Probability and Combinatorics Quiz
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Questions and Answers

What is a sufficient sample size to ensure that the sampling distribution approaches normality, even if the population distribution is not normal?

  • 30 (correct)
  • 10
  • 50
  • 20

    • A Z-test is used when the population standard deviation is known.

    True (A)

    What is the formula for the T-test statistic for two independent samples?

    t = ((x̄1 - x̄2) - (μ1 - μ2)) / √((s1^2 / n1) + (s2^2 / n2))

    To determine the P-value, you can use ______() or ______().

    <p>normalcdf(), tcdf()</p> Signup and view all the answers

    Match the following test statistics with their corresponding sample types:

    <p>Z-test = One sample with known population standard deviation T-test = One sample with unknown population standard deviation Two-sample Z-test = Comparison of means from two independent samples with known standard deviations Paired T-test = Comparison of means from two related samples</p> Signup and view all the answers

    What does the sample space represent in probability?

    <p>The set of all possible outcomes (C)</p> Signup and view all the answers

    Permutations and combinations consider order the same.

    <p>False (B)</p> Signup and view all the answers

    What is the formula for the expected value?

    <p>(outcome 1 * probability of outcome 1) + (outcome 2 * probability of outcome 2) + ...</p> Signup and view all the answers

    The process of using random numbers to model real-world activities is known as __________.

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

    Match the following terms with their definitions:

    <p>Sample Space = The set of all possible outcomes Event = A subset of outcomes from the sample space Trial = A procedure that can be infinitely repeated Venn Diagram = A diagram representing set relationships</p> Signup and view all the answers

    Which symbol is used to represent the intersection of two events in probability?

    <p>A∩B (B)</p> Signup and view all the answers

    The Fundamental Principle of Counting states that if one event has m outcomes and another has n outcomes, there are m + n outcomes for the two events together.

    <p>False (B)</p> Signup and view all the answers

    What does the notation A'​ signify in set theory?

    <p>The complement of event A</p> Signup and view all the answers

    What is the primary characteristic of a binomial experiment?

    <p>Only two possible outcomes (D)</p> Signup and view all the answers

    A simple random sample guarantees that every possible sample has an equal chance of being selected.

    <p>True (A)</p> Signup and view all the answers

    Which of the following statements about independent events is true?

    <p>The occurrence of one event does not affect the occurrence of another. (C)</p> Signup and view all the answers

    What does the Z score represent in a normal distribution?

    <p>The number of standard deviations a value is from the mean.</p> Signup and view all the answers

    The sum of the probabilities of all possible outcomes in a sample space is equal to 0.

    <p>False (B)</p> Signup and view all the answers

    What does the Complement Rule state?

    <p>P(A) + P(A​C​) = 1</p> Signup and view all the answers

    In the Empirical Rule, approximately ___% of values lie within 1 standard deviation of the mean.

    <p>68</p> Signup and view all the answers

    What is the shape of a normal distribution curve?

    <p>Symmetric and bell-shaped (B)</p> Signup and view all the answers

    The probability of event A occurring given that event B has occurred is known as ______.

    <p>conditional probability</p> Signup and view all the answers

    Match the following components with their descriptions:

    <p>Normal Distribution = Symmetric and bell-shaped Z Score = Standardized score indicating standard deviations from the mean Central Limit Theorem = Approximately normal distribution for large sample sizes Simple Random Sample = Every member has an equal chance of selection</p> Signup and view all the answers

    The area under a normal distribution curve is equal to 0.

    <p>False (B)</p> Signup and view all the answers

    What is the formula for the Addition Rule for Mutually Exclusive Events?

    <p>P(A∪B) = P(A) + P(B) (A)</p> Signup and view all the answers

    What happens to the mean of a sample distribution according to the Central Limit Theorem?

    <p>It equals the population mean.</p> Signup and view all the answers

    Cumulative relative frequency gives us an estimate of how many times an event occurred versus how many trials were conducted.

    <p>True (A)</p> Signup and view all the answers

    Define a random phenomenon.

    <p>An event with an uncertain outcome.</p> Signup and view all the answers

    What is the main purpose of standardization in statistics?

    <p>To convert data to a common scale (C)</p> Signup and view all the answers

    A sampling distribution can only be normal if the population distribution is also normal.

    <p>False (B)</p> Signup and view all the answers

    What is the term used to describe an accurate statistic that estimates a population parameter?

