Statistics: Sampling Error and Hypothesis Testing
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Questions and Answers

What defines independent events in probability?

  • Both events must happen together
  • The occurrence or nonoccurrence of one event has no effect on the other (correct)
  • The occurrence of one event negates the occurrence of another
  • The occurrence of one event affects the occurrence of another

Which rule applies to the probability of mutually exclusive events?

  • The average of individual probabilities
  • The absolute difference of individual probabilities
  • The sum of individual probabilities of events (correct)
  • The product of individual probabilities of events

What is subjective probability based on?

  • Mathematical calculations of frequency
  • Predefined probabilities based on historical data
  • Statistical likelihood of events
  • An individual's belief about an event's likelihood (correct)

What does it mean for events to be exhaustive?

<p>They include all possible outcomes (A)</p> Signup and view all the answers

What is the purpose of the multiplicative law of independent events?

<p>To calculate joint probabilities of independent events (B)</p> Signup and view all the answers

When are two random variables considered independent?

<p>When their marginal probabilities are equal to the conditional probabilities for all conditions. (C), When their conditional probabilities are unequal to one another. (A)</p> Signup and view all the answers

What does the general law of probability state about two independent events?

<p>The chance that both events happen at the same time is the product of their probabilities. (D)</p> Signup and view all the answers

What is meant by conditional probability?

<p>The probability of one event occurring given that another event has occurred. (D)</p> Signup and view all the answers

Which of the following statements is true regarding independent events?

<p>The conditional and marginal probabilities are the same for all conditions. (D)</p> Signup and view all the answers

In hypothesis testing, what is being assessed with respect to H0?

<p>The probability of obtaining a score more extreme than the observed score, given that H0 is true. (C)</p> Signup and view all the answers

What does a lower p-value indicate regarding the null hypothesis (H0)?

<p>More evidence against H0. (D)</p> Signup and view all the answers

What is the typical significance level used to decide whether to reject H0?

<p>0.05 (C)</p> Signup and view all the answers

If a researcher commits a Type I error, what have they done?

<p>Rejected H0 when it is true. (C)</p> Signup and view all the answers

What is a critical value in hypothesis testing?

<p>The cutoff point that determines the rejection region. (A)</p> Signup and view all the answers

Which statement correctly describes the power of a test?

<p>Probability of rejecting H0 when it is false. (C)</p> Signup and view all the answers

What characterizes a two-tailed test compared to a one-tailed test?

<p>It tests for extreme values in both directions. (B)</p> Signup and view all the answers

If Type I error rates are reduced, what is the expected impact on Type II errors?

<p>Increase in Type II error rates. (A)</p> Signup and view all the answers

What does it mean to reject the null hypothesis?

<p>Concluding that there is enough evidence against H0. (C)</p> Signup and view all the answers

What is eta squared used for in t-tests?

<p>Measuring the proportion of variance accounted for (A)</p> Signup and view all the answers

What does a higher score in absolute effect sizes indicate?

<p>A clear interpretation of the effect (C)</p> Signup and view all the answers

Which aspect influences the precision of confidence intervals?

<p>The level of variation in the sample (C)</p> Signup and view all the answers

For which analysis is the sd of difference scores particularly beneficial?

<p>Paired sample tests (D)</p> Signup and view all the answers

What does a confidence level of 95% suggest about findings?

<p>There is a 5% chance the findings could be incorrect (C)</p> Signup and view all the answers

What is categorical data primarily composed of?

<p>Frequencies of observations in multiple categories (B)</p> Signup and view all the answers

What is the primary purpose of the chi-square test?

<p>To test the independence of two categorical variables (B)</p> Signup and view all the answers

What do observed frequencies represent in a chi-square test?

<p>The actual counts gathered in the sample (A)</p> Signup and view all the answers

In the context of a goodness-of-fit test, what does the term 'expected frequencies' refer to?

<p>Counts we would expect if the null hypothesis were true (A)</p> Signup and view all the answers

What is the interpretation of the degrees of freedom in the chi-square test?

