Statistics 2
Problems from past midterms: midterm 2
1. (10 points) One hundred draws are made at random, with replacement, from the box:
1
1
1
1
2
Find, approximately, the chance of getting 80 1 ’s and 20 2 ’s.
2. (5 points) Ray and Bob play the following game once. A red die and a blue die are
rolled. Ray pays Bob, in dollars, the number on the red die and Bob pays Ray, in dollars,
the number on the blue die. Find the chance that Ray is down more than a dollar
at the end of the game.
3. (10 points) A gambling house offers you the following game. A ticket is chosen at
random from the box below.
W
I
N
O
R
P
A
Y
U
S
If the letter on the ticket comes from WIN, the house gives you $1. If it comes from PAY,
you must pay the house $1. If it doesn’t come from either word, no money changes
hands. The ticket is returned to the box before the next play. Suppose you decide to
play this game 40 times.
(a) In 40 plays, your net gain will be around $__________, give or take $_______or so.
(b) No money will change hands on around _______________ of the 40 plays, give or take
__________ or so.
4. (5 points) A die is rolled twice. Find the chance that
both.
comes up on one roll, but not
5. (5 points) What is the main thing wrong with quota sampling?
6. (5 points) First one ticket, and then a second one, are drawn at random from the box
shown below. The draws are made without replacement.
A
B
C
D
E
Find the chance the letters on the two tickets are next to each other in the alphabet.
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7. (5 points) Box A contains 100 marbles, some are red, the rest are blue. Box B is like
box A except there are twice the number of red marbles and twice the number of blue
marbles. So box B contains 200 marbles. Marbles will be drawn one at a time, at
random, from one of the two boxes. You are given two choices:
(a) 100 draws with replacement will be made from box A.
(b) 200 draws with replacement will be made from box B.
After you make the choice, the draws are made from the box you chose, and you win
a dollar if the number of times a red marble comes up in the draws is the same as the
number of red marbles in the box.
Check (√ ) one of the four options below.
_____ (a) gives a better chance of winning.
_____ (a) gives a smaller chance of winning.
_____ (a) and (b) give the same chance of winning.
_____ Can’t tell without more information.
State your reasoning.
8. (5 points) Studies often involve taking a sample from a population. True or false, and
explain: “In most such studies the investigator can decide, by examining the sample
carefully, whether or not it is a representative cross-section of the population for the
variables under investigation.”
9. (5 points) A city contains 120,000 households and, taken together, there are 180,000
private automobiles registered to these households. As part of a transportation
study, the planning office is going to take a simple random sample of 1,000 households
from the city. Find, approximately, the chance the average number of cars per household
in the sample will be bigger than 1.5.
10. (10 points) A study involves following a large number of newly-married couples through
their child bearing years. It turns out that 1,200 of these couples have two children. For
a two-child family, there is about 1 chance in 4 that both children are girls; 1 chance in
4 that both children are boys; and 1 chance in 2 that one is a girl and the other is a
boy.
Out of the 1,200 families, the number with both children girls will be around _________
give or take _________ or so.
11. (10 points) Twenty-five draws are made at random, with replacement, from the box:
1
1
1
2
2
Find, approximately, the chance that 1 shows up more often than 2 does in the
twenty-five draws.
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12. (5 points) Four people take turns rolling a pair of dice (6-sided).
(a) Find the chance that everyone rolls snake-eyes.
(b) Find the chance that no one rolls snake-eyes.
You do not have to work out the arithmetic. Note: You roll “snake-eyes” when both dice
show a single dot.
13. (5 points) A deck of cards is shuffled and three cards dealt off the top. What is the
chance that none of them are aces or kings?
14. (10 points) A certain town has 10,000 occupied rental units. A local real estate office
does a survey of these units: 400 are chosen by a simple random sample, and the
occupants are interviewed. Among other things, the rent paid in the previous month is
determined. The 400 sample rents average out to $574 with an SD of $380.
(a) Estimate the average rent in the town.
(b) Attach a standard error to the estimate in (a).
(c) Find a 95% confidence interval for the percentage of units in the town which rent for
more than $900. If this is not possible, explain why not.
15. (10 points) Twenty-five draws are made at random with replacement from the following
box:
1
2
3
4
5
6
7
Find, approximately, the chance that the sum of the 25 draws turns out to be 102.
16. (10 points) There are 125,000 households in a certain city. The 10th, 50th and 90th
percentiles of household incomes are $15,300, $52,100 and $130,000, respectively. A
market research organization plans to take a simple random sample of 400 households
from the city.
The number of households in the sample with incomes over $130,000 will be
around
give or take
or so.
17. (10 points) A coin is tossed 100 times. Find, approximately, the chance of getting
55 heads or 55 tails.
18. (5 points) A gambler plays Red and Black three times in a row. Find the chance he only
wins the first time.
(The roulette wheel has 38 pockets: 18 red, 18 black, 2 green. To play “Red and Black”,
let’s say you choose “red”. Then you bet a dollar on red. The croupier spins the wheel
and drops a ball inside. If the ball comes to rest in a red pocket, you get your dollar back
together with another dollar in winnings. If the ball drops in a black or green pocket,
you lose your dollar.)
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like east-side Manhattan or San Francisco’s Nob Hill, of course, rents are much
higher—if you can find an apartment in the first place.)
A certain town has 10,000 occupied rental units. A local real estate office does a
survey of these units: 250 are chosen at random, and the occupants are interviewed.
Among other things, the rent paid in the previous month is determined. The 250
sample rents average out to $568, and the SD is $385. A histogram is plotted for
the sample rents, and does not follow the normal curve.
