Download 1 Psychology 281(001) Assignment 1 1. In your own words, describe

1
Psychology 281(001)
Assignment 1
1.
In your own words, describe the difference between:
a)
b)
c)
d)
Quantitative and Qualitative Variables
Discrete and Continuous Variables
Ordered and Unordered Variables
Descriptive and Inferential Statistics
2.
Classify the following variables as either quantitative or qualitative, and as discrete,
continuous, ordered, or unordered:
a)
b)
c)
d)
e)
Number of books in a library
Marital Status, measured as married, divorced, widowed, single
Types of stores found in a shopping center
Height measured numerically
Height measured as tall, average, short
3.
The following are the test scores of a sample of 20 students:
16, 22, 26, 30, 32, 33, 34, 36, 36, 37, 38, 38, 39, 40, 41, 41, 43, 47, 51, 56
a)
b)
c)
d)
e)
What is the median score?
What is the mean score?
What is (are) the mode(s)
Is there any reason to believe that the distribution of scores is skewed? Why or why not?
Use the computing formula (not the definitional formula) to find the variance and
standard deviation of the scores. Confirm your answers by keying the data into your
calculator and using it to compute the mean, standard deviation and variance for you.
4.
Another sample, consisting of 15 students, obtained a mean score of 45.5, a median score
of 44, and a standard deviation of 4.2 on the same test given to the sample in question 3.
a)
b)
What is the grand mean (i.e., the mean of both sample's scores combined)?
Is there any way to compute the "grand median" of these data? If not, why not? If so,
what is its value?
5.
On another test, a sample of 50 students obtained a mean score of 14.3 with a standard
deviation of 2.0. Unfortunately, it turned out that some of the test questions had been
wrongly scored, and when the tests were re-scored it was discovered that every student's
score had been increased by a factor such that the variance of the scores was now 9.0.
Given this, what is the new mean of the scores?
2
Answers:
3.a) 37.5
b) 36.8
c) 36, 38, 41
e) s = 9.254
4.a) 40.53
5.21.45
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Psychology 281(001)
Assignment #2
1.
The distribution of a sample of 100 test scores is mound-shaped and symmetrical, with a
mean of 50 and a variance of 144.
a)
b)
c)
Approximately how many scores are equal to or greater than 74?
What score corresponds to the 75th percentile?
If the lowest and highest scores in this sample are 14 and 89, respectively, what is the
range of the scores in standard deviation units?
Assume that the 100 test scores still have a mean of 50 and a variance of 144, but now
have a strongly negatively skewed distribution. At least how many of the scores fall
between 32 and 68 in this distribution?
d)
2.
A sample of 200 people were given a test. The distribution of the test scores was moundshaped and symmetrical with a mean of 100. One person, whose test score was 125, was
found to be at the 84th percentile.
a)
b)
c)
d)
What is the Z score of a person whose test score is 70?
Approximately how many people obtained scores greater than 135?
Approximately how many people obtained scores between 65 and 85?
Suppose the distribution was not mound-shaped but was skewed. At least how many
people would be expected to obtain scores between 65 and 135?
If each person's score was multiplied by 3, what would the test score be of a person who
scored 1.2 standard deviation units below the mean?
e)
3.a)
b)
The mean IQ of 200 patients in a psychiatric hospital is 91, with a variance of 16, and the
distribution is highly negatively skewed. Between what two IQ scores would we expect
to find at least 160 of the patients falling?
If the distribution was mound-shaped and symmetrical, with the same mean and standard
deviation as in part 'a', approximately how many patients would have IQ scores at or
below the 5th percentile or above a score of 97?
4.
Charles Darwin recorded the heights, in inches, of a strange (see part b) species of shrubs
that he discovered on a remote island as follows: "I measured 800 plants. The distribution
of their heights is mound-shaped and symmetrical and has a variance of 9. One plant,
whose height was 4 inches, falls at the 33rd percentile of the distribution".
a)
Approximately how many of the 800 plants in this sample are between 8.5 and 10.5
inches tall?
Approximately what percentage of the sample of 800 plants would be expected to have Z
scores above -1.28? (N.B. For this question, assume that these strange plants can grow
upwards or downwards, above or beneath the earth, such that their "heights" can be
positive or negative).
b)
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Answers:
1.a) 2.5 or 3
b) 58.82 or 59
c) 6.25
d) 55.55 or 56
2.a) -1.2
b) 21.2 or 21
c) 38
d) 98
e) 210
3.a) 82 to 100
b) 28.5 or 29
4.a) 72
b) 87.78%
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Psychology 281(001)
Assignment #3
1.
If a student in a statistics course gets an average score of 75% on a number of quizzes,
the probability that she will get an A on the midterm is .80. Based on the quiz scores she
has received so far, the probability that she will not achieve a 75% average is .15. Also,
the probability that she will not get an A on the midterm given that she fails to achieve a
75% average on quizzes is .70.
a)
What is the probability that the student will achieve a 75% average on quizzes and will get
an A on the midterm?
What is the probability that the student will not achieve a 75% average on quizzes and will
get an A on the midterm?
What is the probability that the student will not achieve a 75% average on quizzes or will
get an A on the midterm?
Are the events that a student achieves a 75% average on quizzes and the event that she
gets an A on the midterm independent or dependent? Show how you arrive at your
answer.
b)
c)
d)
2.
Tom, Dick, and Harry often play tennis together. Tom usually wins: in fact, he has
beaten Dick in 6 out of the last 10 games they've played together, and has only lost 3 of
the last 12 games he's played with Harry. One day, all three are at the tennis courts and
Dick and Harry toss a fair coin to decide which one will play Tom.
a)
b)
What is the probability that Tom will win the game?
Suppose Tom were to win. What would the probability have been that he played Harry?
3.
The probability that a rare tropical disease will be diagnosed correctly is .70. If it is
diagnosed correctly, the probability is .90 that it can be cured. If it is not diagnosed
correctly, the probability is .40 that it can be cured. If a patient having this disease were
to be cured, what would the probability have been that the disease was diagnosed
correctly?
4.
People who have been exposed to a particular virus have a .95 probability of contracting
a fatal disease. 80% of the people who have not been exposed to the virus will not
contract the disease. If a doctor sees 200 patients, of whom 130 have been exposed to the
virus, then:
a)
What is the probability that any randomly selected patient from these 200 will contract
the disease?
If, at some point in the future, one of these 200 patients were to contract the disease, what
would the probability have been that he or she had been exposed to the virus?
b)
Answers:
1. a) .68 b) .045 c) .83 d) Dependent
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2. a) .675 b) .556
3. .84
4. a) .6875 b) .898
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Psychology 281(001)
1.
a)
b)
Assignment #4
A general is planning to invade towns A, B, and C and has 20 soldiers at his
disposal (6 officers and 14 privates). After some thought, the general decides to
select 12 soldiers to carry out the invasion and to keep the remaining 8 (and
himself) behind to protect the command post.
