Download Mid1S01V1Sol

SOLUTIONS TO MIDTERM 1
VERSION 1
1) The range is the largest value minus the smallest value, 161=15. Answer is D.
2) The answer is E. You should be able to see this without actually calculating
the sample standard deviation. Note that the sample standard deviation can never
be negative, it isn’t zero unless all the values are the same, .001 is clearly too
small and 15 is clearly too large. (Think about the empirical rule).
3) The median is zero, and the interquartile range is also zero. The mean is
slightly greater than zero. The sample standard deviation is also slightly
greater than zero. The range is 1. Answer is B.
4) Since the sample standard deviation is zero, the values must all be the same.
Since the median is 3, the common value of all the data points must be 3.
Therefore, the sample mean is 3. Answer is C.
5) Since all values in the data set are equal to 3, the interquartile range is 0. (The
75’th percentile is 3, and the 25’th percentile is also 3). Answer is A.
6) The original data set had 10 values, all equal to 3, so they added to 30. The
new data set has 11 data points, summing up to 30+25=55. The mean is
55/11=5. Answer is A.
7) Answer is B, since if we change a point which is between the highest and
lowest values, the range may stay the same. In the data set {0,5,10}, if we
change the 5 to a 7, the range remains unchanged at 10.
8) Answer is C. Since it’s possible that both consumer confidence and consumer
spending will drop next year, we conclude that the events are not mutually
exclusive. Note that the events are not independent, since if we know that
consumer confidence will drop, this makes it more likely that spending will
drop as well. The events are not complements of each other, since knowing
that consumer confidence will not drop is not the same thing as knowing that
spending will drop.
9) Answer is E. “At Least Two Heads” means “Two Heads or Three Heads”.
The events A and B are not mutually exclusive, since they can occur
simultaneously if we get three heads. A and B are not independent, since if we
know that A did not occur, then this makes it impossible for B to occur. So A
contains information about B. A and B are not complements of each other. We
have A ={Zero Heads or One Head}, which is not the same as B.
10) There are eight equally likely possibilities, HHH, HHT, HTH, THH, HTT,
THT, TTH, TTT. Of these, four give at least two heads. (They are HHH,
HHT, HTH, THH). So the probability of getting at least two heads is 4/8=1/2.
Answer is B.
11) The event AB is the same as B here, since the only way we can get both A
and B to occur simultaneously is to get three heads. Therefore,
P(AB)=P(B)=1/8. (HHH is the only possibility). Answer is D.
12) P(B|A)=P(BA)/P(A)=(1/8)/(1/2)=2/8=1/4. Answer is B.
13) The addition rule is P(AB)=P(A)+P(B)  P(AB). Rearranging this
formula gives P(AB)= P(A)+P(B)  P(AB)=.4+.7.8=.3. Answer is D.
You cannot just multiply P(A) and P(B) together, since A and B are not
independent. How do we know they can’t be independent? Because when we
calculated P(AB) above, we got something different than P(A)P(B).
14) There are three distinct possibilities: The price either goes up, goes down or
remains unchanged. The event B is the same as {Price goes up or remains
unchanged}. That is, B=AC. Since A and C are mutually exclusive (the
price can’t go up AND remain unchanged), we get P(B)=P(A)+P(C), so that
P(A)=P(B) P(C)=.7.1=.6. Answer is C.
15) The event AB is the same as B, that is, the price goes up or remains
unchanged. So P(AB)=P(B)=.7. Answer is D.