1)
Based
on census data, 8.2% of households own no vehicles, 69.1% own one, 17.8% own
two and 4.9% own three. Verify that this is a proper probability distribution.
2)
For
the distribution provided in question 1, compute the mean and standard
deviation. (mean = 1.194; standard deviation = 0.6468)
3)
According
to Chebyshev’s Theorem, at least 75% of a distribution lies within 2 standard
deviations of the mean. Verify this for the distribution in question 1.
4)
Sara
is a telemarketer who makes 20 calls per day. Her average success rate is 5%.
What is the probability she will make exactly 2 sales per day? (18.87%)
5)
What
is the probability that on any given day she will make no more than 1 sale?
(73.59%)
6)
Suppose
her goal is to make at least 2 sales per day. What is the probability she will
do that? (26.41%)
7)
Suppose
she earns a commission of $450 per sale. If she works 5 days a week and makes
the same number of calls each day, what is the most number of sales she can
expect to make 99.7% of the time? Round to the nearest whole number. Based on
that, what is the maximum commission she can expect to make? ($5400)
8)
In
an automobile manufacturing plant, approximately 1 gear in 5000 is scrapped. If
the plant produces 500 gears per shift, what is the probability that none of
the gears is scrap? (90.48%)
9)
The
plant has 3 shifts per day. What is the probability that in this time frame
that at least 1 gear is scrap? (25.92%)
10)
The
plant has a quality control goal of having no more than 6 scrap gears in a
7-day period. What is the probability of this? Is this a realistic goal?
(99.42%; it seems realistic)
11)
Using
Chebyshev’s theorem, what is the maximum number of scrap gears they can expect
in a 7-day period at least 88.9% of the time, rounding to the nearest whole number?
Why is Chebyshev’s probability conservative? (6 gears; as we see in the
previous question, the probability of no more than 6 scrap gears is 99.42%
which is considerably higher than the 88.9% under Chebyshev’s theorem.)
12)
In
the same plant, there is 1 accident per 60 days on average. What is the
probability the plant has no accidents in a 30-day period? (60.65%)
13)
What
is the probability that there are at least 2 accidents in a 180-day period?
(80.09%)
14)
Using
Chebyshev’s theorem, what is the maximum number of accidents they can expect in
a 360-day period at least 88.9% of the time, rounding to the nearest whole
number? What is the probability the plant has no more than this many accidents
in this time frame? (13 accidents; 99.64%)
15)
A
census of corporate executives showed that 2% own neither a cell phone nor a
blackberry, 9% own just a cell phone, 3% own just a blackberry and 86% own both
devices. Compute the mean and standard deviation of the total number of devices
that corporate executives own. As well, compute the percentage of variation in
owning one device that is explained by owning the other device. (mean = 1.84;
standard deviation = 0.4176; r2 = 4.52%)