If the general selects the 12 soldiers randomly, and without replacement, what is the
probability that 3 will be officers and 9 will be privates?
If the general now takes the 12 selected soldiers (3 officers and 9 privates) and randomly
selects 4 soldiers (without replacement) to invade each of towns A, B, and C, what is the
probability that exactly 1 officer ends up being sent to each town?
2.
A project director runs a staff consisting of 6 scientists and 3 lab technicians. Three new
projects have to be worked on and the director decides to assign 4 of her staff to the first
project, 3 to the second, and 2 to the third. In how many ways can this be accomplished
if:
a)
b)
c)
The director assigns her staff to the projects randomly?
Each project requires 1 lab technician?
Of the 4 people assigned to the first project, at least 3 are scientists?
3.
Ten officers and 10 regular soldiers are going to be assigned to units of size 4, 6,
and 10 to carry out different missions. If assignment is done at random, what is
the probability that each unit will end up being half officers and half regular
soldiers?
4.
Each week, Bob and three friends from work pool their money together and buy
12 lottery tickets that are randomly and equally divided up among the group.
a)
b)
How many ways can the tickets be divided up among Bob and his friends?
By an incredible stroke of luck, this week there are two winning tickets among the 12
tickets purchased by Bob and his friends. What is the probability that different people end
up with these winning tickets?
In Lotto Mania, the game played by Bob and his friends, six winning numbers are
selected by random sampling without replacement from a bin of balls numbered 1
through 30. A player wins if the six numbers on his/her ticket match at least five numbers
from the six winning numbers (order is irrelevant). If you buy a single ticket, what is the
probability that you will win something?
c)
5.
A sales company purchases 15 new cars for its top salespeople: 6 identical
Chevrolets, 5 identical Fords, and 4 identical Toyotas.
a)
If 5 of these 15 cars are randomly assigned to the salespeople, what is the probability that
none of them is a Ford?
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b)
c)
d)
If 3 of the 15 cars are randomly assigned to the salespeople, what is the probability that at
least 2 of them are Toyotas?
If 3 of the 15 cars are randomly assigned to the salespeople, what is the probability that
they are all a different model?
If the company parking lot has 15 parking spaces in a row and the 15 new cars were
parked at random, what is the probability that all the same-model cars would end up
being parked next to each other?
Answers:
1. a) .3179 b) .2909
2. a) 1260 b) 360 c) 750
3. a) .164
4. a) 369600 b) .8182 c) .0002442
5. a) .084 b) .154 c) .264 d) .000009514
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Psychology 281(001)
Assignment #5
1.
You have a biased or unfair coin, such that, on each toss, the probability of observing a
head is .25 and the probability of observing a tail is .75. Let X be the variable: number of
heads observed in 4 tosses.
a)
Describe the probability distribution of X and compute the mean and the standard
deviation of this distribution.
2.
A multiple-choice test contains 12 questions, 8 of which have 4 answers each to choose
from and 4 of which have 5 answers each to choose from. If a student randomly guesses
all of his answers, what is the probability that he will get exactly 2 of the 4-answer
questions correct and at least 3 of the 5-answer questions incorrect?
3.
A standard deck of 52 cards consists of 4 suits (spades, hearts, diamonds, and clubs),
each of which contains 13 cards. A player selects 3 cards at random, without
replacement, and is interested in knowing the mean and standard deviation of X: the
number of diamonds he might select. Compute the answers for him.
4.
A store has found that 10% of the items it sells are returned after Christmas. On one day,
a total of 50 items are sold by 5 clerks: Clerk A sold 5 items, Clerk B sold 20 items, and
the rest of the items were sold by the other 3 clerks.
a)
b)
c)
What is the probability that none of the items sold by Clerk A will be returned?
What is the probability that exactly 3 of the items sold by Clerk B will be returned?
What is the probability that between 2 and 7 (inclusively) of the items sold by the other 3
clerks will be returned?
What is the probability that between 3 and 6 (inclusively) of the items sold by Clerk B
and that exactly 4 of the items sold by the other 3 clerks will be returned?
d)
5.
A bag contains 4 red marbles and 2 blue marbles. You draw a marble at random, without
replacement, until the first blue marble is drawn. Let X = the number of draws.
a)
If you repeated this experiment a very large number of times, on average how many
draws would you make before a blue marble was drawn?
Express the maximum number of draws you might have to make as a Z score.
Suppose you had sampled with replacement. Is it more likely that you would make 1
draw or 6 draws before you got a blue marble? Why?
b)
c)
Answers:
1. µ = 1.0, σ = .866
2. .2552
3. µ = .75, σ = .736
4. a) .5905 b) .190 c) .727 d) .044
5. a) µ = 2.334 b) Z = 2.138 c) 1 draw: P(X = 1) = .333, P(X = 6) = .0439.
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Psychology 281(001)
1.
Assignment 6
A bag contains 100 marbles. Each marble is either red or blue. One of two conditions
exists with respect to the number of red and blue marbles:
1. There is an equal number of red and blue marbles (H0), or
2. 60% of the marbles are blue (HA)
Your task is to guess which of the two conditions is in fact true. As an aid, you are
allowed to randomly select 10 marbles (one at a time, with replacement) and to record the
colour of each marble as you take it out. You decide that if 7 or more of the 10 marbles
you select are blue, you will conclude that condition 2 is true.
a)
b)
If, in fact, condition 1 is true, what is the probability that you will erroneously conclude
that condition 2 is true?
If, in fact, condition 2 is true, what is the probability that you will erroneously conclude
that condition 1 is true?
2.
You have two coins, one that comes up heads 50% of the time and one that comes up
heads 70% of the time. A friend of yours who wants to win some bets borrows what he
thinks is your 70% coin to use on some unsuspecting people. Before using the coin,
however, he decides to try it out on himself by tossing it 10 times. He decides that if
heads come up 8 or more times out of 10, he will conclude that you really did give him
the biased (70%) coin.
a)
If, in fact, you gave him the fair (50%) coin, what is the probability that he will
incorrectly conclude that you gave him the biased coin?
If, in fact, you gave him the biased coin, what is the probability that he will incorrectly
conclude that you gave him the fair coin?
Suppose he tossed the coin 25 times and decided to use α ≤ .05 to create a new decision
rule. What is the probability that he would now incorrectly conclude that you gave him
the fair coin if, in fact, you gave him the biased coin?
b)
c)
3.
