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What does the Central Limit Theorem state about larger sample sizes?

  • It guarantees the data will be uniformly distributed.
  • It ensures the data will be skewed.
  • It implies that individual data points will have no effect on the distribution.
  • It allows the distribution to approximate a normal distribution. (correct)

    • In a normal distribution, what percentage of data falls within one standard deviation of the mean?

  • 68% (correct)
  • 95%
  • 99%
  • 50%

    • Which measure of central tendency is appropriate for qualitative data?

  • Geometric mean
  • Median
  • Mean
  • Mode (correct)

    • How is the population mean denoted in statistical notation?

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

    What is a key property of the mean regarding deviations from the mean?

    <p>The deviations from the mean always sum to zero.</p> Signup and view all the answers

    Which term describes the most frequently occurring observation in a data set?

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

    In relation to summary statistics, which two types of measures are commonly used?

    <p>Measures of dispersion and central tendency</p> Signup and view all the answers

    What notation is used to represent the sample mean?

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

    What is the primary purpose of inferential statistics?

    <p>To analyze population parameters using sample data.</p> Signup and view all the answers

    Which of the following is not typically a component of hypothesis testing?

    <p>Identifying the mean of sample data.</p> Signup and view all the answers

    What statistical measure indicates the spread of data points in a dataset?

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

    Which term best describes a range of values that is likely to contain a population parameter?

    <p>Confidence interval</p> Signup and view all the answers

    When performing hypothesis testing, what does a p-value indicate?

    <p>The strength of the evidence against the null hypothesis.</p> Signup and view all the answers

    Which visualization technique is not used to summarize data?

    <p>Logistic regression model</p> Signup and view all the answers

    What foundational concept in statistics helps in understanding uncertainty in data?

    <p>Probability theory</p> Signup and view all the answers

    Which of the following best defines logistic regression?

    <p>A statistical method for predicting binary outcomes.</p> Signup and view all the answers

    What is the primary goal of an experiment in research?

    <p>To demonstrate a cause-and-effect relationship between two variables</p> Signup and view all the answers

    Which of the following best describes discrete variables?

    <p>Variables that consist of indivisible categories</p> Signup and view all the answers

    What distinguishes a sample from a population in research?

    <p>A sample is selected to represent the entire population</p> Signup and view all the answers

    What is the definition of a simple random sample (SRS)?

    <p>A sample where each collection of n population items is equally likely to be selected</p> Signup and view all the answers

    What role do descriptive statistics play in research?

    <p>They summarize and organize data characteristics</p> Signup and view all the answers

    In statistical research, what is inferential statistics concerned with?

    <p>Using sample data to make general conclusions about populations</p> Signup and view all the answers

    Which statement about independent items in a sample is correct?

    <p>Items can typically be treated as independent in most cases</p> Signup and view all the answers

    What is the distinction between a parameter and a statistic?

    <p>A parameter is a descriptive value for a population, while a statistic is for a sample</p> Signup and view all the answers

    What criterion primarily separates biased from unbiased estimators?

    <p>Unbiased estimators consistently equal the population value across samples.</p> Signup and view all the answers

    Which of the following is true regarding sample measurements?

    <p>Sample measurements can possess both variance and bias.</p> Signup and view all the answers

    How is variance defined in a statistical context?

    <p>The mean squared deviation of points from the mean.</p> Signup and view all the answers

    What effect does increasing sample size have according to the Central Limit Theorem?

    <p>The distribution of sample means becomes normally distributed.</p> Signup and view all the answers

    Which symbol represents a sample mean in statistical notation?

    <p>$x̄$</p> Signup and view all the answers

    What is the standard deviation in relation to variance?

    <p>It is the square root of variance.</p> Signup and view all the answers

    Which of the following statements about normal distribution is accurate?

    <p>Mean and variance uniquely define the normal distribution.</p> Signup and view all the answers

    What factor most impacts biased estimators in terms of reliability?

    <p>Sample size, with bias worsening on smaller samples.</p> Signup and view all the answers

    What ensures that statistical tools like t-tests and ANOVA are valid?

    <p>Data should be normally distributed.</p> Signup and view all the answers

    How can data that is log-normally distributed be transformed?

    <p>By applying a logarithm.</p> Signup and view all the answers

    What clue indicates that data may not be normally distributed when using box-and-whisker plots?

