At Hanukkah (coming up in December) Jewish children play a
game for pennies with the four sided spinning top shown here. When
it's your turn you spin the top. This is what happens:
letter showing result
gimmel take all the pennies in the pot
hey take half the pennies in the pot
shin put one penny in the pot
nun nothing (next player's turn)
-
What is the probability that a gimmel appears?
Answer.
The probability of a gimmel is 1/4 since each of the four sides of the dreidel is equally likely to shwo.
- What is the probability that the person spinning the dreidel
wins something on her turn?
Answer.
Half the outcomes (gimmel and hey) are wins for the spinner, so the probability is 1/2.
- What is the probability that the person spinning the dreidel
loses a penny twice in a row?
Answer.
The probability of losing twice in a row is (1/4)*(1/4) = 1/16.
-
Suppose there are ten pennies in the pot when you spin. What would be
the "fair price" to play? That is, on the average, how much would you
expect to win?
Answer.
If you spin 100 times then on average there will be
25 gimmels worth 25*10 = 250 pennies 25 heys worth 25*5 = 125 pennies 25 shins worth 25*(-1) = -25 pennies 25 nuns worth 10*0 = 0 pennies
so in 100 spins you collect (on average) 250+125-25 = 350 pennies. Therefore a fair price to pay for each spin is 350/100 = 3.5 pennies. If you did that you'd just break even (in the long run). That means each turn you collect (on average) - (Extra credit) Why do you think the picture caption describes
the four sides of the dreidel from right to left?
Answer.
Hebrew is written from right to left.
The picture is from the web site www.myjewishlearning.com/holidays/Hanukkah/TO_Hanukkah_Home/Dreidel/Dreidel_Rules_357.htm#, where you can read more if you like. But you won't find anything else there to help with this question.
WHL has received 10 million shares of CAJT stock ... WHL has paid an advertising cost of nine hundred ninety thousand dollars to produce and distribute this public awareness mailer.
- Suppose CAJT stock is selling at $0.41/share.
How much profit will WHL make if they sell their shares now?
Answer.
They will collect
$ 10 million shares * 0.41 ----- = $4.1 million shareso their profit is$4.1 million - 0.99 million ~ $3.1 million . - What is likely to happen to the stock price when WHL sells its
10 million shares and walks away with the profits?
Answer.
The stock price will fall rapidly when lots of shares are sold at one time.
- Estimate how many copies of the flyer WHL sent out.
(Hint: you know how much they say they spent on production
and distribution.)
Answer.
It costs 41 cents now to mail a letter. Presorted bulk mail is cheaper, so I'll estimate postage at 25 cents per flyer. For two sided color printing on heavy paper I'll estimate 50 cents per flyer. WHL spent about a million dollars ($990 thousand) so they sent out
1 flyer $1 million * --------- = 1.3 million flyers $0.75 - (Extra Credit)
WHL believes that the price of CAJT stock will rise
significantly if at least 5,000 people buy some when they get the
mailer. Use your answer to the previous problem to determine what
percentage of the people receiving the mailing must respond in order
for this to happen.
Answer.
5000 flyer responses -------------------- = 0.00384615385 ~ 0.4% 1.3 million flyersA response rate of four tenths of one percent (four per thousand) would suit WHL just fine. - (Extra Credit) WHL hopes their mailer will succeed in
encouraging people to buy CAJT, so the price will go up.
How much extra profit will they make for each penny increase in
the stock price?
Answer.
Each extra penny is worth 10 million pennies (one per share), so an extra $100,000 in profit.
year
age range 2000 2050
0-9 39.70 55.60
10-19 40.86 53.56
20-29 39.90 52.89
30-39 43.24 52.85
40-49 42.77 49.98
50-59 31.35 46.26
60-69 20.40 42.83
70-79 16.28 32.57
80+ 9.25 33.70
You can download that table directly into Excel from the link
USPopulationProjection.xls.
Answer the following questions. (Excel will help, of course!) Write the answers in your blue book (except for the chart).
Turn in a printout of your entire spreadsheet (not just the chart) and also send the spreadsheet as an email attachment to yourself and to eb@cs.umb.edu. Make sure the data are not hidden by the chart.
Answer.
USPopulationProjectionAnswer.xls.