(From 1994 Midterm Exam). In a recent survey, 480 of 600 Canadians polled stated that
they were dissatisfied with politicians. Of the remaining 120 who were polled, 75% were
satisfied with politicians and 25% had no opinion. Assuming that these findings can be
generalized to all Canadians, then:
a)
In a random sample of 4 Canadians, what is the probability that no more than 1 would be
satisfied with politicians?
If two random and independent samples of Canadians were taken, one consisting of 20
people and the other of 25 people, what is the probability either that more than 18 of the
sample of 20 or that between 19 and 23 (inclusively) of the sample of 25 would state a
definite opinion (for or against) about politicians?
b)
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3c)
3d)
Among people who initially have no opinion, 50% will typically form an opinion (for or
against an issue) after reading a relevant news report. In an attempt to get more people to
form an opinion and take one side or the other on political issues, a news service has
developed a new approach to presenting information on the issues. It tries this out on a
random sample of 20 people who initially claimed that they had no opinion about an
issue. The news service will conclude that the new approach is more effective if at least
15 of these 20 report a definite opinion about the issue after reading about it. What is the
probability that the news service will conclude that the new approach is more effective
even if it is in fact no better than previous methods?
In a second study, the news service tries out the new approach to presenting information
on a random sample of 25 people who initially expressed no opinion about an issue. The
news service will test the hypothesis that the new approach is more successful than
previous methods in getting people to form an opinion, using α ≤ .03. Suppose that the
new approach would actually cause 80% of initially no-opinion people to form a definite
opinion. Given this, what is the probability that the news service will draw the correct
conclusion based upon their second study?
Answers:
1.
a) .172
b) .618
2.
a) .055
b) .617
c) .488
3.
a) .89
b) .8305
c) .021
d) .891
12
Psychology 281(002)
Assignment #7
1.
Scores on a particular test are normally distributed in the population, with a mean of 100
and a standard deviation of 15. What percentage of the population has scores....
a)
b)
c)
d)
e)
Between 100 and 125
Between 82 and 106
Between 110 and 132
Above 132
Equal to 132
2.
Archeologists excavating at a Viking site in Newfoundland have determined that the age
of the artifacts they have discovered is normally distributed with a mean of 1000 years.
Ninety-five percent of the artifacts are between 900 and 1100 years old.
a)
b)
What proportion of the artifacts is between 940 and 980 years old?
One archeologist is particularly interested in collecting the oldest 5% of the artifacts.
How old does an artifact have to be to fit in this collection?
The age of artifacts found at a neighboring site is also normally distributed, with a
standard deviation of 60, but the artifacts at this second site are older, on average, than
those at the first site. In fact, only 33% of the artifacts at the first site are older than the
mean age of artifacts at the second site. What percentage of artifacts at the second site are
less than 1200 years old?
c)
3.
The population distribution of scores on a commonly used IQ test is normal with a mean
of 100 and a standard deviation of 15. The test was recently administered to a sample of
psychology undergraduates and to a sample of mildly depressed psychiatric patients.
Both groups yielded normally distributed IQ scores. The standard deviation of the
students' scores was 12 and that of the patients' was 17. The probability that a psychology
student has a lower IQ than the average person in the population was found to be 10
percent. The probability that a patient has a higher IQ than the average psychology
student is 5 percent.
a)
What is the probability that a randomly selected psychology student will have a lower IQ
than the average psychiatric patient?
b)
What percentage of people from the general population will have a higher IQ than a
student at the 67th percentile of the psychology student distribution?
4.
A supermarket manager finds that 20% of his customers buy brand X cornflakes. The
other 80% buy brand Z.
a)
In a random sample of 10 customers, what is the probability that at least 4 customers buy
brand X?
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b)
In a random sample of 100 customers, what is the probability that between 15 and 30
customers (inclusively) buy brand X?
c)
What is the minimum number of boxes of brand X cornflakes that the manager should
have in stock to ensure, with a probability of at least .95, that he can meet the brand X
demands of a random sample of 225 customers? (Assuming 1 box per customer.)
Answers:
1. a) 45.25
b)
54.03
c)
23.48
d) 1.66
e)
0
2.a)
b)
c)
.2293
1083.93 years old
99.85%
3a)
b)
.0099
8.38
4 a)
b)
c)
.121
.9119
55
14
Psychology 281 (001)
Assignment #8
1.
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.
Compute the mean (µx) and standard deviation (σx) of this distribution.
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.
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.
Compute the mean (µx) and standard deviation (σx) of the sampling distribution in part 'd'.
Confirm that µx̄ = µ and that σx̄ = σ/√ n.
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?
b)
c)
d)
e)
f)
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?
What is the probability that the king would be satisfied, but not until his third messenger
reported?
b)
4.
Paranoid schizophrenics often manifest two different types of problems: depression and
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trembling. Following is the probability distribution for a discrete random variable, X: the
number of these problems that a paranoid schizophrenic might experience:
a)
b)
c)
X
P(X)
0
1
2
.35
.47
.18
Generate the sampling distribution for the mean number of these problems that random
samples of two paranoid schizophrenics might experience.
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?
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.
Construct a confidence interval to estimate the mean weekly expenditure on food in the
entire population of this city in 1979 using α ≤ .01.
b)
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
16
Psychology 281 (001)
Assignment #9
1.
A company is considering purchasing an expensive piece of machinery which it hopes
will reduce the amount of time it takes to manufacture its product, but which might
actually have the reverse effect and increase (slow down) production time. Currently
(without the new machine), it takes an average of 55.5 seconds to manufacture the
product. The company decides to test the new machine for a few days to see whether it
speeds up or slows down production.
a)
b)
c)
State the null and the alternative hypotheses that the company wishes to test.
With α ≤ .05, state the decision rule.
The company randomly selects 50 of the products made by the new machine and finds
that these took an average of 54 seconds to make, with a standard deviation of 6.5
seconds. Test the company's hypotheses and make a decision/conclusion about them.
Use the sample data from part 'c' to form a 99% confidence interval to estimate the mean
time of production of the population of products that could be made by the new machine.
d)
2.
A pain reliever currently being used in a hospital is known to bring relief in a mean time
of 3.5 minutes. To compare a new pain reliever with the one currently being used, a
random sample of 45 patients is given the new drug. The mean time to relief for this
sample is 2.8 minutes with a standard deviation of 1.14 minutes.
a)
Test the research (alternative) hypothesis that the new drug is more effective than the old
drug, using α ≤ .05.
Assuming a standard deviation of 1.14 minutes, approximately how large a sample of
patients would be needed to form a 90% confidence interval to estimate the population
time-to-relief of the new drug accurate to within plus or minus 10 seconds?
b)
3.
The frequency distribution of the heights (measured in feet) of the entire Martian army is
as follows:
Height
4
5
6
7
a)
b)
Relative Frequency
2/5
1/5
1/5
1/5
A squad of 64 Martians is selected at random from the Martian army to train for special
undercover operations. What is the probability that the mean height of this squad will be
greater than 5.5 feet?