19. (10 points) Victor and Conrad play the following game. A ticket is drawn at random
from the box:(a) If possible, find a 68%-confidence interval for the average rent paid in the
previous month on all 10,000 occupied rental units in this town. If this is not
possible, explain
R why
E not.
A D Y
(b) True or false, and explain: For about 68% of all the occupied rental units in
this ticket
town, theisrent
paid in the
previouspays
monthVictor
was between
$592.is a consonant,
If the letter on the
a vowel,
Conrad
$1. If$544
theand
letter
8. (Continues
7; hard.) if
True
false, and
if another
occupied hands. They
Victor pays
Conrad exercise
$1. However,
theor letter
is explain:
a “Y,” no
money250
changes
rental
units
were
taken
at
random,
there
would
be
about
a
68%
chance
for
theanew
agree to this because they can’t remember whether “Y” is a vowel or
consonant.
sample average to be in the range from $544 to $592.
Suppose they play twenty times, replacing the ticket in the box each time. Victor begins
9. (Hard.) Census data are available on 25,000 families in a certain town. For all 25,000
with $30. Find,
approximately,
the
chanceand
hetheends
withAmore
$35.
families,
the average income
is $61,700
SD isup
$50,000.
marketthan
research
firm takes a simple random sample of 625 out of the 25,000 families. The figure
below is a probability histogram for the average income of the sample families;
A medical
organization
serves
families.
The
20. (10 points) the
histogram is
drawn in standard
units. 25,000
The average
income of
the average
625 sampleannual income
of these families
is
$31,700
with
an
SD
of
$20,000.
The
organization
takes a simple
families turned out to be $58,700 and the SD was $49,000.
random sample
of 400
of thesebelow,
families.
The sketch
Options:the probability histo(a) On
the histogram
+1 in standard
units is below .shows
gram for the average income in the sample. When the sample was taken, the average
$60,660
$63,700
annual income of the 400
sample
families$107,700
turned out$111,700
to be $30,700 with a SD of
$19,200.
–3
–2
–1
0
1
STANDARD
(i) UNITS(ii)
2
3
(a) In the sketch, the numbers on the horizontal axis are missing.
What number belongs at (i) ?
(b) About 34% of the total area under the histogram lies between (i) and (ii).
What number belongs at (ii)?
21. (5 points) A Gallup Poll pre-election survey is based on a sample of 1,000 people from a
certain district. In the sample, 650 people say they will vote for the Democratic
candidate in the upcoming election. If possible, find an approximate 95% confidence
interval for the percentage of voters in the district, who, at the time the survey is
carried out, would have said they would vote for the Democratic candidate. If this is not
possible, explain why not.
22. (5 points) Someone tosses a coin ten times every day for a month. Someone else does it
every day for six months. Which person is more likely to have a day where the coin is as
far from five heads and five tails as possible; that is, where the coin comes up all heads
or all tails?
23. (5 points) A pair of dice, one red die and one blue die, are thrown. Find the chance the
outcome on the blue die is one more than the outcome on the red die.
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24. (5 points) A mini-deck of cards consists of only five cards: two are hearts, two are
dia monds, and one is a club. The deck is shuffled. Find the chance that the top two
cards on the deck are hearts and the bottom two cards are diamonds.
25. (10 points) An environmental organization draws a sample random sample of 900
people from the residents of the Bay Area. Each person in the sample is asked, among
other things, how long it takes to commute from home to the workplace. The average
of these 900 reported commute times turned out to be 25 minutes. The SD was 18
minutes.
(a) Estimate what the average reported commute time would be if all Bay Area
residents had been interviewed.
(b) Attach a standard error to the estimate in (a).
(c) True or False, and explain: “The estimate in (a) might be off by as much as
two min utes.”
26. (5 points ) A small undergraduate college has 1,000 students, evenly distributed among
the four classes: freshman, sophomore, junior, and senior. Someone proposes to take a
sample of 100 students by the following procedure: 25 students will be selected at
random, without replacement, from each of the four classes.
(a) Is this a probability method?
(b) Is it the same as simple random sampling?
27. (10 points) A computer file contains medical information on 4,600 men and 5,400
women. A simple random sample of 100 people is chosen from the file. Find, approximately, the chance there are 46 men and 54 women in the sample.
28. (5 points) Five cards are dealt, one at a time, from a deck of cards. Find the chance that
the first and fifth cards are hearts, and these are the only hearts in the five cards. (You
do not have to work out the arithmetic.)
(A deck of cards contains 52 cards: 13 spades, 13 hearts, 13 diamonds, and 13 clubs.)
29. (10 points) A gambing house offers the following game, called MAGIC 10.
A box contains 5 tickets:
3
6
9 12 15
A ticket is drawn at random from the box. If the number on the ticket is bigger than
10, the house pays the gambler, in dollars, the amount by which it is bigger. If it’s
smaller than 10, the gambler pays the house (in dollars) the amount by which it is
smaller.
A gambler plays MAGIC 10 one hundred times.
(a) After the 100 plays, the net gain of the gambler will be around _________ give or
take _________ or so.
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(b) The gambler will win $5 on around ____________ of the 100 plays, give or
take ________ or so.
30. (5 points) A small city contains 60,000 households. The population is 142,500, of which
114,000 are age 16 and over. The health department plans to take a simple random
sample of 200 households, and interview all persons 16 and over living in the sample
households. Find, approximately, the chance that more than 380 people will be interviewed.
©2010 by Roger Purves
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