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4.12: Chapter 4 Solution (Practice + Homework)

  • Page ID
    5559
  • 1.

    \(x\)\(P(x)\)
    00.12
    10.18
    20.30
    30.15
    40.10
    50.10
    60.05

    Table4.6

    3.

    0.10 + 0.05 = 0.15

    5.

    1

    7.

    0.35 + 0.40 + 0.10 = 0.85

    9.

    1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45

    11.

    \(x\)\(P(x)\)
    00.03
    10.04
    20.08
    30.85

    Table4.7

    13.

    Let \(X =\) the number of events Javier volunteers for each month.

    15.

    \(x\)\(P(x)\)
    00.05
    10.05
    20.10
    30.20
    40.25
    50.35

    Table4.8

    17.

    1 – 0.05 = 0.95

    18.

    \(X =\) the number of business majors in the sample.

    19.

    2, 3, 4, 5, 6, 7, 8, 9

    20.

    \(X =\) the number that reply “yes”

    22.

    0, 1, 2, 3, 4, 5, 6, 7, 8

    24.

    5.7

    26.

    0.4151

    28.

    \(X =\) the number of freshmen selected from the study until one replied "yes" that same-sex couples should have the right to legal marital status.

    30.

    1,2,…

    32.

    1.4

    35.

    0, 1, 2, 3, 4, …

    37.

    0.0485

    39.

    0.0214

    41.

    \(X =\) the number of U.S. teens who die from motor vehicle injuries per day.

    43.

    0, 1, 2, 3, 4, ...

    45.

    No

    48.

    1. \(X =\) the number of pages that advertise footwear
    2. 0, 1, 2, 3, ..., 20
    3. 3.03
    4. 1.5197

    50.

    1. \(X =\) the number of Patriots picked
    2. 0, 1, 2, 3, 4
    3. Without replacement

    53.

    \(X =\) the number of patients calling in claiming to have the flu, who actually have the flu.

    \(X = 0, 1, 2, ...25\)

    55.

    0.0165

    57.

    1. \(X =\) the number of DVDs a Video to Go customer rents
    2. 0.12
    3. 0.11
    4. 0.77

    59.

    4. 4.43

    61.

    4

    63.

    • \(X =\) number of questions answered correctly
    • \(X \sim B(32, 13)(32, 13)\)
    • We are interested in MORE THAN 75% of 32 questions correct. 75% of 32 is 24. We want to find \(P(x > 24)\). The event "more than 24" is the complement of "less than or equal to 24."
    • \(P(x > 24) = 0\)
    • The probability of getting more than 75% of the 32 questions correct when randomly guessing is very small and practically zero.

    65.

    1. \(X =\) the number of college and universities that offer online offerings.
    2. \(0, 1, 2, …, 13\)
    3. \(X \sim B(13, 0.96)\)
    4. \(12.48\)
    5. \(0.0135\)
    6. \(P(x = 12) = 0.3186 P(x = 13) = 0.5882\) More likely to get 13.

    67.

    1. \(X =\) the number of fencers who do not use the foil as their main weapon
    2. \(0, 1, 2, 3,... 25\)
    3. \(X \sim B(25,0.40)\)
    4. \(10\)
    5. \(0.0442\)
    6. The probability that all 25 not use the foil is almost zero. Therefore, it would be very surprising.

    69.

    1. \(X =\) the number of audits in a 20-year period
    2. \(0, 1, 2, …, 20\)
    3. \(X \sim B(20, 0.02)\)
    4. \(0.4\)
    5. \(0.6676\)
    6. \(0.0071\)

    71.

    1. \(X =\) the number of matches
    2. \(0, 1, 2, 3\)
    3. In dollars: \(−1, 1, 2, 3\)
    4. \(\frac{1}{2}\)
    5. The answer is \(−0.0787\). You lose about eight cents, on average, per game.
    6. The house has the advantage.

    73.