A second squad of 36 Martians is also selected from the Martian army. These 36 are all
experts at extrasensory perception (ESP) and, on Mars, it is believed that ESP experts are
taller than the average Martian soldier. Is there any truth to this belief, if the sample of 36
has a mean height of 5.75 feet and α is ≤ .01?
17
4.
Over the past 300 years or so, the amount of time it has taken Santa Claus to climb down
chimneys, deposit presents, and get back to his sleigh has been normally distributed with
a mean of 10 seconds per chimney and a standard deviation of 3 seconds. This year, since
he's beginning to feel his age, Santa has engaged in a rigorous physical fitness program
which he believes will allow him to significantly reduce his time. To test his belief, Santa
plans to have one of the elves time him on a random sample of 36 chimneys and then
conduct the hypothesis test with α ≤ .025. If, in fact, the exercise program has reduced his
true mean time to 8 seconds (standard deviation unchanged), what is the probability that
Santa will incorrectly conclude that the exercise had no effect?
5.
One day, it suddenly occurs to you that students at Western seem to be taller than the
National average, which you know to be 66 inches. To test your insight, you take a
random sample of 50 students and discover their mean height is 70.5 inches with a
standard deviation of 8.8.
a)
b)
With α ≤ .05, test the hypotheses of interest.
If, in fact, the mean height of all Western students is actually 68 inches, what is the
probability that your experiment in part 'a' would cause you to incorrectly fail to reject the
null hypothesis?
If you repeated the experiment and took two random samples, one of 30 students and the
other of 40 students, what is the probability that with both samples you will correctly
conclude that Western students are significantly taller than the National average, if both
samples' heights have a standard deviation of 3.5 and the true mean height of all Western
students is actually 68 inches (still using α ≤ .05)?
c)
Answers:
1. c) Z = -1.632, do not reject H0, no evidence that the new machine is faster or slower than what
was used before, d) 51.63 to 56.37
2. a) Z = -4.12, reject H0, new drug is more effective, b) 127
3. a) .0197, b) Z = 2.83, reject H0, the ESP soldiers are significantly taller
4. .0207
5. a) Z = 3.62, reject H0, Western students are significantly taller, b) .5160, c) .9092
18
Psychology 281(001)
1.
Assignment #10
Over the past 50 years, an average of 370 minor earth tremors a day have been
recorded in Northern California. Recently, over a period of nine days, the
following numbers of minor tremors were recorded:
386, 394, 362, 423, 456, 370, 385, 430, 400
a)
b)
Is there evidence for a recent increase in the number of tremors (α ≤ .05)?
Based on the 9-day data, form a 95% confidence interval to estimate the mean number of
minor tremors that might be expected each day over the next several weeks.
2.
In the past, students in a particular course have achieved a mean Xmas exam
score of 72.6, with a standard deviation of 12.5. This year, 15 students in one
section achieved a mean Xmas exam score of 72.2, while 10 students in another
section achieved a mean Xmas exam score of 77.3. Is the overall (grand) mean of
the 25 students in this year's two sections significantly different from the mean
obtained in previous years (α ≤ .05)?
3.
A procedure currently used by doctors to detect a particular disease fails to detect
the disease in 15% of the people who actually have the disease. A new procedure
is developed that researchers hope will provide a more accurate diagnosis. A
random sample of 70 people who are known to have the disease are subjected to
the new procedure. Of these 70, the new procedure failed to detect the disease in
only six.
a)
Do these data provide sufficient evidence to indicate that the new procedure provides a
significantly more accurate diagnosis than the one currently in use (α ≤ .05)?
Based on the sample of 70 patients, form a 95% confidence interval to estimate the
proportion of false negative diagnoses that the new procedure could be expected to make
in the population.
b)
4.
Historically, stores in London that sell wine have noted that Londoners tend to buy three
times as much white wine as red wine. They’ve also noted that 40% of all wine sold (red
or white) tends to be Canadian wine.
a)
This year, the Canadian red wine producers declared November to be “Drink Red
Canadian Wine Month” and ran an ad campaign to try to get Londoners to do exactly
that. During November at one randomly selected store in London, 200 bottles of wine
were sold, 30 of which were Canadian reds. Does it appear that the ad campaign was
successful (α ≤ .05)?
At another randomly selected London store, 250 bottles of wine were sold in November
and 98 of these were red. Calculate a 99% confidence interval for the proportion of red
wine sales in London during November.
b)
19
Answers:
1.
a) t = 3.033, reject Ho: there is evidence for a significant increase in the number of errors
b) 377.35 to 423.99
2.
Z = .656, do not reject Ho: there is no evidence that this year's students are performing
differently than students in previous years.
3.
a) Z = -1.51, do not reject Ho: there is no evidence that the new procedure is more
effective
b) .02 to .152
4.
a) Z = 2.357, reject Ho, the campaign was successful
b) .312 to .472
20
Psychology 281(001)
1.
Assignment # 11
In a psychology experiment, 80 subjects were randomly assigned to either a control or an
experimental group. After the experiment, some subjects from each group had dropped
out and the following data were obtained:
Control Group
nC = 38
x̄C = 75
sC = 14.5
a)
b)
c)
2.
a)
b)
Experimental Group
nE = 34
x̄E = 80
sE = 16.2
Test the research hypothesis that the sample means are significantly different from one
another (α ≤ .05).
If x̄C is still equal to 75, and both groups' sample sizes and standard deviations remain
unchanged, at least how high would x̄E have to be to be considered significantly higher
than x̄C (α ≤ .05)?
Suppose both groups' means and standard deviations are unchanged but nC is now 12 and
nE is 8. Is x̄E significantly higher than x̄C (α ≤ .01)?
Subjects were randomly assigned to one of two treatment conditions in a medical
research project which was investigating the effect of a new drug which was expected to
lower the resting pulse rates of people with high blood pressure. Subjects in group 1 were
administered the new drug, while subjects in group 2 were given a placebo. One hour
later, the subjects' resting pulse rates were measured, yielding the following results:
Group 1
Group 2
n1 = 13
x̄1 = 78.4
s1 = 10.73
n2 = 16
x̄2 = 85.8
s2 = 8.62
Perform the appropriate statistical procedure(s) to determine whether the new drug was
effective (α ≤ .05).
The mean resting pulse rate of the population of people who do not suffer from high
blood pressure is 72.8. Is group 1's mean significantly different from this (α ≤ .01)?
(Hint: think back to last term...)
Answers:
1. a) Z = -1.374, do not reject H0, no evidence that the means differ significantly from each other
1. b) 81
1. c) t = .72, do not reject H0, no evidence that the experimental mean is significantly higher than
21
the control mean.
