Stat quiz 2

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  1. 1. (12 points) There are two envelopes. The first envelope contains a $5 bill and a $10 bill. The second envelope contains a $1 bill and a $50 bill.

    From the first envelope a bill is randomly chosen, and from the second envelope, a bill is randomly chosen, and the outcome is recorded. [For instance, the outcome (5, 1) means $5 bill from the first envelope and $1 bill from the second envelope.]

    (a) List all of the outcomes in the sample space.
    (b) Let A be the event "the sum of the bill values is an even number of dollars."

    What outcomes belong to event A? (Just list them). What is the probability of event A? ______

    (c) Let B be the event "the sum of the bill values is greater than 50 dollars." What outcomes belong to event B? (Just list them).
    What is the probability of event B? ______

    (d) Determine the probability P(A B), where A and B are the events described above. Show work/explanation.

     

     

    2. (20 points) A statistical experiment involves flipping a fair coin three times.

    1. (a)  Determine the sample space for the statistical experiment. The sample space is the set of all possible outcomes; for example, HTH is one possible outcome, where H stands for Heads and T stands for Tails. The format of your sample space (S) in set notation would look like

      S = {HTH, ... }

      Hint: You may use a tree diagram to find all possible outcomes.

    2. (b)  Construct a two-column probability distribution table reflecting the values of random

      variable X, representing the number of possible heads (0, 1, 2, or 3), and their corresponding

      probabilities.

    3. (c)  How can you check to verify that you have constructed a correct probability table for the

      statistical experiment under consideration? Resort to mathematical notations of the governing

      rules to be brief.

    4. (d)  Based on what you have done in part (a) or (b) above, what is the probability of getting heads

      twice. Explain how the probability was determined.

    5. (e)  Findtheanswertopart(d)aboveusingthebinomialprobabilitydistributionformula.

    6. (f)  If the statistical experiment had tossed three fair coins at the same time (instead of flipping a single coin three times), would the probability values obtained have been changed? You must provide statistical reason to support your answer.

    7. (g)  How can one find the probability of getting three heads (HHH) without resorting to your sample space, your probability table, or the binomial probability distribution formula. Show your computation and the statistical basis for it.

    8. (h)  Using your table, find the expected value of X and its standard deviation.

    1. (10 points) A stamp collector has a set of five different stamps of different values and wants to take a picture of each possible subset of his collection (including the “empty set,” depicting just the picture frame!), i.e., pictures showing no stamps, one stamp, two stamps, three stamps, four stamps, or five stamps. In each picture showing two or more stamps, the stamps are in a row. Showing your work, determine the maximum number of different pictures possible, when the difference between two pictures would be either in the number of stamps or in the horizontal order of the stamps. For example, if the stamp collector had just two different stamps (say A and B) of different values, he would have five pictures showing: A, B, AB, BA, and the empty frame.

    2. (12 points) Twenty six people have been invited to a party. Each invitee meets all other invitees, shaking hands.

      1. (a)  Determine the total number of the handshakes.

      2. (b)  If four invitees are from out of state and others are local, what is the probability of randomly

        selecting two invitees and those two are not local. 

         

         

         

        5. (21 points) The table below gives the distribution of blood types by sex in a group of 1,200 individuals.

        Blood Type

        Male

        Female

        Total

        O

        200

        416

        616

        A

        68

        284

        352

        B

        40

        144

        184

        AB

        12

        36

        48

        Total

        320

        880

        page3image31880

        1200

        page3image32752

        (Answers for parts a through f can be stated as fractions, such as 35/46, or as decimals rounded to three decimal places)

        A person is selected at random from the group.
        Showing your work, what is the probability that the person:

        (a) is female?
        (b) has blood type A?
        (c) is a female having blood type A?

        (d) is a female or has blood type A?

        (e) is female, given that the person’s blood type is A?

        (f) has blood type A, given that the person is female?

        Consider the events F = "person is female" and A = "person has blood type A".
        (g) Are the events F and A independent? Show work/explain carefully.

         

         

         

        6. (25 points) Using graph(s), shading, and appropriate probability distribution table, and assuming that the heights of women are normally distributed, with a mean of 65.0 inches and a standard deviation of 2.5 inches, and assuming that men have heights that are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches,

        1. (a)  Calculate and write down your height in inches.

        2. (b)  Determine what percentage of people of your gender would be shorter than you are.

        3. (c)  Determine what percentage of people of your gender would be taller than you are.

        4. (d)  If someone of your gender is randomly selected what is the probability that the

          selected person would be shorter than you are?

        5. (e)  What percentage of population of your gender would be exactly as tall as you are?

        6. (f)  What is the minimum height of a person of your gender to join the Top 15% Tall

          Club?

        7. (g)  If 100 people of your gender are randomly selected, determine the probability that

          their mean height would exceed your height. 

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