    <p>Presence of outliers.</p> Signup and view all the answers

    What is one characteristic of the Poisson distribution?

    <p>It is useful for modeling counts of events occurring at a fixed rate.</p> Signup and view all the answers

    What is a potential consequence of not understanding the distribution of your data before applying a model?

    <p>The model may yield misleading results.</p> Signup and view all the answers

    What aspect of histogram normalization involves scaling?

    <p>Making sure the area of the total bars sums to specific values.</p> Signup and view all the answers

    What could misunderstanding the Rhine Paradox lead to?

    <p>Incorrectly asserting the existence of ESP.</p> Signup and view all the answers

    Which of the following distributions describes the frequency of different terms in a document?

    <p>Zipf/Pareto/Yule distributions.</p> Signup and view all the answers

    What is the significance level (α) set in the power calculation described?

    <p>0.05</p> Signup and view all the answers

    What is the computed power of the test based on the provided calculations?

    <p>0.5160</p> Signup and view all the answers

    Which formula is used to calculate the sample size for a one-sample z test?

    <p>$n = 2(\sigma z_{1-\beta} + z_{1-\alpha})^2 / \Delta^2$</p> Signup and view all the answers

    In the context of hypothesis testing, what does H0 represent?

    <p>The null hypothesis</p> Signup and view all the answers

    Given μ0 = 170 and μa = 190, what is the value of Δ?

    <p>-20</p> Signup and view all the answers

    What is the rounded sample size needed for a one-sample z test with 90% power?

    <p>42</p> Signup and view all the answers

    Which curve assumes the null hypothesis (H0) is true in the power illustration?

    <p>Top curve</p> Signup and view all the answers

    How is the probability of obtaining a value greater than 189.6 interpreted?

    <p>It indicates the power of the test.</p> Signup and view all the answers

    Study Notes

    Inferential Statistics Course Notes

    • Course Objective 1: To equip students with skills to summarize and interpret data using descriptive statistics and visualization techniques.
    • Course Objective 2: To develop a foundational understanding of probability and its applications in data science.
    • Course Objective 3: To enable students to perform hypothesis testing and construct confidence intervals for statistical inference.
    • Course Objective 4: To teach students how to build and assess linear and logistic regression models for predictive analysis.
    • Course Objective 5: To provide hands-on experience with statistical software for data manipulation, analysis, and visualization.

    Course Outcomes

    • CO1: Summarize and describe dataset features (mean, median, mode, variance, standard deviation), using graphs (histograms, box plots, scatter plots).
    • CO2: Understand probability theory (random variables, probability distributions, law of large numbers) to model uncertainty in data.
    • CO3: Apply statistical inference (hypothesis testing, confidence intervals, p-value computation) to draw conclusions from sample data about larger populations.
    • CO4: Apply linear and logistic regression for identifying relationships, making predictions, and evaluating model performance.
    • CO5: Use statistical software for data analysis (cleaning, transformation, visualization, various statistical methods).

    Unit-3: Inferential Statistics Syllabus

    • Inferential Statistics & Hypothesis Testing: Statistical Inference Terminology, Hypothesis Testing, Parametric Tests, Non-parametric Tests
    • Industry Application: Hypothesis Testing using Excel, Industry Practices & Applications of Statistics

    Suggestive Readings

    • Text Books: Hastie, Trevor, et al., The elements of statistical learning, Montgomery, Douglas C., and George C. Runger, Applied statistics and probability for engineers, Probability and Statistics, Jeffrey S. Rosenthal.
    • Reference Books: Practical Statistics for Data Scientists (Peter Bruce et al.), An Introduction to Statistical Learning (Gareth James et al.), Think Stats.

    What is a Statistic?

    • Population: The whole group of individuals of interest.
    • Parameter: A value that describes a population.
    • Sample: A part of a population.
    • Statistic: A value that describes a sample.
    • Note: Psychology (PSYCH) often uses samples.

    Descriptive & Inferential Statistics

    • Descriptive Statistics: Organizing, summarizing, and presenting data.
    • Inferential Statistics: Generalizing from samples to populations, hypothesis testing, and relationships among variables.

    Descriptive Statistics Types

    • Frequency Distributions: Number of subjects that fall into categories.
    • Graphical Representations: Graphs and Tables (histograms, box plots, scatter plots).
    • Summary Statistics: Single numbers (mean, median, mode) to describe data.

