Jan 9, 2017

Probability Distribution — practice test 5A

20 cards
, 56 answers
    Problems with solutions in Elementary Statistics — Random Variables; Binomial Probability Distributions: Mean, Variance, and Standard Deviation for the Binomial Distribution; The Poisson Distribution.

    Directions:
    Give your answers to the short-answer items. Choose the correct answer for multiple-choice items.

  • Provide an appropriate response.
    • 1. Suppose a mathematician computed the expected value of winnings for a person playing each of seven different games in a casino. What would you expect to be true for all expected values for these seven games?

      Possible answer
      The expected values be negative for the person playing, as the casinos are designed to make money.

      1. 2. Previously, you learned to find the three important characteristics of data: the measure of central tendency, the measure of variation, and the nature of the distribution. We can find the same three characteristics for a binomial distribution. Given a binomial distribution with p = 0.4 and n = 8, find the three characteristics.

        Possible answer

        Nature of the Distribution

        Measure of central tendency: mean = 3.2, that is 0.4*8
        Measure of variation: SD = 1.386, that is square root of 8*0.4*0.6

      2. Identify the given random variable as being discrete or continuous.
        • 3. The pH level in a shampoo.
          1. Continuous
          2. Discrete

        • 4. The number of field goals kicked in a football game.
          1. Discrete
          2. Continuous

      3. Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.
        • 5.

          Possible answer
          Not a probability distribution. One of the P(x)’s is negative.

          1. 6. If a person is randomly selected from a certain town, the probability distribution for the number, x, of siblings is as described in the accompanying table.


            Possible answer
            Not a probability distribution. The sum of the P(x)’s is not 1, since 0.98 ≠ 1.00.

          2. Find the mean of the given probability distribution.
            • 7. The number of golf balls ordered by customers of a pro shop has the following probability distribution.
              1. 9.72
              2. 7.8
              3. 9
              4. 5.13

          3. Solve the problem.
            • 8. In a certain town, 70% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults) who have a college degree. Find the variance for the probability distribution.
              1. 0.92
              2. 0.84
              3. 0.72
              4. 8.68

            • 9. The prizes that can be won in a sweepstakes are listed below together with the chances of winning each one:
              $5900 (1 chance in 8600);
              $2700 (1 chance in 5000);
              $600 (1 chance in 4800);
              $200(1 chance in 2500).
              Find the expected value of the amount won for one entry if the cost of entering is 52 cents.
              1. $0.67
              2. $200
              3. $0.91
              4. $1.35

            • 10. Assume that a researcher randomly selects 14 newborn babies and counts the number of girls selected, x. The probabilities corresponding to the 14 possible values of x are summarized in the given table. Answer the question using the table.

              Find the probability of selecting 9 or more girls.
              1. 0.001
              2. 0.122
              3. 0.212
              4. 0.061

          4. Answer the question.
            • 11. Focus groups of 12 people are randomly selected to discuss products of the Yummy Company. It is determined that the mean number (per group) who recognize the Yummy brand name is 9.5, and the standard deviation is 0.69. Would it be unusual to randomly select 12 people and find that fewer than 6 recognize the Yummy brand name?
              1. No
              2. Yes

          5. Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
            • 12. Rolling a single die 34 times, keeping track of the numbers that are rolled.
              1. Not binomial: there are too many trials.
              2. Not binomial: there are more than two outcomes for each trial.
              3. Procedure results in a binomial distribution.
              4. Not binomial: the trials are not independent.

          6. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.
            • 13. n = 10, x = 2, p = 1/3
              1. 0.2156
              2. 0.1929
              3. 0.0028
              4. 0.1951

          7. Find the indicated probability.
            • 14. A machine has 7 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.
              1. 0.996
              2. 0.033
              3. 0.029
              4. 0.967

          8. Find the standard deviation, σ, for the binomial distribution which has the stated values of n and p. Round your answer to the nearest hundredth.
            • 15. n = 22, p = .2
              1. 1.88
              2. 6.00
              3. –0.53
              4. 5.15

          9. Use the given values of n and p to find the minimum usual value μ – 2σ and the maximum usual value μ + 2σ.
            • 16. n = 660, p = 4/7
              1. Minimum: 364.43; maximum: 389.86
              2. Minimum: 402.57; maximum: 351.72
              3. Minimum: 328.45; maximum: 425.83
              4. Minimum: 351.72; maximum: 402.57

          10. Solve the problem.
            • 17. The probability that a person has immunity to a particular disease is 0.1. Find the mean number who have immunity in samples of size 13.
              1. 11.7
              2. 0.1
              3. 1.3
              4. 6.5

            • 18. The probability of winning a certain lottery is 1/59,127. For people who play 514 times, find the standard deviation for the number of wins.
              1. 0.0087
              2. 0.0932
              3. 0.1021
              4. 2.1138

          11. Determine is the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than μ – 2σ or greater than μ + 2σ.
            • 19. According to AccuData Media Research, 36% of televisions within the Chicago city limits are tuned to “Eyewitness News” at 5:00pm on Sunday nights. At 5:00pm on a given Sunday, 2500 such televisions are randomly selected and checked to determine what is being watched. Would it be unusual to find that 925 of the 2500 televisions are tuned to “Eyewitness News”?
              1. Yes
              2. No

          12. Use the Poisson Distribution to find the indicated probability.
            • 20. A computer salesman averages 0.9 sales per week. Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is 1.
              1. 0.4025
              2. 0.3659
              3. 0.4574
              4. 0.3293

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