SOLUTIONS TO MIDTERM 1
VERSION 1
1) We have x 
1 n
 xi , which becomes (6+4+2+6+7)/5=25/5=5. Answer is B.
n i 1
2) The sorted observations are 2,4,6,6,7. The median is the middle value, 6.
Answer is D.
3) The range is the difference between the largest and smallest observations,
72=5. Answer is B.
4) The smallest observation is 2. The middle observation (median) is 4. The third
observation x must satisfy (2+4+x)/3=6, so x=186=12. The full data set is {2,
4, 12}. This has a sample mean of 6 and a sample median of 4, as required.
The sample standard deviation is
1 n
( xi  x ) 2  [( 2  6) 2  (4  6) 2  (12  6) 2 ] / 2  28  5.29.

n  1 i 1
Answer is D.
s
5) The sample mean is multiplied by 2, yielding (2)(0)=0, so it does not change.
The standard deviation doubles. Since not all of the data values are the same,
the original standard deviation is positive, and if you double a positive number
it changes. So the standard deviation does change. Answer is B.
6) Suppose for example that all observations are equal to 5, except the first
observation, which is 1. Since both quartiles will be 5, the IQR will be zero,
but since all observations are not equal the sample standard deviation will not
be zero. Now, consider a different example where all observations are 5. Here,
both the IQR and sample standard deviation will be zero. So if the IQR is zero
then the sample standard deviation may or may not be zero. Answer is B.
7) We have z = (x− x )/s = (300000−200000)/50000=2. Answer is C.
8) The sample median (average of the two middle values) will not change unless
we are changing one of the two middle values. The interquartile range would
not change if we move a point that is below the 25’th percentile further to the
left, or if we move a point that is above the 75’th percentile further to the
right. The range will not change as long as the point we are changing is not
the largest or smallest data point, and we do not change our point enough so
that it becomes the largest or smallest data point. But the sample mean must
change since it is an average, which will change if a single data point is
moved. Answer is A.
9) Let A=Recession, B=Drop in Consumer Confidence. We have P(A)=0.3,
P(A∩B)=0.2. We need P(B|A)=P(B∩A)/P(A)=(0.2)/(0.3)=2/3. Answer is B.
10) Using the definition of conditional probability, since P(B∩A)=P(A∩B), the
given fact, P(B|A)=P(A|B) reduces to P(A∩B)/P(A)=P(A∩B)/P(B). There are
two ways for this to occur. Either both sides are zero (which happens if and
only if A, B are mutually exclusive), or else we can divide on both sides by
P(A∩B) to find that 1/P(A)=1/P(B), that is, P(A)=P(B). Answer is C.
11) Remember the multiplication rule for independent events,
P(A∩B)=P(A)P(B). Remember also the general addition rule for any pair of
events, P(AB)=P(A)+P(B)P(AB). But since here A, B are independent,
this last equation becomes P(AB)=P(A)+P(B)P(A)P(B). Adding P(A)P(B)
to both sides of this we get P(A)+P(B)= P(AB)+P(A)P(B). Answer is D.
Since A, B are independent and therefore not mutually exclusive, we cannot
say that P(A)+P(B)=P(AB). Answer is D.
12) If the range is zero, all values must be the same. But they need not all be
zero, so we cannot conclude that the sample mean is zero. Since all values are
the same, the interquartile range will also be zero, as will the sample standard
deviation. Answer is B.
13) A=Rain, B=Lightening. We need P( A  B )  P( A  B)  1  P( A  B) .
Note that P(AB)=P(A)+P(B)P(AB)=P(A)+P(B) P(A)P(B|A), which
from the given information becomes .3+.2(.3)(.4)=.5.12=.38. So the
probability of neither rain nor lightening is 1.38=.62. Answer is D.
14) Using the rule of complements, we have 1P(A) > 1P(B), which is the
same as P(B) > P(A). Answer is B.
15) Note that P(B)=1/2, while P(B|A)=1, since if we know that the number was a
four, we know that it was even. Since P(B) is different from P(B|A), the
events are not independent. Since A and B can both happen (note that
AB=A), the events are not mutually excusive. Clearly, the events are not
complements of each other. Answer is D.