2. a) t = -2.061, reject H0, the new drug is effective
2. b) t = 1.88, do not reject H0, no evidence for a significant difference between group 1 and
regular blood pressure people
22
Psychology 281(001)
1.
Assignment #12
Sixteen subjects were randomly assigned to one of two groups in an experiment
investigating the effect of stress on test-taking performance. Subjects in group 1
completed a 100-item test with no time limit, while subjects in group 2 were allowed only
30 seconds to complete each item on the same test. The following data were obtained:
Group 1
86
72
78
74
83
72
81
74
x̄1 = 77.5
s1 = 5.35
Group 2
72
57
65
72
86
48
54
41
x̄2 = 61.875
s2 = 14.69
Is there any significant difference between the performances of the two groups? (α ≤ .05)
2.
The same experiment was now repeated with a sample of 25 subjects. Again, group 1 did
the test with no time limit and group 2 was timed, yielding the following data:
Group 1
72
84
81
76
79
83
83
72
81
80
83
80
Group 2
64
85
58
76
83
56
92
81
72
60
64
88
55
n1 = 12
n2 = 13
x̄1 = 79.5
x̄2 = 71.85
s1 = 4.123
s2 = 13.088
Did subjects in group 2 score lower, on average, than subjects in group 1? (α ≤ .01)
23
3.
In a memory experiment, subjects in an experimental group were trained to use a
mnemonic strategy which it was hypothesized would enhance their recall of lists of
words. Control group subjects were given no training but simply read lists of words.
Thirty-two subjects were randomly assigned to one of the two groups but, before testing,
five subjects dropped out, leaving 15 in the experimental group and 12 in the control
group. Subjects' data consisted of the number of words they could recall from a list of 20
words and their scores were rank-ordered from the fewest to the most words recalled
regardless of group. Following are the ranked scores assigned to the 15 experimental
subjects:
4, 12, 19, 6, 6, 12, 25, 27, 22, 6, 14.5, 23.5, 21, 26, 23.5
Perform the appropriate statistical procedure(s) to determine whether the recall of the
control subjects is significantly poorer than that of the experimental subjects (α ≤ .05).
Answers
1.
F = 7.54, reject Ho: the variances differ significantly; TA = 90.5, reject Ho: the groups
differ significantly
2.
F = 10.08, reject Ho: the variances differ significantly; Z = 1.197, do not reject Ho: no
evidence that group 2 scores significantly lower than group 1
3.
(Note: no F test is performed (i) because we don't have the variances or any way to
compute them and (ii) because the data that are provided are ranked scores, therefore the
Wilcoxon is the only correct procedure); Z = -1.83, reject Ho: the control group performs
significantly worse than the experimental group
24
Psychology 281 (001)
1.
Assignment #13
Recent research in mathematics education indicates that teaching children general
reasoning and problem-solving skills may be more effective than old-fashioned methods
that stressed rote-learning. To test this, 15 fourth-graders are randomly assigned to one of
two instructional programs, at the end of which they all answer a common set of 20
questions. Their scores are as follows:
Instructional Methods
Rote-Learning
General Reasoning
17
14
12
4
7
10
6
7
x̄1 = 9.625
s1 = 4.438
a)
b)
c)
d)
14
18
16
18
19
16
16
x̄2 = 16.714
s2 = 1.704
Do these data indicate that acquiring general reasoning skills is more effective than rotelearning? (α ≤ .05)
Is the variance of the scores of the children who were given rote-learning instruction
significantly different from the population variance of this test, which is known to be 25
(α ≤ .05)?
Use the sample data of the children who received general reasoning instruction to form a
95% confidence interval to estimate σ2.
The 8 children who underwent the rote-learning instruction in the previous study were
now taught the general reasoning skills that the other children had learned, and then took
the same 20-item test that they had taken before. Following are these children's scores on
the second test, in the same order that they appear above:
18, 17, 12, 8, 10, 9, 12, 14
Have the children's scores improved (α ≤ .05)?
2.
A new pollution control device for automobiles is claimed not only to reduce carbon
monoxide emission but also to increase a car's miles per gallon. To test the latter claim,
the mileage per gallon of 10 randomly selected cars is measured first without and then
with the device, yielding the following data:
25
Miles per gallon
Car
Without Device
1
2
3
4
5
6
7
8
9
10
28
20
44
7
20
11
11
34
9
48
x̄ = 23.20
s = 14.79
With Device
32
21
36
10
20
14
9
32
14
46
x̄ = 23.40
s = 12.47
a)
b)
Does the device significantly increase cars' mileage per gallon (α ≤ .05)?
Typically, the variance of the mileage per gallon of cars not fitted with the device is 175.
Is the variance of the 10 cars in this study significantly less than this when they were
driven with the device? (α ≤ .05) Is their variance significantly greater than 175 when
they were driven without the device? (α ≤ .05)
3.
The lower value of a 90% confidence interval for the variance in maze-running times of a
sample of 31 rats is 4.62. Based on this, form a 98% confidence interval for σ2.
Answers:
1.
a) F = 6.783, Reject H0; variances differ significantly, therefore perform the Wilcoxon
test: TA = 79.5, Reject H0; general reasoning is significantly more effective than rote
learning.
1b)
chi-square = 5.515, Do not reject H0; no evidence that the variance of the children's
scores differs significantly from 25.
1c)
1.206 < σ2 < 14.08
1d)
t = 2.904, Reject H0; the children's scores have increased significantly.
2. a)
2b)
t = .163, Do not reject H0; no evidence that the device increases cars' mpg.
χ2 = 7.997, Do not reject H0; no evidence that the variance is significantly less than 175.
χ2 = 11.250, Do not reject H0; no evidence that the variance is significantly greater than
175.
3.
3.973 to 13.5219
26
Psychology 281(001) Assignment #14
1.
A psychiatrist is interested in a new form of drug treatment (melatonin) for depression.
She compares the effect of melatonin treatment with a control group and two other types
of therapy: lithium treatment and psychotherapy. The following are subjects' data, after
their respective treatments, on a behavioural measure of depression: larger scores
indicating more depression.
Control
Psychotherapy
Lithium
Melatonin
ni
12
10
14
12
x̄i
16.8
17.2
11.6
10.3
si
5.2
6.5
4.0
3.8
Do the different treatments result in significantly different measured amounts of
depression (α ≤ .05)?
2.
After the first study, the psychiatrist randomly assigned 15 manic-depressive subjects to
one of 3 groups, which were administered 0, 5, or 10 ml. of melatonin. After treatment,
the subjects' scores on the depression measure were:
Control Group
(0 ml.)
_____________
23
17
21
25
20
Treatment 1
(5 ml.)
___________
16
9
13
12
15
Treatment 2
(10 ml.)
