for Answers Practice Questions for Exam 1
1. B. The tail points to the lower numbers
2. C. About half the scores are lower than 155 and half are higher.
So, the midpoint
of the distribution is close to 155.
3. A. Most of those scores are higher than 120, except 94.
4. B. We should add together all the students who took longer than 90 min to finish
the test. About 6 took between 90 ~105 and about 4 took between 105~120. The
total should be around 10.
5. B. The balancing point of the distribution is between 45~60. The center should be
between 45~60 min.
6. A. The data is slightly skewed right.
median.
7. C. Pos.= (n+1)/2=20/2=10.
So the mean will be higher than the
75 is at position 10. The median is 75.
8. C. IQR= Q3-Q1= 83-62=21.
9. A. Use your calculator. Instructions as :
http://www.geocities.com/calculatorhelp/index.html
10. B Since it follows the standard normal distribution, the standard deviation(s)
equals to one. S 2 = variance = 1.
11. A. Z= (42-50)/10=-0.8. Enter table A under Z=-0.8. The table entry is 0.2119.
This is the area to the left of 0.8. Because the total area under the curve is 1, the
area lying to the right of 0.8 is 1-0.2119=0.7881.
12. A. Standardize first, and we can get 1.7 < x < -0.2.
total area should be .4207 .0446= .3761
Then, look at the table, the
13. D. We want to find the score x with area 0.03 to its right under the normal curve.
Thats the same as finding the score x with area 0.97 to its left. Use the table A
for the entry close to 0.97. It is 1.88. Then, unstandardize it, x = 50 +
1.88*10=68.8.
14. C. The entry corresponding to z= -1.34 is 0.0901 (this is the probability less), so,
this shows most of students scores ( 90%) are higher than you.
15. C Z= (48-60)/12=-1
16. B. Look the table A, the entry corresponding to the z-score is 0.1587.
17. A. Unstandardize the z-score. X= 60+12*5.2=122.4. 122.4 is much larger than the
mean( =60).
18. B. Range= max- min. And here, the first quartile is bigger than the min, so the
IQR = Q3-Q1 is smaller than the range.
19. A. For example, N~(2, 20) is also a normal distribution. The mean can be positive,
negative or zero.
20. D. Because the variance involves squaring the deviations, it does not have the
same unit of measurement as the original observations. However, the standard
deviation s measures spread about the mean in the original scale.
21. A. Z = (9200-7500)/ 1500=1.13. Enter table A under Z=1.13. The table entry is
0.8708. This is the area to the left of 1.13. Because the total area under the curve
is 1, the area lying to the right of 1.13 is 1-0.8708=0.1292
22. C. Standardize first, and we can get 1.4 < x < .47.
total area should be .6808 .0808= .6
Then, look at the table, the
23. D. So, for x = 4500, you find that the z-score = (4500 - 7500) / 1500 = -2
Then for x= 9000, you find that the z-score = (9000 - 7500) / 1500 = 1
The probability less than -2 equals 0.0228.
The probability less than 1 equals 0.8413 .
So, the probability is 0.8413-0.0228 = 0.8185 or about 81.5% is between 4500 and
9000
24. B. Since the median is smaller than the mean, we can conclude its a skewed right
graph.
25. B. Since it is a skewed distribution, the median is better than mean.
26. B. For the skewed right graphs, the mean is farther out in the long tail than is the
median. So, the median is less than the mean.
27. A. Approximately 95% of the observations fall within 2 of .
62-2*12=38. 62+2*12=86. So the range is 38~86.
28. A. Standardize first, and we can get 1 < x < 1. Then, look at the table, the total
area should be .8413 .1587 = .6826. Or use 68. 95. 99.97 rule, 50 and 74 are 1
standard deviation from . So 68.5% of the data is between the points.
29. B. We want to find the score x with area 0.05 to its right under the normal curve.
Thats the same as finding the score x with area 0.95 to its left. Use the table A
for the entry close to 0.95. It is 1.645. Then, unstandardize it, x = 62 +
1.645*12=81.74.
30. B. Z= (45-62)/12=-1.42. Enter table A under Z=-1.42. The table entry is 0.0778.
.0778*100%=7.78%
31. B. The definition of slope. (For every one unit increases in x, y increased by b
units).
32. A. r=
r 2 =0.6862944.
The slope is positive, so r is also positive.
