Math 1070-1, (Practice) Midterm 2 Spring 2003

Math 1070-1, (Practice) Midterm 2
Spring 2003
The actual exam brochure includes a page of questions that includes a “formula sheet,” two pages
of normal tables, and 5 pages of binomial tables.
1. A selective college would like to have an entering college of 1500 students. Because not all
students who are offered admission accept, the college admits more than 1500 students. Past
experience shows that about 55% of the students admitted will accept. The college decides
to admit 1600 students. Assume that students make their decisions independently from one
another. What is the probability that more than 1500 students accept?
2. A multi-year analysis has shown that the computer chips produced by a certain factory are
defective only 2% of the time. If you take an independent sample of 5 computer chips, what
is the probability that somewhere between 1 to 3 of the chips in your sample are defective?
3. The distribution of annual returns on common stocks is roughly symmetric but extreme
observations are more frequent than in a normal distribution. Because the distribution is
not strongly nonnormal, the mean return over even a moderate number of years is close to
normal. In the long run, annual real returns on common stocks have varied with a mean of
about 10% and a standard deviation of about 30%. Andrew plans to to retire in 40 years
and is considering investing in stocks. Assuming that the past patterns of variation continue,
what is the probability that the mean annual return on common stocks over the next 40 years
will be less than 9%?
4. A utility company serves 100, 000 households. As part of a survey of customer attitudes, they
take a random sample 1, 000 of these households.
(a) The average number of T.V. sets in the sample households turns out to be 2.1 and the
standard deviation is 1.2. If possible, find an 75%-confidence interval for the average
number of T.V. sets in all 100, 000 households. If this is not possible, explain why not.
(b) Of the said 1, 000 households in the survey, 900 have at least one television set. If
possible, find a 95%-confidence interval for the percentage of all the 100, 000 households
with at least one T.V. set. If this is not possible, explain why not.
Useful Formulas
• Sample average.
µX̄ = µ
• Binomial count X.
µX = np
σX̄ = √ .
σX =
• Sample proportion.
µp̂ = p
σp̂ =
np(1 − p).
p(1 − p)