    Frequency Distributions Examples

    • Cross-tabulation: Categorizing data based on multiple variables (e.g., Democrats/Republicans, male/female).
    • Calculating percentages and proportions based on totals from frequency distributions.

    Central Limit Theorem

    • The larger the sample size, the closer a distribution will approximate a normal distribution.

    Normal Distribution

    • Half the scores are above the mean, and half are below (symmetrical).

    Summary Statistics

    • Measures of central tendency: Mean for typical average score
    • Measures of variability: Range, Variance, Standard Deviation, and Standard Error of the Mean

    Measures of Central Tendency

    • Quantitative Data: Mode (most frequent), Median (middle value), Mean (average).
    • Qualitative Data: Mode only.

    Mean (Notation)

    • Sample mean = X
    • Population mean = μ
    • Summation sign = ∑
    • Sample size = n
    • Population size = N

    Special Property of the Mean

    • The sum of all deviations from the mean equals zero.

    Inferential Statistics

    • Using sample data to evaluate a hypothesis about a population.

    Null Hypothesis

    • Ho: The claim we're evaluating (no difference between means).

    Alternative Hypothesis

    • Ha: The claim we're trying to find evidence for (there is a difference).

    Hypothesis Testing Decision Possibilities

    • Null hypothesis is true ⇒ Do not reject Ho
    • Null hypothesis is false ⇒ Reject Ho

    Possible Outcomes in Hypothesis Testing (Decision)

    • Null is True: Correct decision / Error (Type I Error)
    • Null is False: Error (Type II Error) / Correct decision

    Alpha (α)

    • Probability of making a Type I error (rejecting a true null hypothesis)

    Beta (β)

    • Probability of making a Type II error (failing to reject a false null hypothesis)

    Power

    • Ability to reduce type II error

    Inferential Statistics Tests for Mean Differences

    • T-test (Independent/Correlated/Within-Subjects): For comparisons between 2 groups, or repeated measurements in one group.
    • Analysis of Variance (ANOVA): For comparing more than 2 groups.

    Meta-Analysis

    • Statistical averaging of results from independent studies about the same phenomenon.

    Other Important Distributions

    • Poisson: Distribution of counts occurring at a rate, e.g., web visits.
    • Exponential: Intervals between events.
    • Binomial/Multinomial: Categorical outcomes, e.g., die tosses.

    Statistical Significance Testing

    • Practical significance vs. statistical significance.

    Method 1: Ablation

    • Train a model with all the features, and then calculate performance Qo.
    • Remove a feature, retrain the model, and calculate the Performance Q1
    • If Q1 is significantly worse than Q0, the feature is useful (keep it). Otherwise, discard.
    • Note: Check significantly worse using a statistical test, e.g., t-test or bootstrap sampling.

    Method 2: Mutual Information

    • MI measures the relationship one feature has to another. Higher MI means potential to be important.

    Method 3: Chi-Squared

    • Used for comparing contingency table counts to determine feature dependence.

    Measurement

    • Basic properties (min, max, mean, standard deviation).
    • Relationships (scatter plots, regression).
    • Model accuracy and performance.

    Variables

    • Characteristics or conditions that change.
    • Values can vary.
    • Categorical (discrete or ordinal classes).
    • Numerical (can be continuous).

    Population

    • Total group of all possible values.

    Sample

    • Portion of the population.

    Statistical Notation

    • Uppercase (X) represents a random variable.
    • Lowercase (x) represents specific samples from a population.

    Normal Distributions

    • Complete characteristic are based on mean and variance.
    • Standard Deviation = square root of variance

    Central Limit Theorem

    • The theoretical foundation behind generalizing sample results to populations.

    Bootstrap Sampling

    • A method to estimate the variability of a statistic based on resampling.

    Types of Data

    • Numerical/Quantitative: Numerical quantities.
      • Continuous: Can take on any value within a range. (Height, Weight).
      • Discrete: Can take on only certain values. (Number of students in a class, number of equipment in a project).
    • Categorical/Qualitative: Placed into groups.
      • Nominal: No inherent order (Gender, Hair Color).
      • Ordinal: Natural order between categories. (Customer satisfaction surveys, student grades).

    Validation

    • A third set of data used for parameter tuning and validating generalized results based on training.

    Train/Test Split

    • Splitting data into two subsets for training and testing.
    • Prevents overfitting in the model.

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