Review:  Looking for simple step-by-step explanations and solutions
(preferably by Sunday afternoon)

--1. Suppose the manager of a large farm wanted to study the impact of
the fruit flies on the orange crops on a daily basis over a 6-week
period. On each day a random sample of orange trees were selected from
within a random sample of acres. The daily average number of damaged
oranges per tree and the proportion of trees having damaged oranges
were calculated. The two main measures calculated each day (i.e.,
average number of damaged oranges per tree and proportion of trees
having damaged oranges) are called?

--2. This table contains the opinions of a sample of 200 people broken
down by gender about the latest congressional plan to eliminate
anti-trust exemptions for professional baseball.
           For  Neutral Against Totals 
Female     38    54         12     104 
Male       12    36         48     96 
Totals     50    90         60     200 
(a), construct a table of total percentages. 
(b) of those FOR the plan, ________ percent were females. 
(c) of the males in the sample, ________ percent were FOR the plan. 

--3. Health care issues are receiving much attention in both academic
and political arenas. A sociologist recently conducted a survey of
citizens over 60 years of age whose net worth is too high to qualify
for Medicaid by who have no private health insurance. The ages of 25
uninsured senior citizens were as follows:

60 61 62 63 64 65 66 68 68 69 70 73 73 
74 75 76 76 81 81 82 86 87 89 90 92 
*Identify the third quartile of the ages of the uninsured senior
citizens.

--4. The stem-and-leaf display represents the number of vitamin
supplements sold by a health food store in a sample of 16 days.
Stem    Leaves 
1H      99 
2L      0023 
2H      567 
3L      034 
3H      568 
4L       1 

For this sample, the sum of the observations is 448, while the sum of
the squares of the observations is 13,356.
Note: 
1H means “high tens”: 15, 16, 17, 18, or 19; 
2L means “low twenties”: 20, 21, 22, 23, or 24; 
2H means “high twenties”: 25, 26, 27, 28, or 29; etc. 

*What is the standard deviation of the number of vitamin supplements
sold in this sample?

--5. The probability that a new advertising campaign will increase
sales is assessed as being 0.80. The probability that the cost of
developing the new ad campaign can be kept within the original budget
allocation is 0.40. Assuming that the two events are independent, the
probability that the cost is kept within budget and the campaign sales
increases is?

--6. Suppose A and B are mutually exclusive events where P(A) = 0.4
and P(B) = 0.5. Then P(A and B) = ?

--7. A survey is taken among customers of a fast-food restaurant to
determine preference for hamburger or chicken. Of 200 respondents
selected, 75 were children and 125 were adults. 120 preferred
hamburger and 80 preferred chicken. 55 of the children preferred
hamburger.  What is the probability that a randomly selected
individual is a child or prefers hamburger?

--8. Suppose there are three events A, B, and C with probabilities: 
P(A) = 0.50, P(B) = 0.40, P(C) = 0.30. 
Also, A and B are independent; A and C are mutually exclusive; P(B and
C) = .05.

(a) P(C') = 
(b) P(A and C) =

--9. For a potential investment of $5000, a portfolio has an EMV of
$1000 and a standard deviation of $100. What is the coefficient of
variation?

--10. If we know that the length of time it takes a college student to
find a parking spot in the library parking lot follows a normal
distribution with a mean of 3.5 minutes and a standard deviation of 1
minute, find the point in the distribution in which 75.8% of the
college students exceed when trying to find a parking spot in the
library parking lot.

--11. A food processor packages orange juice in small jars. The
weights of the filled jars are approximately normally distributed with
a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the
proportion of all jars packaged by this process that have weights that
fall below 10.875 ounces.

--12. Suppose Z has a standard normal distribution with a mean of 0
and standard deviation of 1. The probability that Z is between -2.89
and -1.03 is?

--13. The amount of time necessary for assembly line workers to
complete a product is a normal random variable with a mean of 15
minutes and a standard deviation of 2 minutes. So, 70% of the products
would be assembled within how many minutes?

Hello sunshine2,
Glad to see you again! Here are the answers to your questions.

1. The average number of damaged oranges per tree is called, by
definition, the "sample mean". It's calculated by taking the number of
observed damaged oranges and dividing it by the number of trees in the
sample.

