviation below the mean and 2 standard deviations above the
mean.
e. more than 2 standard deviations away from the mean.
42. Refer to the graph of the triangular probability distribution of the continuous random
variable x. (See the margin graph.)
a. Find the value of the constant c.
b. Find the probability that x is between 0 and 3.
c. Find the probability that x is between 2 and 9.
P (x)
c
0
0
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5
x
10