Introduction to Statistics - Statistical Inference
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Questions and Answers

What is the significance of a p-value less than 0.05?

  • It proves that the alternative hypothesis is true.
  • It indicates the null hypothesis is false.
  • It shows the results are statistically significant. (correct)
  • It signifies a potential error in data collection.

    • In the context of the provided example, what does a z-score represent?

  • The average IQ of the population.
  • The total number of samples taken.
  • The number of standard deviations a sample mean is from the population mean. (correct)
  • The probability that a specific result occurs.

    • What does a null hypothesis test assess regarding the population parameter?

  • It compares results only within the same sample group.
  • It evaluates data collection methods for bias.
  • It assumes the sample mean is equal to the population mean. (correct)
  • It determines what the sample parameter could be.

    • What is a fundamental goal of statistical inference?

    <p>To make inferences about the population based on samples. (C)</p> Signup and view all the answers

    What is the standard deviation of the sampling distribution also known as?

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

    What does sampling variation refer to?

    <p>Random differences between samples and the population. (A)</p> Signup and view all the answers

    How does the standard error change as the sample size increases?

    <p>It decreases. (B)</p> Signup and view all the answers

    Which of the following is an example of sampling bias?

    <p>Only surveying individuals who have the ability to respond to surveys online. (D)</p> Signup and view all the answers

    What does the Central Limit Theorem state about the mean of the sampling distribution?

    <p>It is the same as the population mean. (B)</p> Signup and view all the answers

    What statistical perspective is primarily discussed in the provided content?

    <p>Frequentist statistics. (D)</p> Signup and view all the answers

    What is the key focus of frequentist statistics?

    <p>Sampling variability and repetition. (A)</p> Signup and view all the answers

    What is a common feature of Frequentist statistics?

    <p>Application of p-values and null hypotheses. (C)</p> Signup and view all the answers

    In the context of a normal distribution, what does the symbol 'µ' represent?

    <p>Mean of the distribution (B)</p> Signup and view all the answers

    Which statement describes Bayesian statistics?

    <p>It relies on prior beliefs and updates them with new evidence. (D)</p> Signup and view all the answers

    When observing sampling distributions, what can generally be expected about their shape?

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

    What is an example of a method used in Bayesian statistics?

    <p>Credible intervals. (C)</p> Signup and view all the answers

    What term describes the variability of a distribution's data points around its mean?

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

    Why is it often impractical to measure an entire population?

    <p>Resources and time are often limited. (B)</p> Signup and view all the answers

    Which of the following describes a primary function of the sampling distribution of the mean?

    <p>It estimates the probability of obtaining the sample mean. (D)</p> Signup and view all the answers

    Which of the following correctly describes the normal distribution?

    <p>It forms a symmetric bell-shaped curve. (D)</p> Signup and view all the answers

    What is the purpose of the Central Limit Theorem?

    <p>To demonstrate how the mean of a sample approximates the population mean as the sample size increases. (A)</p> Signup and view all the answers

    What does the standard error measure?

    <p>The standard deviation of the sampling distribution of a statistic. (D)</p> Signup and view all the answers

    What is a confidence interval?

    <p>An estimate of a population parameter that includes margins for error. (B)</p> Signup and view all the answers

    What does a p-value indicate in statistical testing?

    <p>The probability of observing the data under null hypothesis conditions. (B)</p> Signup and view all the answers

    Why is understanding standard deviation important in statistics?

    <p>It explains how far individual data points deviate from the mean. (B)</p> Signup and view all the answers

    Which statistic is typically used to summarize the spread of a data set?

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

    What happens to the shape of the sampling distribution as the sample size increases?

    <p>It becomes normal. (A)</p> Signup and view all the answers

    What is the standard deviation of the sampling distribution called?

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

    In a normal distribution, approximately what percentage of estimates lie within 1.96 standard deviations of the mean?

    <p>95% (A)</p> Signup and view all the answers

    Which of the following statements correctly describes a 95% confidence interval?

    <p>It will include the population mean in 95% of repeated samples. (A)</p> Signup and view all the answers

    Which of the following formulas correctly describes the computation for a 95% confidence interval?

    <p>samp_mean – (1.96<em>SE) ≤ pop_mean ≤ samp_mean + (1.96</em>SE) (B)</p> Signup and view all the answers

    What is the role of the null hypothesis in hypothesis testing?

    <p>It posits that there is no effect. (D)</p> Signup and view all the answers

    How does an increase in sample size affect the confidence interval?

    <p>It decreases the confidence interval width. (B)</p> Signup and view all the answers

    Which statement best describes the concept of p-values in frequentist statistics?

    <p>They measure the likelihood of observing the sample assuming the null hypothesis is true. (A)</p> Signup and view all the answers

    What does a p-value of 0.023 indicate in this scenario?

    <p>There is a 2.3% probability of observing a sample mean of 105 or higher. (D)</p> Signup and view all the answers

    What would happen if a smaller sample was collected?