__________
14
7
8
12
11
Do the different dosages of melatonin result in significantly different measured amounts
of depression? (α ≤ .01)
3.
In a larger-scale replication study, 60 subjects were randomly assigned to one of 3
treatment conditions (0 ml., 5 ml., or 10 ml. of melatonin). After treatment, the total sum
of squares was found to be 6504.75. In addition, 32% of the variance between the groups'
measured depression could be accounted for by the different dosages of melatonin they
had been administered.
Does this study show significant differences between the groups' mean depression scores
(α ≤ .025)
4.
Forty eight subjects were randomly assigned to one of 4 treatment conditions in a study
27
of different methods of coping with statistics exam anxiety. After the experiment, 12
subjects remained in group 1, 8 in group 2, 10 in group 3, and 10 in group 4. The mean
anxiety score of group 1 was found to be 5.5 points higher than the mean of group 2, 3.5
points higher than the mean of group 3, 5.3 points higher than the mean of group 4, and
3.3 points higher than the grand mean. Interestingly, the sum of squares error was exactly
12.0 times larger than its degrees of freedom.
Did the different treatments result in significantly different amounts of anxiety? (α ≤ .01)
Answers
1.
F = 6.15, Reject H0: the different treatments do result in significantly different levels of
depression
2.
F = 19.07, Reject H0: the different dosages result in significantly different levels of
depression
3.
F = 13.41, Reject H0: there are significant differences between the groups' mean levels of
depression
4.
F = 5.83, Reject H0: the different treatments result in significantly different levels of
anxiety
28
Psychology 281(001)
1.
Assignment #15
Subjects were randomly assigned to one of four groups in an exploratory study of the
effect of different drugs on motion sickness. The data provided below were subjected to
an analysis of variance, resulting in FOBT = 10.0.
n1 = 8
n2 = 7
n3 = 10
n4 = 7
x̄1 = 12.0
x̄2 = 6.5
x̄3 = 13.1
x̄4 = 8.5
First, determine whether the ANOVA is significant (with α ≤ .01), and, if so, determine
whether there is a significant difference between the mean scores of groups 2 and 3, 2
and 4, and 2 and 1, using the post hoc procedure that allows you to use α ≤ .01 for each
comparison.
2.
Subjects were randomly assigned to one of four treatment conditions. After treatment, the
following data were obtained:
n1 = 8
n2 = 10
n3 = 7
n4 = 9
x̄1 = 6.5
x̄2 = 5.5
x̄3 = 8.75
x̄4 = 12.0
In addition, the standard deviation of the scores of all 34 subjects was 3.50.
a)
b)
Before the experiment, the researcher predicted that group 3 would obtain a significantly
higher mean than group 2, and that there would be no significant difference between the
means of groups 1 and 2. Perform the appropriate test(s) to determine whether these
predictions were borne out. (α ≤ .05)
After the data had been collected, the researcher wished to determine whether there was a
significant difference between x̄4 and x̄1, and between x̄3 and x̄1. Using α ≤ .05, perform
the necessary test(s) and, when comparing these means, use whichever valid procedure
gives you the smallest critical value.
3.
A researcher begins with a sample of 60 subjects and randomly assigns 12 to each of 5
treatment conditions. Based on his reading of other studies in the area, the researcher
predicts that group 3 will obtain a significantly different mean from group 1, and that
group 4 will obtain a significantly higher mean than group 1.
a)
Given the following data (note that some subjects have dropped out) are the researcher's
predictions supported with α ≤ .01?
29
b)
Group 1
Group 2
Group 3
Group 4
Group 5
ni
8
10
10
12
12
x̄i
43.5
45.2
61.8
77.3
66.7
si
5.6
6.3
5.4
8.7
7.5
After collecting and inspecting the data, the researcher decides to perform three more
tests. Specifically, he wishes to see whether x̄3 is significantly different from x̄2; whether
x̄5 is significantly different from x̄2; and whether x̄4 is significantly greater than x̄2.
Perform whichever statistical analyses are necessary and, if appropriate, use the
procedure that allows each comparison to be tested with α ≤ .01.
Answers
1.
2 vs. 3, q = 6.74, Reject H0, the mean scores of groups 2 and 3 differ significantly; 2 vs.
4, q = 2.04, Do not reject H0, there is no evidence for a significant difference between the
mean scores of groups 2 and 4; 2 vs. 1, q = 5.62, Reject H0, the mean scores of groups 2
and 1 differ significantly.
2.
a) 3 vs. 2, t = 2.720, Reject H0, group 3 obtained a significantly higher mean score than
group 2; 1 vs. 2, t = .870, Do not reject H0., there is no evidence for a significant
difference between the mean scores of groups 1 and 2.
b)
F = 12.92, Reject H0, at least 2 means differ significantly; 4 vs. 1, q = 6.56, Reject H0, the
mean scores of groups 4 and 1 differ significantly; 3 vs. 1, q = 2.68, Do not reject H0,
there is no evidence for a significant difference between groups 3 and 1.
3a)
3 vs. 1, t = 5.53, Reject H0, prediction is supported: group 3 obtained a significantly
higher mean score than group 1; 4 vs. 1, t = 10.608, Reject H0, prediction is supported:
group 4 obtained a significantly higher mean score than group 1; b) F = 43.696, Reject
H0, at least 2 means differ significantly; 3 vs. 2, q = 7.583, Reject H0, the mean scores of
groups 3 and 2 differ significantly; 5 vs. 2, q = 9.822, Reject H0, the mean scores of
groups 5 and 2 differ significantly; 4 vs. 2, no test valid because post hoc tests cannot be
1-tailed.
30
Psychology 281(001)
1.
Thirty six subjects are randomly assigned to one of twelve groups in a study investigating
the separate and joint effects of imagery instruction and imagery vividness on recall. Four
kinds of imagery instruction (passive, active, reactive, and control) and three kinds of
imagery vividness (low, medium, and high) are used. Following treatment, all subjects
are given a 30-item recall test, yielding the following data:
Control
L
M
4
5
5
7
5
6
a)
b)
c)
d)
e)
2.
a)
Assignment #16
H
7
9
11
Passive
L
M
3
9
5
9
4
8
H
10
14
12
Active
L
5
4
4
M
9
11
13
H
18
19
17
Reactive
L
M
5
13
6
14
3
15
H
22
24
26
Fill out a complete 2-way ANOVA table and determine if the main effects for imagery
instruction and imagery vividness, and the instruction x vividness interaction, are
significant (each with α ≤ .05).
If a main effect for imagery vividness is found, perform Newman-Keuls post hoc tests to
determine which means differ from one another.
If the instruction X vividness interaction is significant, draw a graph of the cell means
and write a brief interpretation of the interaction.