Sample Mean
http://www.animatedsoftware.com/statglos/sgxbar.htm

The proportion of trees having damaged oranges, is called "sample
proportion". It's calculated in the same way as the sample mean, as
you take the number of observed trees that contain damaged oranges,
and divide thsi number by the number of trees in the sample.

2. In the following table, I take the total sample size. Thus, the
percentage you see in the square (Female, For), would mean "what
percentage of all people in the mean is BOTH a female and for the
anti-trust plan?".

           For  Neutral Against Totals  
Female     19%   27%        6%     52%  
Male       6%    18%        24%    48%  
Totals     25%   45%        30%    100%
  
These percentages are calculated in the following way. For example,
the percentage in (Female, For) is calculated by taking the number of
females that are for the plan (38), dividing it by 200 (total number
of people in the sample) and multiplying it by 100. The same goes for
the other elements in the table.

3. The third quartile is "the value of a set of data at the third
quarter, 75%, when that data has been arranged in ascending order. It
is the median value of the upper half of all the values in the data
set"

Glossary O-U
http://www.tta.gov.uk/training/skillstests/numeracy/glossary/o-u.htm

In your case, we first have:
60 61 62 63 64 65 66 68 68 69 70 73 73 74 75 76 76 81 81 82 86 87 89
90 92

The upper half of your data is:
73 74 75 76 76 81 81 82 86 87 89 90 92

And the median of this subset is 81. Thus, the third quartile is 81.

For more information on what the median is, check

Glossary H-N
http://www.tta.gov.uk/training/skillstests/numeracy/glossary/h-n.htm

5. First let's see what "independent" means. Events A and B are said
to be independent if Prob(A and B) = P(A)*P(B).

Independent Random Variables
http://www.uwm.edu/~stockbri/14indep.pdf

Therefore,
A = The campaign will increase sales
B = The campaign can be kept within the original budget

We know that P(A)=0.8 and P(B)=0.4 and that A and B are independent.
Thus,
P(A and B) = 0.8*0.4 = 0.24 = 24%

There is a 24% probability that both events will happen.

6. First, let's see what the definition of mutually exclusive events
is.

"Two events are mutually exclusive if it is not possible for both of
them to occur..."

Mutually exclusive events
http://davidmlane.com/hyperstat/A132677.html

Thus, events A and B cannot happen at the same time; so, 
P(A and B)=0

7. In order to solve this one, we must first build a table like the on
in question 2. From the problem, we know the following data:

            Chicken    Hamburguer    Totals
Children                 55            75
Adult                                 125
Totals        80         120          200

Now, it's easy to fill the blank spaces, since we know that the sum of
lines and columns whould add up to the values in "Totals". For
example, we know that there are 75 children, 55 of which prefer
burguers; therefore 75-55=20 children prefer chicken. Now, we know
that a total of 80 people prefer chicken, 20 of which are children, so
80-20=60 adults prefer chicken. Lastly, since there are 125 adults,
and 60 prefer chicken, we know that 125-60=65 adults prefer
hamburguers. Thus we get the following table

            Chicken    Hamburguer    Totals
Children      20         55            75
Adult         60         65           125
Totals        80         120          200

From this table, we make one with percentages, just like in question
3. We get

            Chicken    Hamburguer    Totals
Children      10%        27.5%        37.5%
Adult         30%        32.5%        62.5%
Totals        40%         60%         100%

Finally, let's get to the question. What is the probability that a
randomly selected
individual is a child or prefers hamburger? Let's see which events
fulfill the condition (Children or Hamburguer). Children that like
chicken fulfill it, so do children that like hamburguers, and so do
adults that like hamburguers. Adults that like chickens do not fulfill
the condition. So now we just have to add the probability of the
events that fulfill the conditions in order to get the solution. So,
the probability of randomly sleecting someone who is a children or
likes hamburguers is 10% + 27.5% + 32.5% = 70%

8. First I'll solve P(C'). C' is the notation for "the complement of
C", which is equivalent to saying "not C". Thus P(C')=P(not C). Check
the following definition:

"Definition:  The complement of an event A is the set of all outcomes
in the sample space that are not included in the outcomes of event A.
The complement of event A is represented by A'.
Rule:  Given the probability of an event, the probability of its
complement can be found by subtracting the given probability from 1.
P(A') = 1 - P(A)"

Therefore,
P(C') = 1-P(C) = 1-0.3 = 0.7

Regarding P(A and C), this is just the same as in question 6. We know
from the question that A and C are mutually exclusive, so
P(A and C) = 0