- What was the total population of the United States in 2000?
What is the projected value for 2050?
Answer.
Excel summed the columns for me and told me the total in 2006 was 283.74 million and the projected total for 2050 was 420.22 million.
- Verify the result of your year 2000 calculation
by finding the same number
on a reliable web site. Do the numbers match exactly?
Answer.
From www.u-s-history.com/pages/h980.html:
The official count of the U.S. population was 281421906 on April 1, 2000
That's close enough to the 283.74 million in the data presented on the exam. - What are the absolute and percentage increases in the
U.S. population between 2000 and the projected value for 2050?
Answer.
Again Excel did the work. The absolute change was 136.48 million, an increase of 48.10%.
- Which age range shows the smallest percentage increase between
2000 and 2050? What are the absolute and percentage increases in that
age range?
Answer.
The smallest percentage increase is 16.87% in the 40-49 age group. The absolute change for that group is 7.21 million people.
- Which age range shows the largest percentage increase between
2000 and 2050? What are the absolute and percentage increases in that
age range?
Answer.
The greatest percentage increase is 264.21% in the 80+ age group! The absolute change for that group is 24.45 million people.
- Construct and properly label a column chart displaying the
information in the table.
Answer.
See the spreadsheet. Since my name and the data source are on the spreadsheet and the instructions say to submit the whole spreadsheet I chose not to take the time to put them on the chart.
I did arrange the chart so that it and the data would all print on one page.
- (Extra credit) Estimate the mean and median ages for each of
the years 2000 and 2050. Describe how you did the work.
Answer.
To find the median age I computed (in Excel) the fraction of the population younger than each age, and looked for where 0.5 came. That told me the median age was about 30 in 2006 and would be about 35 in 2050.
To find the mean age I assumed that each person in each cohort was at the middle of the age range for that cohort. Then I computed the total number of years those people had lived, by multiplying. Then I added up all the years and divided by the total number of people. That told me the mean ages in 2000 and 2050 are 36.3 and 40.8 respectively.
I was surprised that the mean and median for 2050 were not even larger, given how dramatically the population will have aged.
Newspaper circulation change --------- ----------- ------ USA Today 2,293,137 up 1.04% The Wall Street Journal 2,011,882 down 1.53% The New York Times 1,037,828 down 4.51% Los Angeles Times 779,682 up 0.50% New York Daily News 681,415 down 1.73% New York Post 667,119 down 5.24% The Washington Post 635,087 down 3.23% Chicago Tribune 559,404 down 2.90% Houston Chronicle 507,437 down 0.13% Newsday, Long Island 387,503 down 5.62% The Arizona Republic 382,414 down 3.75% The Dallas Morning News 373,586 down 7.68% San Francisco Chronicle 365,234 down 2.29% The Boston Globe 360,695 down 6.66% The Star-Ledger (Newark, NJ) 353,003 down 2.78% The Philadelphia Inquirer 338,260 up 2.31% Star Tribune (Minneapolis) 335,443 down 6.53% The Plain Dealer (Cleveland) 334,195 down 0.81% Detroit Free Press 320,125 down 2.61% Atlanta Journal-Constitution 318,350 down 9.08%
We've downloaded them to Excel and put them in spreadsheet NewspaperCirculation.xls.
Answer.
For the answers, see NewspaperCirculationAnswer.xls.
- What was the mean circulation for these newspapers?
Answer.
The mean daily circulation is 652089.95 or about 6.5 million. (I used the Excel AVERAGE function.)
- What was the median circulation for these newspapers?
Answer.
The median daily circulation is 384958.5 or about 3.8 million. (I used the Excel MEDIAN function.)
- Why are the mean and the median so different?
Answer.
The median is much less than the mean because there are a few papers with a very high daily circulation. This is exactly what we saw happen with the Wing Aero salary structure in class.
- If asked to report the mode for these circulation figures, what would you say?
Answer.
I'd say the mode was somewhere in the mid 300 thousands. That would clearly be the most common value if we grouped the circulation figures into ranges 100K-200K, 200K-300K, 300K-400K and so on.
Heres the link to the complete article: www.boston.com/business/globe/articles/2007/11/06/newspaper_circulation_still_on_decline/. There's nothing there that will help you with the questions on this exam.