<p>The number of categories minus one (D)</p> Signup and view all the answers

How do we determine the significance of the chi-square statistic obtained?

<p>By using the chi-square distribution table and degrees of freedom (D)</p> Signup and view all the answers

What does the expected frequency in a contingency table represent?

<p>What we would expect if the two variables were independent (D)</p> Signup and view all the answers

How is the degrees of freedom calculated for a contingency table?

<p>Df = (R - 1)(C - 1) (C)</p> Signup and view all the answers

What does a low p value indicate in statistical testing?

<p>We have found evidence against the null hypothesis (D)</p> Signup and view all the answers

What does a high p value indicate in statistical hypothesis testing?

<p>There is an absence of evidence against the null hypothesis (B)</p> Signup and view all the answers

Which statement about statistical significance is accurate?

<p>Statistical significance does not equate to practical relevance (C)</p> Signup and view all the answers

What does the term 'nonocurrences' refer to in statistical analysis?

<p>Responses indicating disagreement (D)</p> Signup and view all the answers

Flashcards

Frequentist probability

The probability of an event is determined by the proportion of times it occurs in many repetitions of an experiment.

Subjective probability

An individual's personal belief about the likelihood of an event.

Independent events

Events whose outcomes do not affect each other.

Additive rule

The probability of one event OR another (mutually exclusive) is the sum of their individual probabilities.

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Multiplicative rule

The probability of two independent events BOTH happening is the product of their individual probabilities.

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p-value

The probability of obtaining results as extreme as or more extreme than those observed, assuming the null hypothesis (H0) is true.

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Type I error

Rejecting the null hypothesis (H0) when it is actually true.

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Type II error

Failing to reject the null hypothesis (H0) when it is actually false (and the alternative hypothesis (H1) is true).

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Significance level (alpha)

The probability of making a Type I error.

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Power

The probability of correctly rejecting the null hypothesis (H0) when it is false.

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One-tailed test

A hypothesis test where the alternative hypothesis specifies a direction (e.g., greater than or less than).

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Two-tailed test

A hypothesis test where the alternative hypothesis does not specify a direction (e.g., not equal to).

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Null Hypothesis (H0)

A statement of no effect or difference between groups. It's assumed to be true, unless the data shows otherwise.

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Conditional Probability

The probability of one event occurring, given that another event has already happened.

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Conditional Probabilities for Independent Events

For independent events, conditional probabilities are the same as the marginal probability of the event.

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Co-occurrence Probability

The probability of two or more events happening at the same time.

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General Law

The probability of two events happening together is the probability of one event multiplied by the probability of the other event, given the first event has already occurred.

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Contingency Table

A table used to analyze the relationship between two categorical variables. It shows the observed frequencies of each combination of categories.

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Expected Frequencies

The frequencies we would expect if the two variables in a contingency table were independent. Calculated by multiplying row and column totals and dividing by the total sample size.

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Chi-Square Test

A statistical test used to determine if there is a significant association between two categorical variables. It compares observed frequencies to expected frequencies.

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Degrees of Freedom (Chi-Square)

The number of independent pieces of information used to calculate the chi-square statistic. Calculated as (Number of rows - 1) * (Number of columns - 1).

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Measurement of Agreement (Kappa)

A statistic used to assess the reliability of ratings or judgments made by two or more observers. It examines how much their agreement exceeds what's expected by chance.

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Nonoccurrence Effect

The impact of negative responses or 'non-occurrences' on statistical analysis. They can significantly affect the rejection of the null hypothesis.

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P-Value Pitfalls

A statistical test can show a significant result even with a tiny difference between populations if the sample is large enough. A low p-value doesn't mean a big effect.

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Effect Size in Categorical Variables

Measures the magnitude of the relationship between categorical variables. It helps determine the practical significance of a statistical result beyond just statistical significance.

