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Prob. & Dist. Test, 4th 9-weeks, May 20, 2004

No textbooks allowed, but please use your two notecards and your graphing calcu-
lator. Each of the 21 question numbers has equal weight (5 points each). Read the
questions carefully. Hand in any used scratch paper with the test. SHOW YOUR WORK.}

  1. Form the best match among the following items (lesson 1) (10 points).
        experiment A. examples: rolling die, flipping coin, drawing card
        random experiment B. more than one roll, flip, or draw
        sample space C. each element has an equal chance of being chosen
        impossible D. method by which observations are made
        certain E. set of all possible outcomes
        simple event F. where empirical approches actual probability
        compound event G. each outcome is equally likely
        random sample H. P(A) = 0
        law of large numbers I. outcome which can't be broken down
        fair J. P(A) = 1
  2. Form the best match among the following items (lessons 2 and 3) (10 points).
    TermsDefinitions
        P(At least one) A. event outcomes/total outcomes
        Mutually exclusive B. 1 -
        Exhaustive C. P(A) + P(Ã)=1
        Addition rule D. 1 - P(none)
        Def. of probability E. Sum of all x•P(x)
        Expected Value F. P(A or B)=P(A) + P(B) - P(A and B)
        P-value G. no overlap
        complementary rule H. Everything enumerated
        Bayes Theorem I. An area like
        Power J. P(A|B) = P(A)•P(B|A)/(P(A)•P(B|A) + P(Ã)•P(B|Ã))
  3. Find the number of circular permutations using the letters of the word: P O I S S O N (lesson 2).
  4. Telephone numbers in North America have three groups of digits which must meet certain requirements. Before 1995, the three digit Numbering Plan Area (NPA) code, (commonly known as an area code) had the format NBX, where N could not be 0 or 1, B had to be 0 or 1, and X could be any digit 0 through 9. How many different NPA codes were there then (lesson 2)?
  5. Discuss the meaning of type I and type II errors within the context of an air bag switch, whether or not it triggered, and whether or not it should have (lesson 5).
  6. Green, Blue, Red, and Plaid are running a race. The following odds against are listed: Green: 1 to 1; Blue: 2 to 1; Red: 8 to 1; and Plaid: 17 to 1. Give each contestant's corresponding probability of winning. Did we account for all opponents (lesson 6)?
  7. A couple having fun one evening decide to simulate various families of three children. Help them create eight families of three children. Initialize your TI-83+ random number generator as follows: 0=>rand (rand is math/prob/1), then do int(2rand) 24 times (or randInt(0,1,24)) to find out whether they got a boy (1) or a girl (0). Arrange these in order into eight "families" of three children and calculate the average number of boys per family (lesson 7). (NonTI-83+ users, document your procedure to ensure reproducibility.)
  8. A box contains three $1 bills, four $2 bills, four $5 bills, six $10 bills, and three $100 bills. A person is charged $20 to select one bill. Find the expected value (lesson 8).
  9. Farmer Calkins planted 100 hills of corn per row and 10 rows. Each hill received three kernels. Although three varieties were used, each variety had a published germination of 90%. Using only germination rate and number or kernels per hill (ignoring climatic factors, mechanical damage, crows, cut worms, gophers, etc.) what is the probability for no stalks, one stalk, two stalks, or three stalks per hill (lesson 9).
  10. Farmer Calkins is watching crows landing in his newly planted corn field. He estimates about one crow arrives every minute, the probability of two arriving in a minute is small enough to be ignored, and the crows arrived independently. Find the probability that exactly two crows arriving in any given one-minute interval (lesson 10).
  11. Scientist Calkins knows a certain resonance is convolved with a Gaussian Distribution to produce a Voigt Profile but fits it anyway with y1 = 1/((100-x)2+102). Graph this in a 50 < x < 150 by 0 < y < 0.01 window and find the FWHM. Show a sketch of your work (lesson 11). (Hint: Let y2=0.005 and use CALC (2nd TRACE) Intersect (5) to find the boundary values.)
  12. Gosset discovered how to model sample distributions without knowing the population standard deviation a priori. How many times bigger is the area under his distribution more than 1.96 standard deviations away from the mean for a sample of size 6 than that predicted by the empirical rule. Show a sketch of the region (lesson 12).
  13. Farmer Calkins had some old corn seed which he figured had only an 80% germination rate. Help him calculate a 2 for a goodness of fit test if he took the following sample (lesson 15):
    stalks: 0 1 2 3
    observed: 7 100 350 543
    expected: 8 96 384 512
  14. For the old corn seed data sample in the problem above, calculate the mean number of observed stalks per hill together with the standard deviation. Give your results to four significant digits (stat intro) (10 points).
  15. Farmer Calkins calculates the expected mean number of stalks per hill for the old seed as µ=np=3(0.8)=2.4 and expected standard deviation as = (npq) = (3•0.8•0.2) 0.693. Using either these values or preferably the similar values from the previous problem, calculate the margin of error (for alpha=0.05) and corresponding 95% confidence interval for the true average number of stalks. Clearly indicate the standard error of the mean for his n = 1000 (lesson 13).
  16. E.T. is sitting by the estuary with his trinoculars trained on the bank display. Help E.T. interpret the following data by doing a linear regression between the first and second and between the second and third data value from each ordered triple. You suspect the first value is time so please convert it to: minutes after the first observation. (6:00,69,20), (6:31,57,14), (6:59,49,10), (7:30,40,5), (8:00,32,0). Be sure to include and interpret r and r2 for both regressions (lesson 14) (10 points).
  17. Problems 9-13 above all involve different distributions. In order, give the name of each distribution (10 points bonus).