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ISP 121 -- Activity A7

Probability and Expected Value

Reference: Apr 28 Lecture Notes  

Submit: A Word file with your answers. The usual name convention (Smith-Jones.doc).
Work in groups of two or three.

  1. The survival time in years for a certain type of surgical operation is given by this probability distribution:

    Survival Time
    Time Probability
    0 0.25
    1 0.35
    2 0.20
    3 0.07
    4 0.03

    1. Compute the expected value of the survival time.

    2. Compute the expected value of the average survival time for one thousand patients.

    3. Compute the SD of the survival time.

    4. Compute the SE of the average survival time for 1000 patients.

  2. What is the a priori probability of obtaining 0, 1, 2, 3 or 4 heads out of four spins? Determine this by listing all sixteen outcomes of spinning a coin four times, then count how many outcomes have 0, 1, 2, 3 or 4 heads. A table is required. Your table will be like the one in 1 above, except that you will have Number of Heads instead of Time.

  3. Flip a coin 10 times and report the results to me. The point of this experiment is to investivate whether the probability of head vs. tail is really 50-50 when flipping a coin. I'll give you the results of the class next time. Then I'll ask you a number of related questions - this will complete activity A7, which you will do on Wednesday, Apr 30.

    Here are the results for the class: coin-flips.xls.

    Note: Label all your answers in your Word document (3a, 3b, etc.). Show supporting work for all answers.

    1. Use Excel to compute the number of heads obtained by finding the sum of the 0s and 1s in column A (Excel function SUM).

    2. Use Excel to compute the proportion of heads obtained by finding the average of the 0s and 1s in column A (Excel function AVERAGE).

    3. Assuming a fair coin modeled by a 0-1 random variable (0 for tail, 1 for head), what is the probability distribution?

    4. What is the expected value for one flip?

    5. What is the expected value for the sum of 70 flips?
      Note:   the sum for a 0-1 random variable is the number of heads.

    6. What is the expected value for the average of 70 flips?
      Note: the average for a 0-1 random variable is the proportion of heads.

    7. What is the SD for one flip?

    8. What is the SE for sum of 70 flips?
      Note:   the sum for a 0-1 random variable is the number of heads.

    9. What is the SE for the average of 70 flips?
      Note:   the average for a 0-1 random variable is the proportion of heads.

    10. For this problem, use the SE for the proportion in Part 3i and the expected value EV for the proportion in Part 3f. How many SEs away from the EV is the actual proportion P of heads obtained? Solve the equation P = EV + x(SE).

      If x is greater than 2 or less than -2, we have evidence that the probability of obtaining a head by spinning a quarter is different than 0.5. Is there such evidence?