6.2 Sampling and hypothesis testing

Sampling and hypothesis testing are crucial tools in statistical inference. They allow us to draw conclusions about populations based on smaller, manageable samples. These methods help us make educated guesses about the bigger picture using limited data.

In this section, we'll explore how to select samples, formulate hypotheses, and conduct tests in R. We'll also learn to interpret results, considering p-values, confidence intervals, and potential errors. These skills are essential for making data-driven decisions in various fields.

Sampling and Sampling Distributions

Sampling Concepts

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• Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population
• The goal is to obtain a representative sample that accurately reflects the population
• A sample statistic is a single measure, such as mean or proportion, calculated from a sample
• A population parameter is the actual value of a characteristic in the population, which is typically unknown

Sampling Bias and Variability

• Sampling bias occurs when some members of the population are systematically more likely to be selected in a sample than others, resulting in a sample that is not representative of the population
  • Common types of sampling bias include selection bias, self-selection bias, and non-response bias
• Sampling variability refers to the extent to which a statistic varies in repeated sampling
  • The sampling distribution shows how sample statistics are distributed when a large number of samples are drawn from a population
• The Central Limit Theorem states that, with a large enough sample size, the sampling distribution of the mean will be approximately normal regardless of the shape of the population distribution
  • This allows for the use of parametric tests that assume normality
• Standard error measures the variability of a statistic in repeated sampling
  • It is calculated as the standard deviation of the sampling distribution and decreases as the sample size increases, indicating more precise estimates

Formulating Hypotheses

Hypothesis Testing Concepts

• A hypothesis test is a formal procedure to determine whether sample data provide sufficient evidence against a null hypothesis in favor of an alternative hypothesis
  • It allows for making inferences about population parameters based on sample statistics
• The null hypothesis (H0) is a statement of no effect or no difference, suggesting that any observed differences are due to random chance or sampling error alone
  • It is assumed to be true unless there is strong evidence against it
• The alternative hypothesis (Ha or H1) is a statement that contradicts the null hypothesis, suggesting that there is a real effect or difference in the population
  • It is the hypothesis the researcher wants to support based on the data

Formulating Hypotheses

• Hypotheses should be mutually exclusive and exhaustive, covering all possible outcomes
  • They are typically stated in terms of population parameters, such as means, proportions, or correlations
• One-tailed hypotheses specify the direction of the effect or difference (e.g., $μ > 0$ or $μ < 0$)
• Two-tailed hypotheses are non-directional and only state that there is an effect or difference (e.g., $μ ≠ 0$)
• The choice between one-tailed and two-tailed tests depends on the research question and prior knowledge
  • One-tailed tests provide more power but are only appropriate when there is a clear directional hypothesis based on theory or previous research

Hypothesis Testing in R

Conducting Tests in R

• R provides various functions for conducting hypothesis tests, such as

  t.test()

  ,

  [prop.test](https://www.fiveableKeyTerm:prop.test)()

  ,

  [cor.test](https://www.fiveableKeyTerm:cor.test)()

  , and

  [chisq.test](https://www.fiveableKeyTerm:chisq.test)()

  , depending on the type of data and the specific test required
• To conduct a t-test for comparing two independent groups, use the

  t.test()

  function with the formula notation (

  response ~ group

  ) or provide the response variable and the grouping factor as separate arguments (

  t.test(response, group)

  )
  • Specify the alternative hypothesis (

    alternative = "two.sided"

    ,

    "greater"

    , or

    "less"

    ) and whether to assume equal variances (

    var.equal = TRUE

    or

    FALSE

    )
• For one-sample tests or paired tests, use the

  t.test()

  function with the response variable and

  mu

  (hypothesized mean),

  paired = TRUE

  , or formula notation (

  response ~ 1

  )

Additional Tests in R

• To test the equality of proportions, use the

  prop.test()

  function with the counts and total observations for each group
  • For one-sample tests, provide the count and total observations along with the hypothesized proportion (

    p

    )
• To test the significance of a correlation coefficient, use the

  cor.test()

  function with the two variables and specify the alternative hypothesis and correlation method (e.g.,

  "pearson"

  ,

  "spearman"

  , or

  "kendall"

  )
• For chi-square tests of independence or goodness-of-fit, use the

  chisq.test()

  function with a contingency table or a vector of observed frequencies and expected probabilities
• Set the confidence level for the test using the

  conf.level

  argument (default is 0.95) and obtain the p-value from the output to make decisions based on the significance level (alpha)

Interpreting Hypothesis Test Results

P-Values and Significance

• The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the observed one, assuming the null hypothesis is true
  • It measures the strength of evidence against the null hypothesis
• The significance level (alpha) is the threshold for making decisions in hypothesis testing, typically set at 0.05
  • If the p-value is less than alpha, the null hypothesis is rejected in favor of the alternative hypothesis, suggesting a statistically significant result
• Failing to reject the null hypothesis does not prove it to be true; it only suggests that there is not enough evidence to support the alternative hypothesis
  • Absence of evidence is not evidence of absence

Confidence Intervals and Errors

• The confidence interval provides a range of plausible values for the population parameter with a certain level of confidence
  • It is an alternative way to express the results of a hypothesis test and provides information about the precision and uncertainty of the estimate
• Type I error (false positive) occurs when the null hypothesis is rejected when it is actually true
• Type II error (false negative) occurs when the null hypothesis is not rejected when it is actually false
  • The significance level (alpha) controls the probability of making a Type I error

Limitations and Considerations

• Statistical significance does not necessarily imply practical or clinical significance
  • The effect size and contextual factors should be considered when interpreting the results and making decisions based on hypothesis tests
• Hypothesis testing has limitations, such as the dependence on sample size, the arbitrary nature of the significance level, and the potential for misinterpretation
  • It should be used in conjunction with other statistical methods and subject matter knowledge to make informed inferences