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Categorical Data

Data that represents frequencies of observations falling into different categories. It uses counts of observations within each category.

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Chi-Square Test (χ²)

A statistical test used to determine if there's a relationship between two categorical variables, or if the observed frequencies differ significantly from expected frequencies.

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Chi-Square Distribution

A statistical distribution used in the chi-square test. It's a one-tailed distribution with a single parameter (degrees of freedom) based on the number of categories in the experiment.

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Goodness-of-Fit Test

A specific application of the chi-square test that determines if the observed frequencies in a single sample differ significantly from the expected frequencies.

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Observed Frequencies

The actual number of observations counted in each category of a variable.

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Degrees of Freedom (df)

Number of categories minus 1 in a chi-square test. It determines the shape of the chi-square distribution.

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p-value (for χ²)

The probability of obtaining results as extreme as or more extreme than the observed frequencies, assuming the null hypothesis (of independence) is true.

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Standard Deviation (SD) for T-tests

The measure of variability used in t-tests, reflecting the typical spread of data points around the mean.

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Paired Sample T-test SD

In paired sample t-tests, the standard deviation is calculated for the difference scores (the difference between paired observations).

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One-Sample T-test SD

In one-sample t-tests, the standard deviation is calculated for the single sample being compared to a known population mean (mu).

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Independent Sample T-test SD

In independent sample t-tests, the standard deviation can be calculated for each sample separately or pooled into a single estimate for both groups.

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Effect Size: Eta Squared

A measure of effect size in t-tests, representing the proportion of variance explained by the independent variable.

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Study Notes

Sampling Error

  • Sampling error refers to the variability of a sample statistic from a population parameter.
  • A sample statistic will likely deviate from the population parameter due to chance.
  • For example, if a population has a mean of 50, a sample of ten people might have a mean of 49 or 51.

Hypothesis Testing

  • Hypothesis testing is used to determine if observed differences in data are due to chance.
  • A hypothesis is a statement about a population parameter.
  • Sampling distributions can be used to understand the variability of a statistic.

Sampling Distributions

  • Sampling distributions show how sample statistics vary over repeated samples.
  • They are distributions of sample means.
  • They are not distributions of scores.

Distribution of Values

  • The distribution of values obtained from a sample statistic is used over repeated sampling.
  • This distribution can help determine if chance is the reason for the differences or not.

Probability of Population Means

  • Probability of having the same sampling difference if population means were equal.
  • A low probability suggests independent variable change.

Hypothesis Testing Process

  • Begin with a research hypothesis.
  • Establish null hypothesis (no change/no effect).
  • Create sampling distribution under the null hypothesis.
  • Collect some data.
  • Compare sample statistic to established distribution.
  • Reject or retain null hypothesis based on probability calculations.

Error Types

  • Type I error: Rejecting a true null hypothesis (false positive).
  • Type II error: Failing to reject a false null hypothesis (false negative).
  • Significance level (alpha) is the probability of a Type I error.

Using Conventions

  • Rejection levels/significance levels indicate the differences are statistically significant.
    • For instance, a .05 level means a difference that occurs less than 5% of the time under the null.

Other Statistical Concepts

  • Standard error: The standard deviation of a sampling distribution.
    • Used to assess variability in sample means from repeats.
  • Central Limit Theorem: Sampling distribution of the mean tends towards a normal distribution as sample size increases.
  • Test statistics: Associated with specific statistical procedures, they have their own sampling distributions.

Additional Concepts

  • Bootstrapping: Sampling with replacement from the obtained data/sample.
  • A method helpful for assessing distributions of samples with characteristics similar to the examined sample.
  • Confidence interval: the range within which the true population mean is likely to fall with a given confidence level.

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Inferential Statistics PDF

Description

This quiz covers key concepts in statistics, focusing on sampling error, hypothesis testing, and sampling distributions. Understand how sample statistics can vary from population parameters and learn about the implications of these concepts in data analysis. Engage with scenarios to test your knowledge on statistical variability and inference.

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