9. We know that EMV is the mean of the value of the portfolio, and the
formula for the coefficient of variation is given in the following
page

Coefficient of Variation
http://engineering.uow.edu.au/Courses/Stats/File1586.html

Therefore, the coefficient of variation is: 100/1000=0.1

10. Let X be the number of minutes it takes a student to find a
parking spot. We know that X hasa normal distribution with mean 3.5
and standard deviation 1. The question here is find the number a that
solves

P(X > a) = 0.758

In order to solve this we must use a table for the normal
distribution. The tables you will find in books and on the web are for
standard normal distribution (that is, with mean 0 and standard
deviation 1). So we have to convert our random variable X into a
standard normal distributed variable. This is easily done: we have to
substract the mean and divide by the standard deviation. Thus, we have
that

P(X > a)
=P(X - 3.5 > a-3.5)

But X - 3.5 has a standard normal distribution, so I'll rename it as
Z. We are looking for:

P(Z > a-3.5) = 0.758

In the following page, you will find an applet that calculates P(Z <
a)

Z table
http://davidmlane.com/hyperstat/z_table.html

Thus, we need to rewrite your problem
P(Z > a-3.5)= 1-P(Z < a-3.5) = 0.758
So,
P(Z < a-3.5) = 1-0.758= 0.242

So, now we write 0.242 in the applet box called "Area below Z", and we
find that the z-value that solves this is -0.6996. Thus,
a-3.5=-0.6996
a=3.5-0.6996=2.8004

Therefore, 75.8% of the students take more than 2.8004 minutes to find
a parking spot.

11. This problem is similar to the previous one. Here we know that X
is distributed normally with mean 10.5 and standard deviation 0.3. We
have to find P(X<10.875). Again, we have to convert X to standard
normal, so we have that:

P(X < 10.875)
=P(X-10.5 < 10.875-10.5)
=P( (X-10.5)/0.3 < (10.875-10.5)/0.3)
=P( Z < 1.25)

We now have to plug 1.25 in the applet box called Z to find that
89.44% of the observations fall below this value; and thus 89.44% of
the jars weigh less than 10.875 ounces.

12. We will use the same applet here. We have to find out:
P(-2.89 < Z < -1.03)

But this is equal to say
P( Z < -1.03) - P( Z < -2.89)

So now we just have to plug these numbers in the applet box called Z,
finding that:

P(Z<-1.03)=0.1515
P(Z<-2.89)=0.0019

So, 0.1515-0.0019=0.1496. Thus, the answer is 14.96%


I hope these answers are clear enough. If you have any doubts, please
don't hesitate to request a clarification, I'll be glad to explain
further.


Best regards,
elmarto

---

Clarification of Answer by elmarto-ga on 19 Jun 2003 11:52 PDT

I'm sorry, I neglected questions 4, 13, part of question 2 and the
search terms I used. Here they are.

--------------------------------------------

2. I'll repost the whole answer to question 2 here and then add the
missing answers:

In the following table, I take the total sample size. Thus, the
percentage you see in the square (Female, For), would mean "what
percentage of all people in the mean is BOTH a female and for the
anti-trust plan?".

           For  Neutral Against Totals  
Female     19%   27%        6%     52%  
Male       6%    18%        24%    48%  
Totals     25%   45%        30%    100%
  
These percentages are calculated in the following way. For example,
the percentage in (Female, For) is calculated by taking the number of
females that are for the plan (38), dividing it by 200 (total number
of people in the sample) and multiplying it by 100. The same goes for
the other elements in the table.

(b) of those FOR the plan, ________ percent were females.

From the table in the question, we have that there are 50 people for,
38 of which are women. Thus the percentage of those FOR the plan who
were females would be (38/50)*100 = 76%

(c) of the males in the sample, ________ percent were FOR the plan

We know from the table that there are 96 males, 12  of which are FOR
the plan. Thus the percentages of males who are FOR the plan is
(12/96)*100 = 12.5%


4. In order to find the standard deviation (SD), we must first find
the variance. Then we will take the square root of the variance and
that will be the SD. The formula for the sample variance is:

          N
         ___
(1/N-1)* \    (xi-x)^2
         /    
         ---
         i=1

where N is the sample size (16), xi is each observation and x is the
sample mean. The sample mean in this case would be 448/16= 16. This
formula was taken from

