Normal Distributions
Worksheet 1
Open the interactive
version of the normal tables. (Opens
a separate browser window.)
Obtain some pictures
of normal curves. Print them out
and use them to help solve problems.
Exercise I
Part A
The playing time X of jazz CDs has the normal
distribution with mean 52 and standard deviation 7; the N(52, 7)
distribution.
 According to the 689599.7 rule, what percentage of jazz
CDs play between 45 and 59 minutes?
 What is the relative frequency of jazz CDs with playing
time X less than 40 minutes (40 minutes is a typical
playing time for an LP record)? That is, find the
relative frequency of the event X < 40.
 What is the relative frequency of jazz CDs with playing
time X exactly 45 minutes?
 What is the relative frequency of jazz CDs with playing
time over 1 hour?
Part B
The playing time X of classical CDs has the normal
distribution with mean 54 and standard deviation 5; the N(54, 5)
distribution.
 A density curve has 3 important features: shape, center
and spread. Compare each of these features for the
distributions given in problems 1 and 2.
 What is the relative frequency of classical CDs with
playing time X less than 40 minutes? That is, find the
relative frequency of the event X < 40.
 What relative frequency of classical CDs have playing
time over 1 hour?
 What is the relative frequency of classical CDs with
playing time between 45 and 59 minutes?
Exercise II
SAT (combined) scores of collegebound seniors in high school
has the normal distribution with mean 1050 and standard deviation
150.
 What is the relative frequency collegebound seniors who
have SAT score X less than 756?
 Find the value x such that the 0.025 of all
seniors have SAT score below x. (Hint: see part
a.). 2.5% of all seniors would have SAT below this value.
 Find the value x such that the 0.20 of all
seniors have SAT score below x.
 What are the 2.5^{th} and 20^{th}
percentiles of this distribution?
 What is the 97.5^{th} percentile?
 Use your results to form a 95% probability interval for
the SAT score of a collegebound senior.
Exercise III
A small company records budgeted and actual expenses for each
research project it takes on. For instance, a project is budgeted
for $100,000; later the actual cost for the project is found to
be $103,428. This is a difference of $3,428 or 3.428% over budget
(projects finishing under budget are recorded as negatives). You
work in accounting where the percentage over/underbudgeted (the
"budget discrepancy") has historically been modeled
with the normal distribution, mean 1.53% and standard deviation 1.06%.
 There's a variable X here. Describe X.
 What is the relative frequency of projects that come in
under budget?
 Find the 99^{th} percentile of budget
discrepancies. What percentage of budgets exceed this
amount?
 You examine a year's worth of recent project data: 24 of
498 projects finished with a budget discrepancy over 4%.
Does this information agree with the answer of part c?
Part A

 68%
 0.0436
 0
 0.1271

Part B

 They have the same shape (bellshaped or normal).
The distribution of playing times for classical
CDs has center to the right of (above), and has
less spread than, the distribution of playing
times for jazz CDs.
 0.0026
 0.1151
 0.8054

 0.0250
 756
 924 (20% of all seniors have SAT below 924)
 924 and 756 respectively
 1344
 (756, 1344)
 X is the percentage budget discrepancy.
 0.0749 (about 7.5% of all jobs).
 4%; 1
 12/498 = 4.08%. From part c we see that about 1% of all
projects should come in at over 4% budget
discrepancy. The model (normal, mean 1.53%, s.d. 1.06%)
projects a result that is at odds with actual practice;
therefore the model should be revised.