Download Section 2.5 Normal Distributions

Section 2.5 The Normal Distribution
Statistics
(Problem numbers refer to those on pages 92 – 95 of the text)
P32.) Find the percentage of values below each given z-score in a standard
normal distribution.
a.) -2.23
b.) -1.67
c.) -0.40
d.) 0.80
P33.) Find the z-score that has the given percentage of values below it in a
standard normal distribution.
a.) 32%
b.) 41%
c.) 87%
d.) 94%
P34.) What percentage of values in a standard normal distribution fall between:
a.) -1.46 and 1.46?
b.) -3 and 3?
P35.) For a standard normal distribution, what interval contains
a.) the middle 90% of z-scores?
b.) the middle 95% of z-scores?
P36.) Refer to the table in the example on page 87 of the text.
a.) California had 196 deaths from heart disease and 154 deaths from
cancer per 100,000 residents. Which rate is more extreme compared
to other states? Why?
c.) Colorado had an unusually low rate of heart disease, 143 deaths per
100,000 residents. Hawaii had an unusually low rate of cancer, 156
deaths per 100,000 residents. Which is more extreme?
E59.) What percentage of values in a standard normal distribution fall
a.) below a z-score of 1.00?
below a z-score of 2.53?
b.) below a z-score of -1.00?
below a z-score of -2.53?
c.) above a z-score of -1.5?
d.) between z-scores of -1 and 1?
E61.) Standardizing. Convert each of these values to standard units, z (Do not
use a calculator. These are meant to be done in your head.)
a.) x = 12, mean 10, SD 1
b.) x = 12, mean 10, SD 2
c.) x = 12, mean 9, SD 2
d.) x = 12, mean 9, SD 1
e.) x = 7, mean 10, SD 3
f.) x = 5, mean 10, SD 2
E62.) Unstandardizing. Find the value of x that was converted to the given zscore.
a.) z = 2, mean 20, SD 5
b.) z = -1, mean 25, SD 3
c.) z = -1.5, mean 100, SD 10
d.) z = 2.5, mean -10, SD 0.2
E63.) SAT I critical reading scores are scaled so that they are approximately
normal, with mean about 505 and standard deviation about 111.
a.) Find the probability that a randomly selected student has an SAT I
critical reading score
i.) between 400 and 600
ii) over 700
iii) below 450
b.) What SAT I critical reading scores fall in the middle 95% of the
distribution?
E64.) SAT I math scores are scaled so that they are approximately normal, with
mean about 511 and standard deviation about 112. A college wants to
send letters to students scoring in the top 20% on the exam. What SAT I
math score should the college use as the dividing line between those who
get letters and those who do not?
P37.) The heights of 18- to 24- year-old males in the United States are
approximately normal, with mean 70.1 in. and standard deviation 2.7 in.
The heights of 18- to 24-year-old females are also approximately
normally distributed and have mean 64.8 in. and standard deviation 2.5
in.
a.) Estimate the percentage of US males between 18 and 24 who are 6 ft
(72 inches) tall or taller.
b.) How tall does a US woman between 18 and 24 have to be in order to
be at the 35th percentile of heights?
P38.) Refer to the table in the example on page 87.
a.) The middle 90% of the states’ death rates from heart disease fall
between what two numbers?
b.) The middle 68% of death rates from heart disease fall between what
two numbers?
P39.) Refer to the information in P37.
For the following choices:
A. A male who is 79 in. tall.
B. A female who is 68 in. tall.
C. A male who is 65 in. tall.
D. A female who is 65 in. tall.
Which of the heights for the above choices are outside the middle 95% of
the distribution?
__________________________________________
Which of the heights for the above choices are outside the middle 99% of
the distribution?
__________________________________________
E67.) Puzzle Problems. Problems that involve computations with the normal
distribution have four quantities: mean, standard deviation, value x, and
proportion P below value x. Any three of these values are enough to
determine the fourth. Think of each row in this table as little puzzles, and
find the missing value in each case. This is not the sort of thing you are
likely to run into in practice, but solving the puzzles can help you become
more skilled at working with the normal distribution.
Mean SD x Proportion
3
1 2
-a10
2 -b0.187
-c3 6
0.09
10 -d- 12
0.60
a.) is _________
b.) is _______
c.) is _______
d.) is _____
E68.) More puzzle problems. In each row of this table, assume the distribution
is normal. Knowing any two of the mean, standard deviation, Q1, and Q3
is enough to determine the other two. Complete the table.
Mean
10
-c-e10
SD
5
-d10
-g-
Q1 Q3
-a- -b120 180
100 -f-h- 11
a.) ______
b.) ______
c.) ______
d.) ______
e.) ______
f.) ______
g.) ______
h.) ______
E69.) ACT scores are approximately normally distributed, with mean 18 and
standard deviation 6. Without using your calculator, roughly what
percentage of scores are between:
a.) 12 and 24?
b.) 6 and 30?
c.) above 24?
d.) below 24?
e.) above 6?
f.) below 6?
E71.) The heights of 18- and 24-year-old males in the United States are
approximately normally distributed with mean 70.1 in. and a standard
deviation 2.7 in.
a.) If you select a US male between ages 18 and 24 at random, what is
the approximate probability that he is less than 68 in. tall?
b.) There are roughly 13 million 18- to 24-year-old males in the US.
About how many are between 67 and 68 in. tall?
c.) Find the height of 18- to 24-year-old males that falls at the 90th
percentile.
E73.) The British monarchy. Over the 1200 years of the British monarchy, the
average reign of kings and queens has lasted 18.5 years, with a standared
deviation of 15.4 years.
a.) What can you say about the shape of the distribution based on the
information given?
b.) Suppose you made the mistake of assuming a normal distribution.
What fraction of the reigns would you estimate lasted a negative
number of years?
c.) Use your work in part b to suggest a rough rule for using the mean
and standard deviation of a set of positive values to check whether it
is possible that a distribution might be approximately normal.