In 2012, 1,664,479 students took the SAT exam. The distribution of scores in the verbal section of the SAT had a mean
µ = 496 and a standard deviation
σ = 114. Let
X = a SAT exam verbal section score in 2012. Then
X ~
N (496, 114).
Find the
z -scores for
x1 = 325 and
x2 = 366.21. Interpret each
z -score. What can you say about
x1 = 325 and
x2 = 366.21?
The
z -score for
x1 = 325 is
z1 = –1.14.
The
z -score for
x2 = 366.21 is
z2 = –1.14.
Student 2 scored closer to the mean than Student 1 and, since they both had negative
z -scores, Student 2 had the better score.
Suppose
x has a normal distribution with mean 50 and standard deviation 6.
About 68% of the
x values lie between –1
σ = (–1)(6) = –6 and 1
σ = (1)(6) = 6 of the mean 50. The values 50 – 6 = 44 and 50 + 6 = 56 are within one standard deviation of the mean 50. The
z -scores are –1 and +1 for 44 and 56, respectively.
About 95% of the
x values lie between –2
σ = (–2)(6) = –12 and 2
σ = (2)(6) = 12. The values 50 – 12 = 38 and 50 + 12 = 62 are within two standard deviations of the mean 50. The
z -scores are –2 and +2 for 38 and 62, respectively.
About 99.7% of the
x values lie between –3
σ = (–3)(6) = –18 and 3
σ = (3)(6) = 18 of the mean 50. The values 50 – 18 = 32 and 50 + 18 = 68 are within three standard deviations of the mean 50. The
z -scores are –3 and +3 for 32 and 68, respectively.
Try it
Suppose
X has a normal distribution with mean 25 and standard deviation five. Between what values of
x do 68% of the values lie?
between 20 and 30.
From 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let
Y = the height of 15 to 18-year-old males in 1984 to 1985. Then
Y ~
N (172.36, 6.34).
About 68% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________, respectively.
About 95% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________ respectively.
About 99.7% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________, respectively.
About 68% of the values lie between 166.02 and 178.7. The
z -scores are –1 and 1.
About 95% of the values lie between 159.68 and 185.04. The
z -scores are –2 and 2.
About 99.7% of the values lie between 153.34 and 191.38. The
z -scores are –3 and 3.
Try it
The scores on a college entrance exam have an approximate normal distribution with mean,
µ = 52 points and a standard deviation,
σ = 11 points.
About 68% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________, respectively.
About 95% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________, respectively.
About 99.7% of the
y values lie between what two values? These values are ________________. The
z -scores are ________________, respectively.
About 68% of the values lie between the values 41 and 63. The
z -scores are –1 and 1, respectively.
About 95% of the values lie between the values 30 and 74. The
z -scores are –2 and 2, respectively.
About 99.7% of the values lie between the values 19 and 85. The
z -scores are –3 and 3, respectively.
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Source:
OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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