    <p>Unbiased estimator</p> Signup and view all the answers

    The variability of the sampling distribution decreases as the sample size n increases by a factor of ______.

    <p>1/√(n)</p> Signup and view all the answers

    When is a sample result considered statistically significant?

    <p>When P &lt; 0.05 (A)</p> Signup and view all the answers

    Chance error refers to the systematic bias in the sampling procedure.

    <p>False (B)</p> Signup and view all the answers

    What is the central limit theorem (CLT) regarding the sampling distribution of the sample mean?

    <p>The sampling distribution of the sample mean is approximately normal if n is large, regardless of the population distribution.</p> Signup and view all the answers

    What formula represents the standard deviation of the sampling distribution of the sample mean (x̄)?

    <p>σ​x̄ = σ / √(n) (C)</p> Signup and view all the answers

    The margin of error is the full width of the confidence interval.

    <p>False (B)</p> Signup and view all the answers

    What is the interpretation of a confidence level of 95%?

    <p>If I created confidence intervals for all possible samples, the population mean will fall in 95% of them.</p> Signup and view all the answers

    The formula for calculating the confidence interval when the population standard deviation is unknown is x̄ ± t * ( ________ / √(n)).

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

    Match the following hypothesis tests with their definitions:

    <p>One sample test = Test comparing one sample mean to a population mean Two sample test = Test comparing two sample means Paired test = Test comparing paired differences between two related samples</p> Signup and view all the answers

    Which of the following statements accurately describes statistical confidence?

    <p>A higher statistical confidence level decreases the statistical significance. (B)</p> Signup and view all the answers

    In a null hypothesis, H​0​: μ = x, the parameter being tested is the sample mean.

    <p>False (B)</p> Signup and view all the answers

    What does SRS stand for in the context of technical conditions for significance testing?

    <p>Simple Random Sample</p> Signup and view all the answers

    Flashcards

    Sample space

    The set of all possible outcomes in an experiment.

    Event

    A subset of outcomes in the sample space.

    Permutation

    The arrangement of items in a specific order.

    Combination

    A selection where order doesn't matter.

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    Expected value

    The long-term average value of a numerical random process.

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    Simulation

    A representation of a random process used to study its long-term properties.

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    Probability

    The probability that an event will occur.

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    Venn diagram

    A visual representation of the relationship between sets.

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    Theoretical Probability

    An event whose probability is determined in advance by a process of reasoning.

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    Empirical Probability

    An event whose probability is determined by examining the results of an experiment.

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    Mutually Exclusive Events

    Two events that cannot occur simultaneously.

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    Independent Events

    An event that does not influence the outcome of another event.

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    Dependent Events

    An event whose outcome is influenced by the outcome of another event.

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    Random Phenomenon

    An event whose outcome is unpredictable.

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    Random Variable

    The numerical outcome of a random phenomenon.

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    Probability Histogram

    A graphical representation of a random variable's probability distribution.

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    Simple Random Sample (SRS)

    A way to avoid a biased sampling method. Gives every member of the population the same chance of being selected for the sample. Ensures that every possible sample has an equal chance of being in the sample ultimately selected.

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    Treatment

    An explanatory variable group the researcher controls/imposes. Used to compare the effect on the response variable.

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    Unbiased Statistic

    Values of the statistic from different random samples are centered at the actual parameter value.

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    Binomial Distribution

    A frequency distribution of the possible number of successful outcomes in a given number of trials. Needs two possible outcomes, independent trials, and a constant probability of success.

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    Normal Distribution Curve

    Symmetric, single-peaked, and bell-shaped. The mean, median, and mode are equal. The area under the curve is 1. Approaches, but never touches, the x-axis.

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    Z-score

    Shows how many standard deviations above or below the mean a particular value falls. Z = (x - μ)/σ.

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    Empirical Rule (68-95-99.7 Rule)

    A rule saying that 68% of data is within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

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    Central Limit Theorem for a Sample Mean

    For a SRS from a large population, the sampling distribution will be approximately normal if the population is normal or the sample size is larger than 30. The mean of the sampling distribution will be equal to the population mean. The standard deviation of the sampling distribution is σ/√(n).

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    Sampling distribution of the sample mean

    A distribution representing all possible sample means that can be drawn from a population.

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    Probability distribution

    The distribution of all the possible outcomes of a random variable.

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    Population distribution

    A distribution representing the actual values of a characteristic within a population.

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    Sample distribution

    A distribution representing the data from a single sample drawn from a population.

