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The Normal Distribution
1. What is the standard normal distribution and why is it important?
2. Consider data that was obtained from a normal distribution with a mean of 6.3 and
a standard deviation of 3. 17. Convert the following to z-scores.
a. 12.31
b. 8.2
c. 2.1
d. 15.8
3. What is the physical meaning of a z-score of 1.96?
4. What is the physical meaning of a z-score of –1.96?
5. Consider the intelligence quotient (IQ) of a person. IQ’s are approximately normal
with a mean of 100 and a standard deviation of 15. If a person were selected at
random,
a. What is the probability that her/his IQ would be below 120?
b. What is the probability that her/his IQ would be above 120?
c. What is the probability that her/his IQ would be between 90 and 105?
d. What is the probability that her/his IQ would be below 90?
6. Consider data that was obtained from a normal distribution with a mean of 6.3 and
a standard deviation of 3.17. Find the 85th percentile.
7. Find the following probabilities:
a. P(-1.5 < z < 1.1)
b. P(0 < z < 6.2)
c. P(-1 < z < 1)
d. P(-2 < z < 2)
e. P(-3 < z < 3)
f. P(-1.96 < z <1.96)
8. Given that a particular normal random variable, x, has a mean of 13 and a
standard deviation of 6.2, find the following probabilities:
a. P(14 < x < 20)
b. P(4 < x < 17.2)
c. P(x < 7)
9. A job satisfaction index score for nurses is normally distributed with a mean of 50
and a standard deviation of 10. What is the probability that a nurse selected at
random has an index score:
a. Higher than 55?
b. Between 47 and 59?
10. A radar unit is used to measure the speed of automobiles on a busy street in
downtown Santa Cruz that has a speed limit of 35 mph. Suppose the speeds of
individual automobiles are normally distributed with a mean of 37 mph.
a. Find the standard deviation if 5% of the automobiles travel faster than 45
mph.
b. Based on the standard deviation you just calculated, find the 85th percentile
for the random variable “automobile speed.”
c. Based on the standard deviation you just calculated, what percentage of
cars travel within 3 mph of the posted speed limit?