    1. \(X \sim B(15, 0.281)\)
      This histogram shows a binomial probability distribution. It is made up of bars that are fairly normally distributed. The x-axis shows values from 0 to 15, with bars from 0 to 9. The y-axis shows values from 0 to 0.25 in increments of 0.05.

      Figure 4.4

      1. \(\text{Mean }= \mu=n p=15(0.281)=4.215\)
      2. \(\text{Standard Deviation } =\sigma=\sqrt{n p q}=\sqrt{15(0.281)(0.719)}=1.7409\)
    2. \(P(x > 5)=1 – 0.7754 = 0.2246\)
      \(P(x = 3) = 0.1927\)
      \(P(x = 4) = 0.2259\)
      It is more likely that four people are literate that three people are.

    75.

    1. \(X =\) the number of adults in America who are surveyed until one says he or she will watch the Super Bowl.
    2. \(X \sim G(0.40)\)
    3. \(2.5\)
    4. \(0.0187\)
    5. \(0.2304\)

    77.

    1. \(X =\) the number of pages that advertise footwear
    2. \(X\) takes on the values 0, 1, 2, ..., 20
    3. \(X \sim B(20, \frac{29}{192})\)
    4. \(3.02\)
    5. No
    6. \(0.9997\)
    7. \(X =\) the number of pages we must survey until we find one that advertises footwear. \(X \sim G(\frac{29}{192})\)
    8. \(0.3881\)
    9. \(6.6207\) pages

    79.

    0, 1, 2, and 3

    81.

    1. \(X \sim G(0.25)\)
      • \(\text{Mean }=\mu=\frac{1}{p}=\frac{1}{0.25}=4\)
      • \(\text{Standard Deviation }=\sigma=\sqrt{\frac{1-p}{p^{2}}}=\sqrt{\frac{1-0.25}{0.25^{2}}} \approx 3.4641\)
    2. \(P(x = 10) = 0.0188\)
    3. \(P(x = 20) = 0.0011\)
    4. \(P(x \leq 5) = 0.7627\)

    82.

    1. \(X \sim P(5.5); \mu = 5.5; \sigma=\sqrt{5.5} \approx 2.3452\)
    2. \(P(x \leq 6) \approx 0.6860\)
    3. There is a 15.7% probability that the law staff will receive more calls than they can handle.
    4. \(P(x > 8) = 1 – P(x \leq 8) \approx 1 – 0.8944 = 0.1056\)

    84.

    Let \(X =\) the number of defective bulbs in a string.

    Using the Poisson distribution:

    • \(\mu = np = 100(0.03) = 3\)
    • \(X \sim P(3)\)
    • \(P(x \leq 4) \approx 0.8153\)

    Using the binomial distribution:

    • \(X \sim B(100, 0.03)\)
    • \(P(x \leq 4) = 0.8179\)

    The Poisson approximation is very good—the difference between the probabilities is only \(0.0026\).

    86.

    1. \(X =\) the number of children for a Spanish woman
    2. \(0, 1, 2, 3,...\)
    3. \(0.2299\)
    4. \(0.5679\)
    5. \(0.4321\)

    88.

    1. \(X =\) the number of fortune cookies that have an extra fortune
    2. \(0, 1, 2, 3,... 144\)
    3. \(4.32\)
    4. \(0.0124\) or \(0.0133\)
    5. \(0.6300\) or \(0.6264\)
    6. As \(n\) gets larger, the probabilities get closer together.

    90.

    1. \(X =\) the number of people audited in one year
    2. \(0, 1, 2, ..., 100\)
    3. \(2\)
    4. \(0.1353\)
    5. \(0.3233\)

    92.

    1. \(X =\) the number of shell pieces in one cake
    2. \(0, 1, 2, 3,...\)
    3. \(1.5\)
    4. \(0.2231\)
    5. \(0.0001\)
    6. Yes

    94.

    4