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    Unbiased estimator

    A statistic that accurately estimates a population parameter.

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    Standardization

    The process of converting data to a standardized scale using z-scores, enabling comparisons between different distributions.

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    Statistical significance

    A significant difference between the sample mean and the population mean is unlikely to be due to chance alone.

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    Central Limit Theorem (CLT)

    The central limit theorem states that the sampling distribution of the sample mean will be approximately normal, regardless of the population distribution, if the sample size is sufficiently large (n ≥ 30).

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    Normal Probability Plot

    Checking the distribution of a sample data to see if it's normally distributed even if we don't know the population distribution. It uses the central limit theorem to approximate the sampling distribution.

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    One Sample T-test

    A statistical test to determine if there's a significant difference between the mean of a single sample and a known population mean.

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    Two Sample T-test

    A statistical test to determine if there's a significant difference between the means of two independent samples.

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    Paired T-test

    A statistical test to determine if there's a significant difference between the means of two dependent samples.

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    P-value

    The probability of obtaining the observed results or more extreme results if the null hypothesis is true.

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    Standard deviation of the sampling distribution of the sample mean (σ​x̄)

    The standard deviation of the sampling distribution of the sample mean (x̄), which measures how much the sample means are likely to vary from the true population mean.

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    Standard error (s​x̄)

    An estimate of the standard deviation of the sampling distribution of the sample mean when the population standard deviation (σ) is unknown. It uses the sample standard deviation (s) instead.

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    Confidence interval

    The range of values within which we are confident that the true population mean lies, based on a sample.

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    Point estimate

    A specific value calculated from a sample (e.g., the sample mean), used as a best guess for the population parameter.

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    Margin of error

    The amount added and subtracted from the point estimate to create the confidence interval. It reflects the uncertainty in the estimation.

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    Significance testing

    A statistical test used to determine whether there is enough evidence to reject the null hypothesis. It involves comparing the observed data to the null hypothesis.

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    Null hypothesis (H​0​)

    A statement about the population parameter that we are trying to disprove. It usually assumes no effect or no difference.

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    Alternative hypothesis (H​A​)

    A statement that contradicts the null hypothesis and describes the alternative scenario we are trying to find evidence for.

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    Study Notes

    Probability

    • Probability is the mathematics of chance behavior
    • Sample space is the set of all possible outcomes
    • Outcomes are the elements of a sample space
    • Event is a subset of outcomes from the sample space
    • Trial is any procedure that can be infinitely repeated and has a well-defined sample space

    Permutations

    • Permutation is an arrangement of items in a particular order
    • P(n,r) = n! / (n - r)! for 0 ≤ r ≤ n

    Combinations

    • Combination is a selection in which order does not matter
    • C(n, r) = n! / r! (n - r)! for 0 < r ≤ n

    Factorials

    • n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1
    • 0! = 1

    Expected Value

    • Expected value is the long-term average value achieved by a numerical random process
    • (outcome 1 * probability of outcome 1) + (outcome 2 * probability of outcome 2) + ...

    Percentile

    • Each of 100 equal groups into which a population can be divided according to a distribution of values
    • of a particular variable

    Simulation

    • Simulation is an artificial representation of a random process used to study the process's long-term properties
    • Steps in creating a simulation:
      • Identify the real-world activity to be repeated
      • Link the activity to one or more random numbers
      • Describe how to use random numbers to complete a full trial
      • State the response variable

    Fundamental Principle of Counting (Multiplication Principle)

    • If one event has "m" possible outcomes and a second event has "n" possible outcomes, then there are m*n possible outcomes for the two events together.

    Venn Diagrams

    • S (rectangle) represents the sample space
    • A and B (circles) represent specific events in the sample space S
    • Not (complement): A' or AC—the probability that an event will fail to occur
    • And (intersection): A∩B—the probability that both events occur
    • Or (union): A∪B—the probability that either of two events occur

    Independent vs. Dependent Events

    • Independent events—the occurrence of one event does not affect the occurrence of a second
    • Dependent events—the occurrence of one event affects the occurrence of a second
    • Mutually exclusive events (disjoint events): Two events that cannot happen at the same time

    Probability of an Event

    • P(E) = number of outcomes in event E / number of outcomes in the sample space

    Multiplication Rule

    • P(A∩B) = P(A) * P(B|A) (General Multiplication Rule)
    • P(A∩B) = P(A) * P(B) (Multiple Rule for Independent Events)

