Download PRACTICE PROBLEMS for Normal Distribution. 1) Suppose that Φ

PRACTICE PROBLEMS for Normal Distribution.
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Suppose that Φ(1.13) = 0.871. If z is a standard normal variable, find the probability that z is less than 1.13.
Suppose that Φ(0.59) = 0.722. If z is a standard normal variable, find the probability that z > 0.59.
In the standard normal distribution, what value z separates the rest in the upper 5 th percentile?
The grades on a chemistry midterm are normally distributed with a mean of 60 and a standard deviation of
12. Jim scored 42. Find the value z corresponding to Φ(z) in order to detarmine the proportion of students
who scored lower than Jim.
5) Human body temperatures have a mean of 98.3° F and a standard deviation of 0.6° F. Express the probability
that a patient got his body temperature of 99.5° F or higher in terms of Φ(z).
6) The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a
standard deviation of 10. Find the area of the shaded region by using the calculated value of Φ(0.1) = 0.54,
Φ(1) = 0.841, or Φ(1.1) = 0.864.
7) Suppose that in 2004, the verbal portion of the Scholastic Aptitude Test (SAT) had a mean score of µ = 500 and
a standard deviation of σ = 100, while in the same year, the verbal exam from the American College Testing
Program (known as ACT) had a mean of µ = 21.0 and a standard deviation of σ = 4.7. Two friends, Mike and
Tom, applying for college took the tests, Mike scoring 650 on the SAT and Tom scoring 30 on the ACT. Which
of these students, Mike or Tom, scored higher among the population of students taking the relevant test?
Justify your answer.
8) Assuming a normal distribution with mean µ and standard deviation σ, what value z separates the interval of
µ±zσ for 90% from the rest of 10% outside?
9) Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation
of 10. Find the probability that a randomly selected adult has an IQ between 90 and 120 by using the
calculated values from Φ(−1) = 0.159, Φ(1) = 0.841, Φ(1.2) = 0.885, or Φ(2) = 0.977.
10) The scores on the Graduate Management Admission Council’s GMAT examination are normally distributed with a mean
of 530 and a standard deviation of 100. What is the probability of an individual scoring above 500 on the GMAT? Use
the calculated values from Φ(−0.3) = 0.382, Φ(0.3) = 0.618, Φ(3) = 0.999.
11) Continue from the previous question, what is the interval on the GMAT in order to contain 95% of the scores?
12) An unknown distribution has a mean of 90 and a standard deviation of 20. Samples of size n = 25 are drawn randomly
from the population. Find the probability that the sample mean is between 85 and 92. Use the calculated values from
Φ(−1.25) = 0.106, Φ(-0.25) = 0.401, Φ(0.1) = 0.54, Φ(0.5) = 0.691.
13) The annual precipitation amounts in a certain mountain range are normally distributed with a mean of 107
inches, and a standard deviation of 12 inches. What is the probability that the mean annual precipitation
during 36 years will exceed 110 inches? What is the standard deviation of mean annual snowfall during those 36
years? Use the calculated values from Φ(0.25) = 0.599, Φ(1.5) = 0.933, Φ(2) = 0.977, Φ(3) = 0.999.
Answer key
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0.871
0.278
1.645
−1.5
1 − Φ(2)
0.841
Tom taking the ACT test performed better because the probability 1 − Φ(1.91) of getting better than Tom is “smaller” than the
probability 1 − Φ(1.5) of the same case by Mike.
1.645
0.818
0.618
(334, 726)
0.585
0.067; the standard deviation is 2 inches.