SCCC 312A – ProSeminar in Statistics
E. A. Peña’s
Class, Spring 2004
Note: Exam will be held in our lab room on Monday, March 29, 2004 from 1:25-3:15. Exam will be designed for approximately 1 hour, but you could use the full lab period if so desired. It will be administered by Mr. Jonathan Quiton and Mr. Jun Han, both doctoral students in the department.
Problem 1: A box contains two coins, one of which is
an ordinary coin (Head on side and Tail on another side), while the other is a
coin with both Heads on each side (a two-headed coin). A coin is randomly
chosen from this box, and without looking at which type of coin it is, this
coin is tossed once. If the coin lands “Head” what is the probability that it
is the two-headed coin? [ANSWER: 2/3]
Problem 2: In a biological study there is a possibility of a genetic mutation occurring in a randomly chosen individual. The probability that the genetic mutation occurs is 0.01. If a mutation occurs, then the probability that the individual will acquire a certain trait (like becoming a “genius”) is 0.90, whereas if the mutation does not occur, the probability that the individual will acquire the trait is 0.40.
a) For a randomly chosen individual, what is the probability that he/she will possess the trait? [ANSWER: .405]
b) Given that the individual possesses the trait, what is the probability that the mutation has occurred for this individual? [ANSWER: .0222]
Problem 3: A tetrahedron is a four-sided “die”. Suppose that two fair tetrahedra (plural for tetrahedron) are rolled, and we let X denote the sum of the number of dots in the outcomes of the two tetrahedra.
a) How many possible outcomes are there in the sample space, and what are the probabilities of each of these outcomes? [ANSWER: 16, 1/16]
b) Construct the probability distribution function of X.
ANSWER: p(2) = 1/16; p(3)=2/16; p(4)=3/16; p(5)=4/16; p(6)=3/16; p(7)=2/16; p(8)=1/16
c) Graph the probability distribution function of X.
d) Find Pr{4 < X < 7}. [ANSWER: .75]
e) Find the mean of X. [ANSWER: 5]
f) Find the variance of X. [ANSWER: 2.5]
g) Find the standard deviation of X. [ANSWER: 1.58]
Problem 4: A man is claiming that he has extrasensory perception (ESP) in that he has a higher probability than ordinary mortals in determining hidden objects. A scientist decides to test this man’s claim and repeats the following experiment ten times:
The scientist has five cards of the same shape, color, and size, but with one of the cards having a red mark. The scientist lays out, face down and in a random fashion, these five cards on a table. Then he asks the man to pick the card with the red mark. They then check if the man picked the right card.
a) If the man does not possess ESP but is just guessing on his choice, what is the probability that he will pick the card with the red mark on any given trial? [ANSWER: 1/5]
b) What is the probability that if he is just guessing, he will get the card with the red mark at least 5 times out of the 10 trials? [ANSWER: .033]
c) If he is just guessing, what are the mean and standard deviation of the random variable denoting the number of times that the man picks the card with the red mark? [Mean = 2; StdDev = 1.265]
d) Supposing that he gets 6 correct guesses/picks, do you think the man really possess ESP? Explain. [Either a very rare event has occurred (the guy is lucky) since P(X > 6) = .007; or he does possess some powers to increase his chances of discovering hidden objects; or the experiment was not done properly to the extent that he might have been able to get info from the card or the face of the scientist! ... though we generally assume that the experiment was done and controlled properly]
Problem 5: Based on past experience, the temperature at noontime on April 1st in a monitoring station at Columbia Metropolitan Airport is known to be normally distributed with mean of 55 degrees Fahrenheit with a standard deviation of 5 degrees Fahrenheit. Suppose that this April 1st at noontime we measure the temperature at this station, and we let X denote the temperature.
a) Find the probability that X exceeds 70 degrees Fahrenheit. [ANSWER: almost 0]
b) Find the probability that X will be between 45 and 50 degrees Fahrenheit. [ANSWER: .1359]
c) What is the highest temperature this April 1st that we could have at noontime in Columbia Airport but still belong to the coldest 5% of all April 1st. [ANSWER: 46.775]
Problem 6: A population has the following probability function: P(X = 1) = .4, P(X = 3) = .2, P(X = 4) = .4.
a) Find the population mean and standard deviation. [ANSWER: 2.6 and 1.3564]
b) Suppose that you consider the experiment of taking a random sample of size n = 2 from this population, and you denote by XBAR the sample mean. Construct the sampling distribution of XBAR, and draw the histogram of this sampling distribution.
Sampling distribution of XBAR:
|
XBAR |
1 |
2 |
2.5 |
3 |
3.5 |
4 |
|
P(XBAR) |
.16 |
.16 |
.32 |
.04 |
16 |
.16 |
c) Determine the mean of the sample mean. [ANSWER: 2.6]
d) Determine the standard error of the sample mean. [ANSWER: .9592]
e) What will become of the shape of the sampling distribution of XBAR if the sample size is increased to n = 30? (Note: of course you should not try to obtain the exact sampling distribution in this case, but you may certainly try, at your own risk!) [ANSWER: By the Central Limit Theorem, the sampling distribution of XBAR will become approximately normal with mean of 2.6 and standard deviation or error of .2476]
Problem 7: The population of income taxes paid by US taxpayers is known to have a very right-skewed distribution with mean of $13,000 and standard deviation of $20,000. Suppose that a random sample of n = 100 households is obtained and the mean income taxes for these 100 households is computed.
a) What is the (approximate) sampling distribution of this sample mean? [ANSWER: The sampling distribution will be approximately normal by virtue of the Central Limit Theorem, and its mean will be $13000 and its standard deviation (error) will be $2000]
b) What is the probability that the sample mean will exceed $18,000? [ANSWER: .0062]
c) What is the probability that the sample mean will be between $12,000 and $14,000? [ANSWER: .3830]
d) Suppose that you are just contemplating on doing your sampling and you want to determine the appropriate sample size in order that the sample mean you obtain will not differ by more than $1,000 from the true mean of $13,000 with probability of 0.95. What is the needed sample size to achieve these requirements? [ANSWER: 1537]