33. D. The use of a regression line for prediction far outside the range of values of the
explanatory variable that you used to obtain the line.
34. C. Our predicted sales is 762+18.53*90=905.7.
So the residual should be res = obs y- predy= 1100-905.7=194.3.
35. B. When you substitute x into the equation of the regression line, the result of
y is always equal to the mean of y. ( p 142)
36. B. The use of chance to divide experimental units into groups is called
randomization.
37. C. Voluntary response sample consists of people who choose themselves by
responding to general appeal.
38. D. An observation study observes individuals and measures variables of interest
but does not attempt to influence the response.
39. A. When we confound two variables means that we cannot distinguish their
effects on the response variable.
40. A. In a matched pairs experiment we have experimental blocks consisting
of two subjects, each one receiving one of the two treatments, or we have
each subject receiving both treatments.
41. A. There is not an apparent trend to the data.
We cannot find any relationship
between those two variables.
42. C. Since we know the correlation is 0.755, theres a fairly strong positive linear
relationship between those two variables.
43. C. We can use those formulas: slope = b = r *
b = r*
Sy
Sx
sy
sx
and intercept = y - b* x . = a
=.755* (11.35/13.66)=.627.
a = y - b* x =78.4- .627*(74.7)=31.5.
44. B. The point doesnt cause the regression lines slope to change, so it is not
influential.
TEACHER EVALUATION
90
80
70
60
55
65
75
85
95
STUDENT GRADE
45. B. The definition of slope. (For every one unit increases in x, y is expected to
change by b units).
46. A. X = 0 makes sense here, so the average GPA of someone who doesnt drink is
3.45.
47. A. The definition of R 2 , is the percentage of the variation of y that is explained by
x.
48. C. Since the slope is negative, the r= - r 2 = -.564.
49. C. (22+6)/(100+100)
50. A. An experiment deliberately imposes some treatment on individuals in order to
observe their responses.
51. C. It shows clearly a strong negative linear relationship between two variables.
52. D. More training is associated with less injuries.
53. C. The use of a regression line for prediction far outside the range of values of the
explanatory variable that you used to obtain the line.
54. C. The point at (4 , 32) is an outlier and it has a large residual.
55. C.
56. A. We can see a positive linear trend in the scatter plot. The data set is positive
because x increases so does y.
57. B. A lurking variable is a variable that is not among the explanatory or response
variables in a study and yet may influence the interpretation of relationships
among those variables.
58. D. Measures an outcome of a study.
59. B. Explains or causes changes in the response variable.
60. C. A lurking variable is a variable that is not among the explanatory or response
variables in a study and yet may influence the interpretation of relationships
among those variables.
61. A. The individuals on which the experiment is done are the experimental units.
62. C. Since the risk of developing heart disease is getting higher either when sleeping
too little or too much, the scatter plot should have a curve trend between those two
variables.
63. D. The individuals on which the experiment is done are the experimental units.
64. A. The explanatory variables in an experiment are called factors.
65. B. A specific experimental condition applied to the units is called a treatment. If
there is one factor, the number of treatments equals to the number of levels.
66. A. A response variable measures an outcome of a study.
67. B. The individuals on which the experiment is done are the experimental units.
68. A. Since we select the random sample to represent the whole population of
women.
69. D.
70. A. Because the experiment enables us to control the effects of outside variables on
the outcome. A blind experiment is an experiment where participant does not
know what treatment he is getting.
71. A. Slope = r *
sy
sx
.
b = (0.9)*(1.2/3.6)= 0.30. So we can calculate the slope
equals to .30
72. B. Find the R 2 = .9*.9=.81. Definition of R 2 : the percentage of the variation of y
that is explained by x.
73. C. The probability P(A) of any event A satisfies 0 P(A) 1.
74. B. Because we infer conclusions about the population data from on selected
Individuals (all sample).
75. B. A parameter is a number that describes the population.
A statistic is a number
that describes the sample.
76. A. P (8 lights are working)=(.9) 8 =.4305 because of independence. OR
This is also a binomial where X= the number of light bulbs that work. So, in this
case, we have n= 8, X=8, and p = 0.90.
So, this is (8 choose 8) * 0.98*0.10. Since (8 choose 8 ) = 1 and 0.10 = 1, this
reduces down to 0.98.
