Download TUTORIAL 2 1. A student takes a 20-question, multiple

TUTORIAL 2
1.
A student takes a 20-question, multiple-choice exam with five choices for each question and
guesses on each question. Find the probability of guessing at least 15 out of 20 correctly. Would
you consider this event likely or unlikely to occur? Explain your answer.
Ans: 0.000; the probability is extremely small.
2.
In a survey, 3 of 4 students said the courts show “too much concern” for criminals. Find the
probability that at most 3 out of 7 randomly selected students will agree with this statement.
Explain your answer.
Source: Harper’s Index.
Ans: 0.071
3.
The probability that a certain machine will produce a defective item is 0.20. If a random sample
of 6 items is taken from the output of this machine, what is the probability that there will be 5
or more defectives in the sample? Explain your answer.
Ans: 0.0016
4.
A statistics quiz consists of 10 multiple questions. There are four choices for each question. One
student is unprepared and decides to guess the answers to every question. Assuming 70% is a
passing grade; find the probability that the student will pass the quiz. Interpret your answer.
Ans: 0.0035
5.
30% of pupils in a school travel to school by bus. From a sample of ten pupils chosen at random,
find and interpret the probability that
i. only three travel by bus.
ii. less than half travel by bus.
Ans: ii. 0.2668 ii. 0.8497
6.
Newsweek reported that 60% of young children have blood lead levels that could impair their
neurological development. Assuming that a class in a school is a random sample from the
population of all children at risk, the probability that at least 5 children out of 10 in a sample
taken from a school may have a blood level that may impair development is:
Ans: 0 .84
7.
Suppose 20% of the marbles packed in a box are red in color. Suppose 4 marbles are chosen at
random. Find the probability that
i. two are red.
ii. three are red.
iii. none are red.
Ans: i. 0.1536 ii. 0.0256 iii. 0.4096
8.
If a keyboard operator averages 2 errors per page of newsprint, and if these errors follows
Poisson process, what is the probability that,
i. exactly 4 erors will be found on a given page?
ii. at least 2 error will be found on a given page?
Ans: i. 0.0902 ii. 0.594
9.
10.
On the average, 12 people per hour approach a decorating consultant with questions in a fabric
store. What is the probability that at least three people will approach the consultant with
questions during a 10-minute period?
Ans: 0.3232
The rate at which a particular defect occurs in lengths of plastic film being produced by a stable
manufacturing process is 4.2 defects per 75 meter length. A random sample of the film is
selected and it was found that the length of the film in the sample was 25 meters. What is the
probability that there will be at most 2 defects found in the sample?
Ans: 0.8335
11.
Each 500-meter roll of sheet includes two flaws, on average. A flaw is a scratch which would
affect the use of that segment of sheet steel in the finished product. What is the probability that
a particular 100-meter segment will include no flaw?
Ans: 0.6703
12.
The marketing manager of a company has noted that she usually receives 10 complaint calls
during a week (consisting of five working days), and that the calls occur at random. Let us
suppose that the number of calls during a week follows the Poisson distribution. The probability
that she gets five such calls in one day is:
Ans: 0.0361
13.
A university found that 20% of its students withdraw without completing the introductory
statistics course. Assume that 20 students registered for the course. Find and interpret
i. the probability that two or fewer will withdraw
ii. the probability that exactly four will withdraw
iii. the probability that more than three will withdraw
iv. the expected number of withdrawals.
Ans: i. 0.2061 ii. 0.2182 iii. 0.5886
iv. 4
14.
Suppose that 0.03% of plastic containers manufactured by a certain process have small holes that
render them unfit for use. Let X represent the number of containers in a random sample of 10000
have this defect. Find


ii. P  X  2 
iii. P 1  X  4 
iv. E  x  and Var  x 
i. P X  3
Ans: i. 0.2240
15.
ii. 0.4232 iii. 0.5974 iv. 3, 1.732
The number of messages received by a computer bulletin board is a Poisson random variable with
a mean rate of 8 messages per hour.
i. What is the probability that five messages are receive in a given hour?
ii. What is the probability that 10 messages are received in 1.5 hours?
iii. What is the probability that fewer than three messages are received in one-half hour?
Ans: i. 0.0916 ii. 0.1048 iii. 0.0005
16.
From a large shipment of transistors from a supplier, 1% of the items are known to be defective.
If a sample of 200 transistors is randomly selected, what is the probability that 4 or more
transistors will be defective?
Ans: 0.1429
17.
Last month your company sold 1000 new watches. Past experience indicates that the probability
that a new watch will need repair during its warranty period is 0.002. Compute the probability
that:
i. At least than 5 watches will need warranty work.
ii. At most than 3 watches will need warranty work.
iii. Less than than 7 watches will need warranty work.
Ans: i. 0.0527 ii. 0.8571 iii. 0.9955
18.
The distribution of weights in a large group is approximately normally distributed. The mean is
80 kg and approximately 68% of the weights are between 70 and 90 kg. The standard deviation of
the distribution of weights is equal to?
Ans: 10
19.
The random variable X is distributed N (μ, σ2) and it is known that P(X > 80) = 0.00113 and P(X <
30) = 0.0287. Find the value of μ and σ.
Ans: 49.2036, 10.1071
20.
Marks on a Chemistry test follow a normal distribution with a mean of 65 and a standard
deviation of 12. Approximately what percentages of the students have scores below 50?
Ans: 10.56%
21.
The number of shirts sold in a week by a shop is normally distributed with a mean of 2080 and a
standard deviation of 50. Estimate
i. the probability that in a given week fewer than 2000 shirts are sold
ii. the number of weeks in a year that between 2060 and 2130 shirts are sold
iii. the least number n of shirts such that the probability that more than n are sold in a
given week is less than 0.02.
Ans: i. 0.0548 ii. 26 iii. 2183
22.
The random variable X is normally distributed with a mean of 45. The probability that X is
greater than 51 is 0.288. Find the standard deviation of the distribution.
Ans: 10.7
23.
The average time it takes a group of adults to complete a certain achievement test is 46.2
minutes. The standard deviation is 8 minutes. Assume the variable is normally distributed.
i. Find the probability that a randomly selected adult will complete the test in less than
43 minutes.
ii. Find the probability that, if 50 randomly selected adults take the test, the mean time
it takes the group to complete the test will be less than 43 minutes.
iii. Does it seem reasonable that an adult would finish the test in less than 43minutes?
Explain.
iv. Does it seem reasonable that the mean of the 50 adults could be less than 43 minutes?
Ans: i. 0.3446 ii. 0.0023 iii. Yes iv. Very unlikely
24.
At a large publishing company, the mean age of proofreaders is 36.2 years, and the standard
deviation is 3.7 years. Assume the variable is normally distributed.
i. If a proofreader from the company is randomly selected, find the probability that his
or her age will be between 36 and 37.5 years.
ii. If a random sample of 15 proofreaders is selected, find the probability that the
mean
age of the proofreaders in the sample will be between 36 and 37.5 years.
Ans: i. 0.1589 ii. 0.4961