Statistics: Mean and Standard Deviation Quiz
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Questions and Answers

What does the mean of a binomial distribution represent?

  • The sum of the probabilities of all outcomes
  • The total number of fixed trials
  • The maximum potential outcome of the distribution
  • The value you would expect after an infinite number of trials (correct)

    • If the number of fixed trials in a binomial distribution is 10 and the probability of success is 0.5, what is the mean (µ)?

  • 10
  • 15
  • 2
  • 5 (correct)

    • For a binomial distribution, what is the formula to calculate the standard deviation?

  • σ = Σ[x P(x)]
  • σ = sqrt(npq) (correct)
  • σ = µ + 2σ
  • σ = np

    • What indicates that a value is unusual according to the range rule of thumb?

    <p>It lies outside the limits of µ + 2σ or µ – 2σ</p> Signup and view all the answers

    In a binomial distribution where the probability of success is 0.3 and there are 20 trials, what is the value of q?

    <p>0.7</p> Signup and view all the answers

    Given that p = 0.4 and n = 15, what is the variance (σ²) of the binomial distribution?

    <p>7.2</p> Signup and view all the answers

    If you are conducting a binomial experiment with 40 trials and a success rate of 0.25, what is the expected number of successes?

    <p>10</p> Signup and view all the answers

    If a binomial distribution has a mean of 8 and a standard deviation of 2, what is the minimum usual value?

    <p>4</p> Signup and view all the answers

    What would be a typical mean for the number of green M&Ms in a sample of 100 if the claimed rate is 16%?

    <p>16</p> Signup and view all the answers

    If a nursing student guesses on an exam with 75 true/false questions, what is the expected standard deviation for the number of correct answers?

    <p>4.7</p> Signup and view all the answers

    Is it unusual for a student to score at least 45 correct answers by guessing on the exam?

    <p>No, it's expected.</p> Signup and view all the answers

    Based on the Experience.com poll, what is the total number of graduates who actually stayed at their first job less than 2 years?

    <p>250</p> Signup and view all the answers

    What range of usual values is calculated for the number of graduates who stay at their first job less than 2 years?

    <p>(142.11, 177.89)</p> Signup and view all the answers

    What statistical methods were discussed for analyzing random samples from a population?

    <p>Mean, variance and standard deviation</p> Signup and view all the answers

    If the claimed rate of graduates who stay less than 2 years is 50%, what is the standard deviation, assuming a sample size of 320?

    <p>8.0</p> Signup and view all the answers

    What conclusion can be drawn about the headline stating that 'most stay at first jobs less than 2 years' based on the results?

    <p>It is unjustified because the actual number is too high.</p> Signup and view all the answers

    What parameter predominantly influences the Poisson distribution?

    <p>Mean μ</p> Signup and view all the answers

    In which situation is the Poisson distribution typically used to approximate the binomial distribution?

    <p>When n is large and p is small.</p> Signup and view all the answers

    How is the mean (μ) calculated in the context of the Poisson distribution?

    <p>μ = np</p> Signup and view all the answers

    What is the probability of observing exactly one occurrence in the Poisson distribution if μ = 0.365?

    <p>0.2534</p> Signup and view all the answers

    In a Poisson process, what does the parameter μ represent?

    <p>The average rate of occurrences</p> Signup and view all the answers

    Which of the following is a requirement for using the Poisson distribution to approximate the binomial distribution?

    <p>n &gt;= 100, np &lt;= 10</p> Signup and view all the answers

    Which scenario best illustrates a real-world application of the Poisson distribution?

    <p>The number of emails received in an hour.</p> Signup and view all the answers

    When computing the probability that three or more parts will fail in ten years under certain conditions, which distribution is appropriate?

    <p>Poisson distribution</p> Signup and view all the answers

    Which characteristic is NOT a requirement for a Poisson probability distribution?

    <p>The interval must always be in units of time.</p> Signup and view all the answers

    When was the Poisson distribution first derived, and by whom?

    <p>1837 by Siméon Poisson</p> Signup and view all the answers

    Which of the following is a suitable application of the Poisson distribution?

    <p>The number of car accidents in a city in one year.</p> Signup and view all the answers

    What is required about the occurrences in a Poisson distribution?

    <p>They must be uniformly distributed over the interval.</p> Signup and view all the answers

    Which of the following scenarios best illustrates a Poisson process?