77. B. P (at least one of the lights is not working)
=1- P(all lights are working) =1-.4305=.5695
78. A We have an X which is the number of successes. The situation follows the
requirements for a Binomial Experiment. X~ Binomial(n, p)
X~ Binomial ( 10,
0.2)
79. C Since X is binomial we can use the binomial probability function to find the
probability that X = 1.
n
10
P ( X = 1)= P ( X = k ) = p k (1 p ) n k = P ( X = k ) = 0.21 (0.8)10 1 = 0.268435
k
1
80. A We have an X which is the number of successes. The situation follows the
requirements for a Binomial Experiment. X~ Binomial(n, p)
X~ Binomial ( 20,
0.6).
81. A Since X is binomial we can use the binomial probability function to find the
probability that X = 2.
n
20
P ( X = 2)= P ( X = k ) = p k (1 p) n k = P ( X = k ) = 0.06 2 (0.94) 20 2 = 0.225
k
2
82. D Since X is binomial we can use the binomial probability function to find the
probability that X 1.
20
P ( X 1)= 1 P ( X < 1) = 1 P ( X = 0) = 1 0.060 (0.94) 20 0 = 0.710 .
0
83. A, A Probability that one cap has a winning symbol is 0.10. Since the caps are
independent, you can multiple the probabilities of getting a winning symbol on the
first cap and on the second cap and on the third cap, fourth, fifth and sixth to get
the probability that they all have winning caps. P(all have winning symbols) =
P(1st cap wins)*P(2nd cap wins) *P(3rd cap wins) * P(4th cap wins)* P(5th cap
wins)*P(6th cap wins) = 0.10*0.10*0.10*0.10*0.10*0.1=0.106
84. B The probability that cap does not have a winning symbol is 0.90. The caps are
once again independent so you can multiply the probabilities that the caps do not
have winning symbols.
P(all have winning symbols) = P(1st cap doesnt win)*P(2nd cap doesnt win)
*P(3rd cap doesnt win) * P(4th cap doesnt win)* P(5th cap doesnt win)*P(6th cap
doesnt win) = 0.90*0.90*0.90*0.90*0.90*0.9=0.906
85. C The P(at least one bottle wins) = 1- P(none of the bottles win) = 1- 0.96
86. D P(at least one of the bottles doesnt win) = 1- P(all of the bottles win) = 1-0.16
87. C, in general a slope k implies that a one unit increase in the X variable will give a
k unit increase in the Y variable.
In this case, for every one unit (ft/s) increase of
the boxcar, the number of damaged cans increases by k.
88. C, (42.9/1.895)*(0.522 ) = 11.8
89. A, since we can technically have a speed of 0 from the boxcar, it follows that
plugging in 0 in an equation with an intercept of 17.343 would imply that at a
velocity of 0, we will have about 17 damaged cans.
90. D,Y(15) = 25.3 + 12.6*(15) = 214.3
91. C, in a binomial experiment, we know that each trial is independent of the other
ones, so it would not make sense for one womans response to depend on another
womans response.
20
92. B,P(X=10) = *(.45)10(.55)10 = 0.1593
10
93. A, E(X) = np = (100)(.45) = 45
94. B, P(X>= 6) = 0.04 + 0.04+ 0.2 = 0.28
95. A
E(X) =
= (1)(.24) + (2)(.18)+(3)(0.13)+(4)(0.10)+(5)(0.07)+(6)(0.04)+(7)(0.04)+ (8)(.20)
= 3.86
96. D; Correlation has no units and is bound between negative one and one.
97. B;
Since correlation has no units, its value will remain the same whether length
is measured in feet or inches. Remember that correlation measures the direction
and the strength of the relationship. Dividing both variables by 12 will not change
the direction or the strength.
98. C;
Parameters describe the population and are unknown. Statistics are estimates
for these parameters calculated from the sample.
99. A;
Since the normal distribution is continuous over the real number line, that
means that there is an infinite amount of numbers on the line. You can think of
infinite as a really really big number, like 9999999.
So, the probability at one
point can be thought of as 1/infinity = almost zero.
(type 1/9999999 into your
calculator and you will see this is approximately zero).
100. A;
P(X<x)=.025
Z= (x-266)/16=.025
Looking up .025 in the Z-Table yields approximately -1.96.
Thus,
x= z * sigma + mu = -1.96*16 + 266 = 234.64
101.
