MATH 20095 Mathematics for Business Decisions II Fall 2005
Section 01 Ms. Kracht
General Normal Distributions
1. Three landmarks of baseball achievement are Ty Cobb’s batting average of .420 in 1911, Ted Williams’ .406 in
1941, and George Brett’s .390 in 1980. These batting averages cannot be compared directly because the distribution
of major league baseball batting averages has changed over the years. The distributions are quite symmetric and
reasonably normal. Although the mean batting average has been held roughly constant through rule changes and
the balance between hitting and pitching, the standard deviation has dropped over time. Here are some examples.
Decade Mean Standard Deviation
1910’s .266 .0371
1940’s .267 .0326
1970’s .261 .0317
Let Tbe the random variable representing the batting average of a major league baseball player in the 1910’s. Then
µTand σTare given in the first row of the table.
Let Rbe the random variable representing the batting average of a major league baseball player in the 1940’s. Then
µRand σRare given in the second row of the table.
Let Sbe the random variable representing the batting average of a major league baseball player in the 1970’s. Then
µSand σSare given in the third row of the table.
(a) Write each of the following first using probability notation and then in terms of the appropriate probability
density function (fT,fR, or fS) and then in terms of the cumulative distribution function (FT,FR, or
FS). Then use your calculator to evaluate.
i. the probability that a 1910’s player batted .200 or worse
ii. the probability that a 1940’s player batted .200 or worse
iii. the probability that a 1970’s player batted .200 or worse
iv. the probability that a 1910’s player batted .300 or better
v. the probability that a 1940’s player batted .300 or better
vi. the probability that a 1970’s player batted .300 or better
vii. the probability that a 1910’s player batted between .220 and .280
viii. the probability that a 1940’s player batted between .220 and .280
ix. the probability that a 1970’s player batted between .220 and .280
(b) Compute the standardized batting averages (z-scores) for Cobb, Williams, and Brett to compare how far each
stood above his peers. Who was the best batter?
2. A job-applicant test consisted of three parts: verbal, quantitative, and logical reasoning. Assume that the distribution
of scores for each part of the test is normal with mean and standard deviation given below.
Verbal Quantitative Logical Reasoning
mean 84 118 14
standard deviation 10 18 4
Suppose Holly’s scores were 90 on verbal, 133 on quantitative, and 18 on logical reasoning.
(a) Compute Holly’s standardized scores (z-scores) for each part of the test.
(b) On which part did she perform relatively highest?
(c) On which part did she perform relatively lowest?
(d) If the overall composite score is the mean of the z-scores of the three parts, what is Holly’s composite score?
3. Amy’s z-score on her state reading proficiency test was 1.27. If the average (raw) score was 61 with standard
deviation 4.2, find Amy’s raw score.