Prior to the experiment, it was predicted that the means in the low and medium imagery
vividness conditions would not differ from one another. Is this prediction borne out with
α ≤ .05?
Prior to the experiment, it was predicted that the mean in the active instruction condition
would be significantly greater than the mean in the passive instruction condition. Is this
prediction borne out with α ≤ .01?
Eighty Canadian citizens of voting age are randomly polled under the restrictions
of (i) equal representation from the West, Ontario, Quebec, and the Maritimes,
and (ii) equal numbers of liberals and non-liberals within each of the four regions.
All subjects rate their attitude toward the Prime Minister. Overall, the mean rating
is 29.25, with a standard deviation of 12.7. Mean ratings by region and political
affiliation are given below:
West
Ontario
Quebec
Maritimes
Liberal
46
40
33
35
Nonliberal
20
28
17
15
Perform the appropriate analyses to determine which effects are significant (each
with α ≤ .05)
31
b)
Regardless of your answer in part 'a', perform Newman-Keuls tests to determine
which regions differ significantly from one another (α ≤ .05).
Answers:
1a) Instruction: F = 52.57, Reject Ho; Imagery: F = 198.17, Reject Ho; Interaction: F = 16.85,
Reject Ho.
1b) High/Low: Qcrit = 3.53, Qobt = 28.12;
High/Med: Qcrit = 2.92, Qobt = 14.47;
Med/Low: Qcrit = 2.92, Qobt = 13.65.
1d) tcrit = ±2.064, tobt = 9.67, Reject Ho, Prediction not supported.
1e) tcrit = 2.492, tobt = 4.399, Reject Ho, Prediction supported.
2a) Region: F = 8.94, Reject Ho; Affiliation: F = 126.15, Reject Ho; Interaction: F = 3.29,
Reject Ho
2b) (s = significant difference; ns = non-significant difference)
Ont vs. Mar: Q = 5.46 (s);
Ont vs. Que: Q = 5.46 (s);
West vs. Mar: Q = 4.86 (s);
Ont vs. West: Q = .607 (ns);
West vs. Que: Q = 4.86 (s);
Que vs. Mar: Q = 0 (ns)
32
Psychology 281(001)
1.
Assignment #17
Verbal ability consists of at least two components: verbal comprehension (e.g., reading
ability) and verbal fluency (e.g., writing ability). A researcher measures these two
components in a sample of 8 seventh-grade children and obtains the following scores:
Comprehension
57
74
42
48
75
62
76
69
Fluency
40
46
38
36
54
42
50
25
a)
b)
Is there a significant correlation between the two variables (α ≤ .05)?
If another seventh-grader obtained a verbal comprehension score of 70, what would his or
her predicted verbal fluency score be?
2.
A sample of 25 subjects was given two tests (labelled X and Y), leading to the following
sample statistics:
x̄ = 65
ȳ = 42
SX̄ = 10
SȲ = 8
In addition, the correlation between X and Y was found to be .65.
a)
b)
c)
d)
3.
What are the values of SSX, SSY, and SSXY?
What are the values of βError! Bookmark not defined.hat1 and ßhat0? Show two ways to
compute ßhat1.
Test rXY and ß1 for significance (α ≤ .01), confirming that you obtain the same tobt in both
analyses.
Between what two values would you expect ß1 to fall with 95% confidence?
A friend of yours is studying the relationship between two variables (X and Y), and has
come to you for some help. He starts by showing you the following sample statistics,
obtained from a group of 20 subjects:
x̄ = 43.9
ȳ = 70.65
33
Sx̄ = 22.391
Sȳ = 18.695
Unfortunately, he lost his only copy of the original raw scores before computing the
correlation between X and Y, but he does remember that a Y-hat score of 41.154
corresponded to an X score of 0.
a)
b)
c)
What is the value of ßhat1 and is it significantly different from 0? (α ≤ .05)
What percentage of the variance in Y can be explained by X? Is this a significant amount?
(α ≤ .05)
Compute the predicted Y-hat score for a person with an X score of 25.
Answers
1.a) rXY = .461, t = 1.272, Do not reject H0: no evidence for a significant correlation between
the two variables
b) 43.66
2.a) 2400, 1536, 1248
b) .52, 8.2
c)
t = 4.102, Reject H0: The correlation and the slope are both significant.
d) .258 to .782
3.a) .672, t = 5.753, Reject H0: the slope is significantly different from 0.
b) 64.8%, t = 5.753, Reject H0: the correlation is significantly different from zero.
c) 57.954
34
Psychology 281(001)
Assignment #18
1.
In Wounded Knee, a suburb populated exclusively by Yuppies, everyone makes quite a
lot of money and everyone jogs. It has been hypothesized that the more money a yuppie
makes, the farther he or she jogs, and, in fact, a random sample of 12 yuppies yielded a
correlation between income and distance-jogged of .85. In addition, in this sample, the
income variable had a mean of 60 and a standard deviation of 5 (in thousands of dollars
per year), and the distance variable had a mean of 15 and a standard deviation of 4 (in
km. per day).
a)
If Fred earns 1000 dollars more per year than Bill, how much farther would he be
expected to jog each day?
Is this a significantly greater distance (α
A large number of Wounded Knee residents earn $56,000 a year. On average, how far
would these people be expected to jog each day and what are the 95% bounds on the
error of this prediction?
b)
c)
2.
a)
b)
c)
Recent studies have suggested that a significant negative correlation exists between
subjects' cerebral glucose metabolic rate (CGMR) and their scores on a standardized test
of intelligence (IQ). That is, subjects who metabolize less glucose obtain higher IQ test
scores. In an attempt to replicate this negative correlation, a researcher measures the
CGMR of 8 young adult subjects as they work on an IQ test. The following data were
obtained, where CGMRs are expressed as Z scores:
Subjects
CGMR
IQ
1
2
3
4
5
6
7
8
.70
1.50
-1.01
-.14
.13
.85
-1.47
-.56
114
101
133
128
111
92
122
107
Has this study successfully replicated the significant negative correlation between CGMR
and IQ (α = .05)?
Regardless of your answer in part 'a', use the given sample data to generate a 99%
prediction interval for the IQ score of a subject with a CGMR Z score of +.44.
Among older adults, the slope for predicting IQ scores from CGMR is -12.4. Is the slope
in the present sample of young adults significantly different from this (α = .05)?
35
Answers
1.
a) .68 km
b)
t = 5.102, Reject H0: it is a significantly greater distance (the slope is significantly greater
than zero).
c)
12.28 +/- 1.852
2.
a) r = -.70, t = -2.401, Reject H0: a significant negative correlation exists.
b)
109.248 +/-
c)
t = .68, do not reject H0: no evidence for a significant difference between the slopes.
36
Psychology 281(001) Assignment #19
1.