A; In a probability distribution, the total probabilities must sum to one. Thus,
1= P(0) + P(1) +P(2)
1 =.25 + Q + .25
1=.5 + Q
Q= 1-.5
Q=.5
102. A; roman numeral one is true because an extremely large value out in the
right-hand tail pulls the mean to the right; roman numeral two is true because the
calculation of the mean uses all the numerical values, so unlike the median, it
depends on how far observations fall from the middle, therefore the median is said
to be resistant to extreme observations; roman numeral three is false because the
standard deviation, which uses the mean, is very sensitive to extreme
observations.
103. A; roman numeral one is false because correlation does not imply causation, it
could be that the junk food causes weight gain, that the weight gain causes
increased junk food, or some variable C causes weight gain; roman numeral two is
false because the question states that it was a survey and that no treatments was
applied; roman numeral three is true because it gives the correct strength and
direction of the correlation.
104. D; Variance=r2=(.7183)2=.516, convert to percent by multiplying by
100=51.6%Interpretation of R-squared:
It is the variability of y (response time)
that is explained by x (number of hours slept).
105.
D; A normal distribution is by definition symmetric which means if there is the
same shaded areas of the upper and lower tails, their probabilities are equal; the
mean is equal to the median because the distribution is symmetric; a normal
distribution is by definition continuous.
106.
D; when rolling a fair dice there is an equal probability (1/6) of getting a 1, a 2,
a 3, a 4, a 5, or a 6, thus the distribution is uniform; This particular type is a
discrete uniform distribution. each of the observations are equally likely because
the problem states that it is a fair dice;
x
1
2
3
4
5
6
P(x)
1/6
1/6
1/6
1/6
1/6
1/6
the mean is (1/6)1 + (1/6)2+ (1/6)3+(1/6)4+(1/6)5+(1/6)6 = (1/6) (1+2+3+4+5+6)
=(1/6) 21=
107.
3.5.
B, To be a big deal, you must score higher than 97.5% of the others. The score at
the 97.5th percentile (z=1.96), is x= 1.96*4.5+100 = 108.82. Since both Veronica
and Ron scored higher than 108.82, both are a big deal.
108.
A, since z(grandaddy) = (4.4-4.82)/.3 = -.42/.3 = -1.4
and z(5.12) = (5.12-4.82)/.3 = 1,
so P(less than -1.4) =0.0808 and P(less than 1) = 0.8413
Probability in between 4.4 and 5.12 is 0.8413 0.808 = 0.7605
109. A since Shelley has only dealt with a SAMPLE of the skin boarders.
110. A
P(X<2) = P(X<=1) = P(X = 0) + P(X = 1)
13
13
= [ *(.15)0*(.85)13] + [ *(.15)1*(.85)12]
0
1
=.121 + .277 = .398
111. A; It is a categorical variable; zip code is a category that just happens to be
numeric. If you averaged 5 zip codes, the average has no meaning.
112. C; To find the missing probability we realize that the total probabilities sum to 1,
so 1 1/36 1/18 -1/12 -1/9 -1/6 -5/36-1/9-1/12-1/18-1/36 = 5/36
113. C;
This is binomial, because we have independent and identical n trials, and
constant proportion p. In this case, p = 0.5 because it is a fair coin and n =10.
n x
5 3 2
nx
= 5/16
p (1 p ) = .5 .5
x
3
114.
A; Y = 29.25 + .57X
Y=29.25+.57*78 = 73.71
115. B; Because each trial is independent and we have 50 trials and we know our
proportion is .70 and each outcome is either a success or failure, we know that X is
Binomial with p=.70 and n=50
116. To do this problem, you want to first identify that this is a "given probability"
problem. The problem asks for us to find the score associated with the top 10%. Our
table doesn't give us top percentages. However, the number that is the top 10% is also
the 90th percentile or the point in which 90% is below. So, look up 0.90 in the middle
of the table. 0.90 is not exactly in the table. The two closest values are 0.8997 and
0.9015. 0.8997 is closest to 0.90. So, we are going to get the z-score to go with that
value. So, look all the way to the right and all the way to the top. You get 1.28. 1.28 is
the z-score.
So,
x= z * sigma + mu = 1.28*10 + 65 = 77.8
OR
1.28 = (x 65) / 10
12.8 = x -65
X = 77.8