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.

  1. According to the 68-95-99.7 rule, what percentage of jazz CDs play between 45 and 59 minutes?
  2. 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.
  3. What is the relative frequency of jazz CDs with playing time X exactly 45 minutes?
  4. 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.

  1. 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.
  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.
  3. What relative frequency of classical CDs have playing time over 1 hour?
  4. What is the relative frequency of classical CDs with playing time between 45 and 59 minutes?

Exercise II

SAT (combined) scores of college-bound seniors in high school has the normal distribution with mean 1050 and standard deviation 150.

  1. What is the relative frequency college-bound seniors who have SAT score X less than 756?
  2. 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.
  3. Find the value x such that the 0.20 of all seniors have SAT score below x.
  4. What are the 2.5th and 20th percentiles of this distribution?
  5. What is the 97.5th percentile?
  6. Use your results to form a 95% probability interval for the SAT score of a college-bound 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/under-budgeted (the "budget discrepancy") has historically been modeled with the normal distribution, mean 1.53% and standard deviation 1.06%.

  1. There's a variable X here. Describe X.
  2. What is the relative frequency of projects that come in under budget?
  3. Find the 99th percentile of budget discrepancies. What percentage of budgets exceed this amount?
  4. 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?

Solutions

Exercise I

Part A
  1. 68%
  2. 0.0436
  3. 0
  4. 0.1271

Part B
  1. They have the same shape (bell-shaped 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.
  2. 0.0026
  3. 0.1151
  4. 0.8054

Exercise II

  1. 0.0250
  2. 756
  3. 924 (20% of all seniors have SAT below 924)
  4. 924 and 756 respectively
  5. 1344
  6. (756, 1344)

Exercise III

  1. X is the percentage budget discrepancy.
  2. 0.0749 (about 7.5% of all jobs).
  3. 4%; 1
  4. 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.