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1.
Psychology 281 (001)
1.
Assignment #8
November 19, 2007
A bag contains 12 gambling chips: 8 of these chips have a zero marked on them, the
other 4 chips have a one marked on them. Another bag also contains 12 chips: 4 marked
zero and 8 marked one. Players randomly select one chip from each bag and their score
is the larger of the two numbers on the chips that they select.
a) Present the probability distribution of the scores that could be achieved by a player in
one game.
b) Compute the mean (µx) and standard deviation (σx) of this distribution.
c) Present the sampling distribution of the total score that a player could achieve in
three games, assuming chips are returned to their bags after each game.
d) Present the sampling distribution of the mean score that could be achieved by a
player in three games, assuming chips are returned to their bags after each game.
e) Compute the mean (µx) and standard deviation (σx) of the sampling distribution in
part 'd'. Confirm that µx = µx and that σx = σx/ n.
f) What is the mean and standard deviation of the sampling distribution of the mean
score that could be achieved by a player in 35 games?
2.
A wine producer is trying to decide if it is time to harvest his grapes. He sends out two
of his pickers to collect several bunches of grapes in order to measure the sugar content
of each bunch. If the mean sugar content of both the first and the second picker's
bunches exceeds 24.0o BRIX, the producer will harvest, otherwise he will wait. The
first picker collects 50 bunches and the second picker collects 100 bunches. The
bunches are chosen randomly and independently. If the true mean sugar content of all
grape bunches in the vineyard is 23.75o with a variance of .781, what is the probability
that the harvest will begin?
3.
Once upon a time an evil king decided his subjects might not be paying enough taxes.
Since the average yearly income in his kingdom was 2000 drotneys he decided his
subjects should pay an average of 1000 drotneys in taxes (see why I said he was evil!).
The king sent 2 messengers forth: one to ask 100 people, the other to ask 400 people
what they pay in taxes. They were to each report the average of the sample they took.
If both reported an average of 1000 or over, the king felt he would be satisfied. If
neither did he would immediately raise taxes and the issue would be settled. If only one
reported a mean of 1000 or more he would send out a third messenger to sample 900
people to break the tie.
Suppose that in fact the average tax is 1020 with a standard deviation of 300.
a) What is the probability that the king would be satisfied after his first two messengers
reported?
2.
b) What is the probability that the king would be satisfied, but not until his third
messenger reported?
4.
Paranoid schizophrenics often manifest two different types of problems:
depression and trembling. Following is the probability distribution for a discrete
random variable, X: the number of these problems that a paranoid schizophrenic
might experience:
X
P(X)
0
1
2
.35
.47
.18
a) Generate the sampling distribution for the mean number of these
that random samples of two paranoid schizophrenics might experience.
problems
b) What are the mean and standard deviation of the sampling distribution of the mean
number of these problems that samples of 40 paranoid schizophrenics might
experience?
c) What is the probability that the total number of these problems in a random sample
of 40 paranoid schizophrenics will exceed 36?
5.
In 1979, a random sample of 200 families in a large city was found to spend an average
of $85.44 per week on food, with a standard deviation of $8.12.
a) Construct a 90% confidence interval to estimate the mean weekly expenditure on
food in the entire population of this city in 1979.
b) Construct a confidence interval to estimate the mean weekly expenditure on food in
the entire population of this city in 1979 using α ≤ .01.
Answers: 1b) µx = .78, σx = .414 1e) µx = .78, σx = .239 1f) µx = .78, σx = .07
2. .0000524
3a) .6799 3b) .2903
4b) .83, .1119 4c) .2643
5a) 84.495 to 86.385 5b) 83.962 to 86.918