Variance
http://mathworld.wolfram.com/Variance.html

Let's "play" with the term:
 (xi-16)^2

We can expand it to get:
(xi-16)^2
=xi^2 - 2*xi*16 + 16^2
=xi^2 - 32*xi + 256

So,
          N
         ___
         \    (xi-x)^2
         /    
         ---
         i=1

is equal to
          N
         ___
         \    xi^2-32*xi+256
         /    
         ---
         i=1

which is equal to

  N
 ___         ___           ___
 \    xi^2 - \   32*xi  +  \    256
 /           /             /
 ---         ---           ---
 i=1

(All sums are between i=1 and N)

We know what the first term is: it's the sum of the squares of the
observations, which we know is 13,356. The third term is simply 256
summed N times, so it's 256*16=4096. The second term can be rewritten
as

     N
    ___
 32*\    xi
    /    
    ---
    i=1

But we know that the sum of xi is 448, so the second term would be
32*448=14,336

Therefore, we have that

  N
 ___         ___           ___
 \    xi^2 - \   32*xi  +  \    256
 /           /             /
 ---         ---           ---
 i=1

equals 13,356 - 14,336 + 4096 = 3,116

So, the variance is (1/N-1)*3,116 = 3,116/15 = 207.73... Finally, the
standard deviation is the square root of the variance, which would be
14.412...


13. In this question, we are looking for "a" such that
P( 15-a < X < 15 +a ) = 0.70

Again, we have to convert X to a standard normal in order to use
tables. Thus
P( 15-a < X < 15 +a )
=P(-a < X-15 < a)
=P(-a/2 < (X-15)/2 < a/2)
=P(-a/2 < Z < a/2)

Now,
P(-a/2 < Z < a/2) = 0.70
=P(Z<a/2)-P(Z<-a/2)   (*)

But, because the standard normal distribution is symmetric around 0,
we have that
P(Z<a/2)=P(Z>-a/2)

And we also know that
P(Z<-a/2)=1-P(Z>-a/2)

So, by replacing this in (*), we get that
P(Z<a/2)-P(Z<-a/2)
=P(Z<a/2)- (1-P(Z>-a/2))
=P(Z<a/2)- (1-P(Z<a/2))
=2*P(Z<a/2) - 1 = 0.70

Now,
2*P(Z<a/2) = 1.70
P(Z<a/2) = 0.85

So now it's easy to calculate a, we've done it in the previous
questions. We just plug 0.85 in the applet box "Area below Z", finding
that Z=1.0364. Therefore, a/2=1.0364, so a=2.0728.

The answer is then that 70% of the products will be assembled in
between 15-2.0728 and 15+2.0728 minutes (between 12.92 and 17.07
minutes).


And that's all the answers. The Google search terms I used were:
sample mean
proportion
normal table
quartile
"coefficient of variation"
complement
"mutually exclusive"


Again, if you need any further assistance, don't hesitate to request
clarification. Otherwise, I await your rating and final comments.


Best wishes!
elmarto

---

Request for Answer Clarification by sunshine2-ga on 19 Jun 2003 12:55 PDT

#5
Hi.  Thanks for the fast reply.

For #5 is it also possible for the answer to be 0.20?

---

Clarification of Answer by elmarto-ga on 19 Jun 2003 13:25 PDT

Hi sunshine2,
Unless the question statement is wrong, it's not possible for the
answer to be 0.20. However, I did make a calculation mistake and gave
the wrong answer to this question. I'm very sorry. The correct answer
is not 24%. It is 32%, or 0.32.
 
By definition, to say "A and B are independent" implies P(A and B)
equals P(A)*P(B). Therefore, since P(A)=0.8 and P(B)=0.4 and they are
independent, then P(A and B) MUST be equal to 0.8*0.4=0.32.

Somewhere along the line I said to myself "8 times 4 is 24", which, as
most people know, is wrong :-)

---

Request for Answer Clarification by sunshine2-ga on 19 Jun 2003 14:08 PDT

Thanks again elmarto.  I am working through the rest of the problems. 
After I am finished I will post your rating and tip :-)

Sunshine

---

Clarification of Answer by elmarto-ga on 19 Jun 2003 18:05 PDT

Thanks a lot!! :-)

Just ask if there's anything you don't understand.

Another great help session.  Thank you.