A hospital keeps a record of the number of males and females in different age groups
who contract a particular disease. The data are summarized as follows:
Age Groups
10-25
26-40
41-55 56 +
____________________________________________
Male
23
34
64
29
Female
20
31
55
44
____________________________________________
a)
b)
Are gender and age of contracting the disease independent of one another (α ≤ .05)?
A researcher believes that people in the 41-55 age group are three times as likely to
develop the disease as are those in each of the two younger age groups, while those in the
56 + age group are only twice as likely (relative to those in the two youngest groups). The
two youngest age groups are not expected to differ from each other. Do the data support
these beliefs (α ≤ .01)?
c)
It is also believed that, relative to males, the proportion of females in the 56 + age group
that develops the disease is significantly different from the proportion of females in the
10-25 age group that develops the disease. Test this belief with α ≤ .05.
2.
Fred is testing a large prize wheel which has the numbers one, two, three, and four
repeated along its circumference. When Fred spun the wheel 80 times, the wheel stopped
at a one 12 times, at a two 20 times, at a three 20 times, and at a four 28 times. Fred then
oiled the wheel and spun it an additional 120 times. This time he obtained frequencies of
12, 20, 32, and 56 respectively. Fred was careful to spin the wheel in the same manner
each time.
a)
b)
Does the wheel behave differently after being oiled? (α ≤ .05)
When the wheel is oiled, is the probability of occurrence of a number proportional to the
number (i.e., does the wheel stop at a two twice as often as at a one, etc.)? Use α ≤ .05.
3.
There are still many local chapters of the Grateful Dead fan club throughout the United
States and Canada. Each chapter has 10 members and by coincidence the gender
composition of each is exactly the same: 7 males and 3 females. Each local chapter
sends a committee of 3 to the annual national convention. This year, the composition of
these committees is as follows:
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# of such committees
a)
b)
# Males
1
750
0
60
2
2742
3
1448
Do these numbers indicate that this year's committees were not formed randomly with
respect to gender (α ≤ .01)?
Grateful Dead fan clubs also exist in Europe and, like those in N. America, contain
exactly 7 males and 3 females each. The European national convention records the
following gender composition for local committees:
# of such committees
# Females
1
1049
0
578
2
216
3
157
Do these and the previous data indicate that the gender composition of committees differs
significantly between N. America and Europe (α ≤ .05)?
4.
A sports magazine wanted to determine whether any relationship existed between the
kind of sports different athletes participated in and the speed with which the athletes
could run a 100 metre dash. 400 athletes were classified by their sport (Baseball,
Football, or Soccer) and by their running speed: rated as Slow (S: more than 1 standard
deviation below the mean), Somewhat Slow (SS: between -1 standard deviations and the
mean), Somewhat Fast (SF: between the mean and +1 standard deviations), and Fast (F:
more than 1 standard deviation above the mean). The results are presented in the
following table:
Baseball
Football
Soccer
a)
b)
S
SPEED
SS
SF
8
12
20
46
43
41
65
48
37
F
41
27
12
Is there a significant relationship between the two variables, with α ≤.10?
Ignoring sports, does it appear that the running speed of the athletes has a normal
distribution (α ≤ .01)?
Answers
1.
a) χ2 = 4.112, do not reject H0: gender and age of contracting the disease are independent;
2
b) χ = 14.03, reject H0: the data do not support the researcher's beliefs; c) χ2 = 2.068, do not
reject H0: it appears that this belief is not supported (there is no evidence to suggest that the
belief is correct).
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2.
a) χ2 = 10.133, reject H0: the wheel does behave differently after being oiled; b) χ2 =
2.44, do not reject H0: it appears that the probability of occurrence of a number is proportional to
the number (there is no evidence to reject this).
3.
a) χ2 = 31.208, reject H0: the numbers indicate that the committees were not formed
randomly with respect to gender; b) χ2 = 223.56, reject H0: the gender composition of the N.
American and European committees differs significantly.
4.
a) χ2 = 20.898, reject H0: there is a significant relationship between the variables; b) χ2 =
14.63, reject H0: the distribution of running speed is significantly non-normal.
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Psychology 281(001)
1.
Assignment 20
Is there a significant difference between the Psychology 281 grades earned by students
from different faculties? To address this question, the Psychology 281 final exam scores
of a random sample of 40 Social Science students were compared to those of 50 Science
students. The means and variances of these scores are as follows:
Social Science
n = 40
x̄ = 70.5
s2 = 200
a)
b)
Science
n = 50
x̄ = 72.1
s2 = 250
Determine whether a significant difference exists between the mean scores of these
samples (α ≤ .05).
Suppose that, in fact, the mean final exam scores of the populations of Social Science and
Science students differ by 3 points. Given this, what is the probability that you would fail
to detect a significant difference based on the sample sizes and variances used in part 'a'
(and using α ≤ .05)?
2.
Past research indicates that people who feel good about themselves will be more prosocial than those who feel bad about themselves. Specifically, on a standard measure of
pro-social behaviour, high self-esteem subjects obtain an average score of 58, with a
standard deviation of 12, and low self-esteem subjects obtain an average score of 46 with
a variance of 225.
a)
If you were to randomly select 80 high self-esteem subjects and 50 low self-esteem
subjects, what is the probability that the mean pro-social score for the high self-esteem
subjects would exceed that of the low self-esteem subjects by more than 15 points?
You suspect that if people are made more aware of how their self-esteem levels influence
behaviour, the population difference between high and low self-esteem subjects would
decrease. You test out your theory by administering the Esteem Awareness Procedure to
35 high self-esteem subjects and 40 low self-esteem subjects. If this procedure actually
reduces the population difference in pro-social behaviour by 6 points, what is the
probability that you would correctly detect this using an alpha level of .01?
b)
3.
A procedure currently used by doctors to detect a particular disease fails to detect the
disease in 15% of the people who actually have the disease. A new procedure is
developed that researchers hope will provide a more accurate diagnosis. A random
sample of 70 people who are known to have the disease are subjected to the new
procedure. Of these 70, the new procedure failed to detect the disease in only six.
a)
Do these data provide sufficient evidence to indicate that the new procedure provides a
40
b)
significantly more accurate diagnosis than the one currently in use (α ≤ .05)?
Suppose that, in fact, the new procedure would fail to detect the disease 10% of the time.
Given this, what is the probability that the researchers would draw the correct conclusion
based on a study of 70 people known to have the disease and using α ≤ .05?
Answers:
1.
a) Z = -.506, Do not reject H0, no evidence for a significant difference between the
faculties b) ß = .8420
2.
a) .1151, b) .3409
3.
a) Z = -1.506, Do not reject Ho, no evidence that the new procedure is more effective b)
1 - ß = .2877