1. Lie detectors are based on measuring changes in the nervous system. The assumption is that
lying will be reflected in physiological changes that are not under the voluntary control of the
individual. When a person is telling the truth, the galvanic skin response scores have a
distribution that is normal with a mean of 49.4 and a standard deviation of 3.
(a) What is the probability that a person will have a score less than 45?
(b) Suppose a SRS of 36 people is taken, what is the distribution of X ?
(c) What is the probability that the average score will be more than 50?
2. Suppose a sample of 50 items is drawn from a population of manufactured products and the
random variable X is the weight of the item. Prior experience has shown that the weight has
a distribution with mean 6.0 ounces and standard deviation of 2.5 ounces.
(a) What is the probability that the weight of an item randomly selected will be more than 7
ounces?
(b) Using proper notation, what is the distribution of X ?
(c) What is the probability that the manufacturer’s sample has a mean weight between 5.75
and 6.25 ounces?
3. The lifetimes (in months) of ten randomly selected automobile batteries of a particular brand
are:
22
17
20
21
17
23
The manufacturer claims, however, that this particular make of battery has an 18 month
lifetime. Do you believe the manufacturers claim? Hint: compute a 95% confidence
interval and compare with claim.
4. The caffeine content of a random sample of 81 cups of black coffee dispensed by a new
machine is measured. The mean and standard deviation for the sample are 110 mg and 5.0
mg, respectively. The manufacturer of the machine claims that the average caffeine content
per cup is 109 mg. Do you believe that the manufacturer’s claim is valid or invalid?
5. A simple random sample of 852 South Dakotans is taken to determine the proportion p of all
South Dakotans who eat prairie dog. Of the 852 sampled, 126 admit they do.
(a) Identify the sample proportion 𝑝̂ .
(b) Use your answer to (a) to estimate ¾𝑝̂ , the standard error for all sample proportions
for samples of this size.
(c) Suppose we wish to construct a 99% confidence interval for p. Find the critical zscore for this confidence level.
(d) Construct the 99% confidence interval for p based on the given sample.
(e) Given no knowledge of p, how large a sample would I need to take in order to
construct a 99% confidence interval for p with margin of error no great than 0:02?
6. Assume that 2% of all products that come off a company's assembly line are defective. The
company produces thousands each day. A simple random sample of 120 of these products is
drawn.
a. What is the probability that more than 5% of the products are defective?
7. Experience has shown that 1/200 of all CDs produced by a certain machine are
defective. If a quality control technician randomly tests twenty CDs, compute each of
the following probabilities. What is the probability that no more than 2 are defective?
8. A fair coin is tossed 5 times. What is the probability that it lands tails up
exactly 3 times?
9. A board game has a spinner on a circle that has five equal sectors, numbered 1,
2, 3, 4, and 5, respectively. If a player has four spins, find the probability that
the player spins an even number no more than two times on those four spins.
10. A brake inspection station reports that 15% of all cars tested have brakes in
need of replacement pads. For a sample of 20 cars that come to the inspection
station, what is the probability that exactly 3 have defective breaks?
11. The probability that a person recovers from a particular type of cancer
operation is 0.7. Suppose 8 people have the operation. What is the probability
that
a. exactly 5 recover?
b. they all recover?
c. at least one of them recovers?
12. Scores on the SAT-I test are normally distributed with a mean of 1800 and a standard
deviation of 212.
a. What proportion of students who take the test score above 1910?
b. The director of admissions for a junior college in Mississippi wants to admit as many
students as possible, but knows that his college is not right for all students. Therefore, he
decides that he will not admit students who score in the lower 4% or the upper 4% on the
SAT. What are the cutoff scores on the SAT for admission to this college?
13. The manager of a local soft drink bottling company believes that when a new beveragedispensing machine is set to dispense 7 ounces, it in fact dispenses an amount x at random
anywhere between 6.5 and 7.5 ounces inclusive. Suppose x has a uniform probability
distribution.
a) Find the mean and standard deviation for the distribution.
b) What is the probability that for a randomly selected bottle the machine will dispense at
least 6.75 ounces?
c) What is the probability that for a randomly selected bottle the machine will dispense
more than 7.25 ounces?
d) What is the probability that for a randomly selected bottle the amount the machine
dispenses is within one standard deviation of the mean?
14. If heights of adult males are normally distributed with a mean of 70.2 inches and a standard
deviation of 4.2 inches, and 95% of all adult males satisfy a minimum height requirement for
police officers, what is that minimum height requirement?
15. In studying ocean conditions, the Bureau of Fisheries found that for one location, the August
water temperatures (in degrees Fahrenheit) were normally distributed with a mean of 83.6 and
a standard deviation of 2.4. Find the temperature that is exceeded 10% of the time.
16. The amount of money spent weekly on cleaning, maintenance and repairs at a large restaurant
was observed over a long period of time to be approximately normally distributed with mean of
$615 and standard deviation of $42. The amount of money spent weekly on salaries is also
approximately normally distributed with a mean of $5000 with standard deviation of $205. If
the restaurant budgets $6000 for cleaning, maintenance, repairs and salaries, what is the
probability that the actual costs will exceed the budgeted amount?