    <p>The number of orders at a restaurant during lunch.</p> Signup and view all the answers

    In the context of the Poisson distribution, what does the variable 'e' represent?

    <p>The base of the natural logarithm, approximately 2.71828.</p> Signup and view all the answers

    What is the probability of an event occurring 'x' times over a specific interval in a Poisson distribution?

    <p>Based on a formula involving 'e' raised to the power of negative mean.</p> Signup and view all the answers

    Which of the following events is considered a rare occurrence suitable for Poisson distribution analysis?

    <p>The number of customer complaints at a bank per month.</p> Signup and view all the answers

    Which of the following scenarios would likely be modeled by a Poisson distribution?

    <p>The number of car accidents on a road in a year</p> Signup and view all the answers

    What is characteristic of a Poisson distribution in terms of the mean and variance?

    <p>Mean is equal to variance</p> Signup and view all the answers

    How would you determine the probability of a specific number of events occurring in an interval using a Poisson formula?

    <p>By calculating the mean and using the Poisson probability formula</p> Signup and view all the answers

    Which of these examples is NOT well modeled by a Poisson distribution?

    <p>Number of heads in 10 coin flips</p> Signup and view all the answers

    Which of the following statements about the Poisson distribution is true?

    <p>It can be used for events that occur continuously over time</p> Signup and view all the answers

    In a Poisson distribution, if the average rate of occurrences is 4 events per hour, what is the expected probability of observing exactly 2 events in a 30-minute interval?

    <p>0.1975</p> Signup and view all the answers

    When considering the scenarios provided, which of the following can be most accurately modeled using a Poisson distribution?

    <p>Number of cars passing a toll booth in an hour</p> Signup and view all the answers

    Study Notes

    Section 5.4: Mean and Standard Deviation of Binomial Probability Distributions

    • Binomial distributions involve important characteristics like center, variation, and distribution.
    • Given a binomial distribution, you can calculate its mean, variance, and standard deviation.
    • Emphasis is placed on interpreting these values.

    Formulas for any Discrete Probability Distribution

    • Mean (μ): μ = Σ [x • P(x)]
    • Variance (σ²): σ² = Σ [x² • P(x)] – μ²
    • Standard Deviation (σ): σ = √[Σ x² • P(x)] – μ²

    Formulas for Binomial Distributions

    • Mean (μ): μ = np

    • Standard Deviation (σ): σ = √(npq)

    • n = number of fixed trials

    • p = probability of success in one trial

    • q = probability of failure in one trial

    Range Rule of Thumb

    • 95% of data lies within 2 standard deviations of the mean.
    • [μ – 2σ, μ + 2σ]
    • Values outside this range are considered unusual.

    Example 1

    • Probability of a pea having a green pod is 0.75.
    • Expected number of green peas in 5 offspring: μ = np = 5(0.75) = 3.75

    Example 2

    • Michael tested a theory that 75% of peas have green pods.
    • Collected 580 offspring; 428 had green pods.
    • Calculation shows this result is usual, not unusual.

    Example 3

    • Mars Inc. claims 24% of its M&Ms are blue.
    • Independent researcher collected 100 M&Ms.
    • Expected mean and standard deviation for a sample of 100 M&Ms are calculated.
    • Finding the number of blue M&Ms that would be unusual, given the claim.

    Example 4

    • Drug designed to increase the probability of a baby boy.
    • Monitored births of 152 babies.
    • 127 were boys.
    • The example shows that this result is unusual.

    Other Examples

    • SAT multiple choice questions, random guessing.
    • A headline in USA Today about job tenure, based on Experience.com poll of 320 college graduates.

    Recap

    • Mean, variance and standard deviation formulas for any discrete probability distribution.
    • Mean, variance and standard deviation formulas for the binomial probability distribution.
    • Interpreting results.

    Homework

    • Pg. 232: 11, 13, 16, 18

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    Related Documents

    Section 5.4 Binomial Probability Distributions PDF
    Poisson Probability Distributions PDF

    Description

    This quiz focuses on the mean and standard deviation of binomial probability distributions. You will learn how to calculate these measures using provided formulas and interpret their significance in the context of discrete probability distributions. Brush up on your understanding of binomial distributions and their characteristics.

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