    <p>The p-value would increase, indicating a less significant result. (D)</p> Signup and view all the answers

    How does the alpha level of 0.05 relate to the interpretation of the p-value?

    <p>It indicates the maximum probability allowed for rejecting the null hypothesis. (D)</p> Signup and view all the answers

    What is the primary purpose of using samples in statistics?

    <p>To manage sampling variation and draw conclusions about the population. (C)</p> Signup and view all the answers

    What does the term 'standard error' refer to in the context of sampling distributions?

    <p>The variability among different sample means drawn from the same population. (D)</p> Signup and view all the answers

    Which of the following is a consequence of the central limit theorem?

    <p>Larger sample sizes help in approximating normal distributions. (D)</p> Signup and view all the answers

    In hypothesis testing, what does it mean to conclude that a sample is 'statistically significantly different'?

    <p>The sample statistic falls outside the range of typical random variation. (B)</p> Signup and view all the answers

    What is a key limitation when working with non-normally distributed outcome variables?

    <p>Inference about population parameters becomes more complicated. (D)</p> Signup and view all the answers

    Study Notes

    Introduction to Statistics - Statistical Inference

    • The lecture was about statistical inference
    • Learning objectives included understanding and defining key concepts, like sampling variation, sampling distributions, normal distributions, the Central Limit Theorem, standard error, standard deviation, confidence intervals, and p-values.
    • The lectures assume prior knowledge of means, standard deviations (from descriptive statistics). Basic R functions are also assumed.
    • Students should consult Chapters 3, 4, and 5 of the Navarro textbook for a refresher on these topics. (link provided)

    British Social Attitudes Survey

    • A British Social Survey from 2010-2021 showed that climate change is the most important environmental problem.
    • Data regarding the survey's methodology and sample sizes was provided.

    Sampling Variation

    • Differences between samples and the population are random.
    • This is called sampling variation
    • Sampling bias is different from this variation (sampling bias occurs due to systematic forces like only interviewing people who answer their phones).
    • This involves how best to estimate population values, given sampling variation.

    Frequentist vs. Bayesian Statistics

    • The lecture introduces frequentist statistics, which is the perspective of this course.
    • The alternative approach is Bayesian statistics, which differs fundamentally.
    • Frequentist statistics are more commonly used in practice for things like, p-values, significance, null hypotheses, and confidence intervals.
    • Bayesian statistics involve things like Bayes Factors, priors, and credible intervals.

    Normal Distribution

    • The normal distribution (also known as the Gaussian distribution or bell curve) was covered.

    • Key characteristics of the normal distribution, including:

      • Mean (μ)
      • Standard Deviation (σ)

    • Key formulas regarding calculating standard deviation: Σ(Χί – μ) 2 /N

    • Visual representations of normal distributions and probability densities were shown.

    Sampling Distributions

    • The sampling distribution of the mean refers to the distribution of means when repeatedly sampling from a population.
    • The mean of the sampling distribution is the same as the population mean.
    • The standard deviation of the sampling distribution is the standard error, which decreases with an increase in sample size.
    • Sampling distributions approach a normal distribution as sample size increases (Central Limit Theorem).

    Standard Error

    • The standard deviation of the sampling distribution is called standard error
    • Its value decreases as the sample size increases. This is a helpful measure of sampling error variation and precision

    Confidence Intervals

    • A 95% confidence interval means that 95% of repeated samples will contain the population mean within the upper and lower bounds.
    • Confidence intervals are calculated using the mean of the sample and the standard error.
    • The size of the confidence interval decreases with increasing sample size.

    p-values

    • p-values are used to test hypotheses in frequentist statistics.
    • The null hypothesis assumes no effect or no difference.
    • The p-value gives the probability of observing a sample result as extreme or more extreme, assuming the null hypothesis is true.
    • Results with p-values less than the alpha level (e.g., 0.05) are typically considered statistically significant. A p-value in this range means the sample result is unlikely to have occurred if the null hypothesis were true.

    One-Sample Example

    • A one-sample example concerning 36 children who have completed a program is described.
    • The population parameters for IQ tests are known: mean is 100 and standard deviation is 15.
    • Calculation steps for finding a p-value are detailed.

    Summary

    • We collect samples to make inferences about the wider population
    • Sampling variation is unavoidable but can be reduced with larger sample sizes.
    • Frequentist statistics rely on the Central Limit Theorem which describes how sampling distributions are normally distributed.
    • Confidence intervals and p-values can be calculated from the Central Limit Theorem.
    • p-values are useful for hypothesis testing, comparing an observed outcome to an expected outcome under the null hypothesis.

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    Related Documents

    Lecture Statistical Inference MH 2024 PDF

    Description

    This quiz covers key concepts in statistical inference, including sampling variation, normal distributions, and the Central Limit Theorem. Students should have prior knowledge of means and standard deviations and are encouraged to refer to Chapters 3, 4, and 5 of the Navarro textbook. Engage with the content to enhance your understanding of statistical methodologies.

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