    Addition Rule

    • General Addition Rule: P(A∪B) = P(A) + P(B) - P(A∩B)
    • Addition Rule for Mutually Exclusive Events: P(A∪B) = P(A) + P(B)

    Conditional Probability Rule

    • P(B|A) = P(A∩B) / P(A)

    Random Phenomenon

    • An event with an uncertain outcome

    Random Variable

    • The numerical outcome of a random phenomenon

    Probability Histogram of a Random Variable

    • X-axis: possible values (x) of the random variable
    • Y-axis: probability of x

    Theoretical (Exact) Probability

    • Ratio of the number of favorable outcomes to the total number of possible outcomes

    Empirical (Estimate of) Probability

    • Ratio of the number of favorable outcomes to the total number of trials

    Cumulative Relative Frequency

    • Ratio of cumulatively, how many times an event occurs to the maximum amount of times it could have occurred

    Tables

    • Group observational units based on two categorical variables (e.g., education and salary)

    Tree Diagrams

    • Describe the probability of A happening
    • Then, given A, describe the probability of B happening

    Binomial Distributions

    • A frequency distribution of possible numbers of successful outcomes in a given number of trials
    • Conditions for a binomial experiment:
      • Two possible outcomes (typically "success" and "failure")

    Normal Distributions

    • Shape: Symmetric, single-peaked, and bell-shaped
    • Mean, median, and mode are equal
    • Area under the curve is 1
    • Curve approaches but never touches the x-axis

    Normal Quantile Plots

    • Normal: The scatter plots resemble a straight line
    • Skewed to the left: The points curve downward
    • Skewed to the right: The points curve upward

    Z-score

    • z = (x - μ) / σ ; x is a particular value

    68-95-99.7 Rule

    • Normal distribution: Values within 1 standard deviation of the mean: 68%
    • Within 2 standard deviations: 95%
    • Within 3 standard deviations: 99.7 %

    ShadeNorm() and normalcdf()

    • Used to find areas under the curve
    • Given population mean and standard deviation, you can find the area between the lower and upper bounds

    invNorm()

    • Given an area and population mean/standard deviation, find the z-value.

    Central Limit Theorem for a Sample Mean

    • Shape: The distribution will be approximately normal if population distribution is normal or the sample size is greater than 30
    • Center: The mean will equal μ, regardless of population distribution
    • Spread: The standard deviation will equal σ / √(n).

    Population Distribution

    • Distribution of all members of a population

    Sample Distribution

    • Distribution of a single sample of a population

    Sampling Distribution

    • Distribution of sample means of all or many possible samples of a population

    Standardization

    • Calculate a z-score to determine how many standard deviations a value falls above or below the mean

    Unbiased Estimator

    • An accurate statistic used to estimate a population parameter

    Sampling Distribution of x̄

    The sampling distribution of the sample mean is approximately normal when the sample size is large (n>30) regardless of the shape of the population distribution.

    Significance Testing

    • Hypothesis testing to determine if there is enough evidence to reject a null hypothesis

    Unbiased Estimator of μ

    • The variability of the sampling distribution decreases as the sample size increases.

    Central Limit Theorem (Shape of the Sampling Distribution)

    • Distribution approximated by a normal distribution for large sample sizes

    Sources of Variation

    • Bias in sampling procedure
    • Chance error
    • Significant event = A difference between the population mean and the sample mean that cannot reasonably be attributed to chance.

    Statistical Significance

    • A sample result is considered statistically significant if it's unlikely to occur due to random variability alone (P-value < 0.05)

    Standard Error

    • A close estimate of the standard deviation of the sampling distribution

    Confidence Intervals

    • Provides a range of values within which the true population parameter is likely to fall with a certain level of confidence

    Confidence Level

    • The probability that the confidence interval contains the true population parameter.

    Margin of Error

    • Half the width of the confidence interval.

    Confidence Interval for a Sample Mean (σ known)

    • x̄ ± z * (σ / √n)

    Confidence Interval for a Sample Mean (σ unknown)

    • x̄ ± t * (s / √n)

    Interpretation of a Confidence Interval

    • Interpretation of a confidence level, how confident we are that the interval captures the true population.

    Statistical Confidence vs. Statistical Significance

    • They are opposites; if a statistical test has a 95% confidence level then a statistical significance level is 5%

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    Test your understanding of probability, permutations, combinations, and expected values. This quiz covers essential concepts and calculations related to chance behavior and arrangements in mathematics. Challenge your knowledge of factorials, events, and simulations.

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