Inference Using T-Distributions

Objectives

This chapter introduces confidence intervals and hypothesis tests based on t-distributions, which are used when population standard deviations are unknown. You will learn to:

• Apply the Central Limit Theorem (CLT) to inference problems.
  
• Construct and interpret confidence intervals using the t-distribution.
  
• Perform hypothesis tests and compute t-statistics and p-values.
  
• Link confidence intervals to hypothesis test results.
  
• Distinguish statistical from practical significance.
  
• Use R to calculate critical values and p-values.

Central Limit Theorem (CLT)

If we draw a large enough sample from a normal (or approximately normal) distribution, the sampling distribution of the sample mean will also be normal. The mean of this sampling distribution of the sample means equals the population mean from which those sample means were taken. This allows us to use the sampling mean, \(\bar{X}\), to make inferences about the population mean, \(\mu\).

Interpretation of a Confidence Interval

If we took many random samples from the same population and constructed a confidence interval for the mean from each one, then approximately 95% of those intervals would contain the true mean. For a single interval, we interpret this as: we are 95% confident that the interval contains the true mean.

Confidence Intervals

• Measure how large or small a parameter value could plausibly be.
  
• Random sampling leads to slightly different sample means (estimates of the population mean).
  
• A confidence interval is constructed around \(\bar{X}\) with a given level of confidence that it contains \(\mu\). In other words, it contains the plausible values of \(\mu\).
• From normal distribution theory:
  • Approximately 68% of values lie within \(1\sigma\) of \(\mu\)
  • Approximately 95% within \(2\sigma\)
    
  • Approximately 99.7% within \(3\sigma\)
    
• In practice, we usually estimate both the population mean (\(\mu\)) and standard deviation (\(\sigma\)) using sample values. We estimate \(\sigma\) using the sample standard deviation, \(s\).
  
• The t-distribution has fatter tails than the normal distribution to account for additional uncertainty from estimating \(\sigma\) with the sample standard deviation \(s\).

The formula for a confidence interval is: \[ CI = \bar{X} \pm t_{\text{crit}} \left( \frac{s}{\sqrt{n}} \right) \] where \(t_{\text{crit}}\) is based on the desired confidence level and degrees of freedom.

Example confidence interval statement: “With \((1-\alpha) \times 100\%\) confidence, the true mean \(\mu\) is between [lower bound] and [upper bound], based on a sample of size \(n\) from this population.”

Hypothesis Testing

Steps for Testing Significance

1. Choose a significance level (\(\alpha\)):
   • \(\alpha\) is the probability of a Type I error (rejecting \(H_0\) when it is true).
     
   • Smaller values of \(\alpha\) reduce the chance of a Type I error.
2. Specify the test direction and check assumptions:
   • Choose a one-sided or two-sided test.
     
   • Confirm that assumptions are met (e.g., approximate normality, random sampling).
3. State the hypotheses (i.e., assumptions about the population parameter or parameters):
   • \(H_0\): Null hypothesis (no effect or difference)
     
   • \(H_a\): Alternative hypothesis (effect or difference exists)
4. Follow these six steps:
   1. Identify \(H_0\) and \(H_a\).
      
   2. Find the critical value(s) and draw and shade the rejection region.
      
   3. Calculate the test statistic (i.e., the evidence).
      
   4. Compute the p-value.
      
   5. Compare the p-value to \(\alpha\) and decide whether to reject or fail to reject \(H_0\).
      
   6. Write a conclusion in non-technical terms, including the p-value, confidence interval, and scope.

Test Statistic

A test statistic is a sample estimate divided by its variability (standard error): \[ t = \frac{\bar{X} - \mu_0}{\frac{s}{\sqrt{n}}} \]

• For a one-sided test, the sign of the numerator matters. Always subtract \(\mu_0\) from the sample mean \(\bar{X}\).

p-Value

• Calculated assuming \(H_0\) is true.
  
• A small p-value suggests the observed difference is unlikely due to random variation alone.
  
• The p-value is the area in one or both tails beyond the observed t-statistic, depending on the form of \(H_a\).
  
• A p-value is a probability - not a binary cutoff. Report the value and let the reader decide.

Linking Hypothesis Tests and Confidence Intervals

• A \((1 - \alpha) \times 100\%\) confidence interval corresponds to a two-sided hypothesis test at significance level \(\alpha\).
  
• If the null value is outside the interval, we reject \(H_0\) at level \(\alpha\).
• Confidence intervals provide an alternative way to evaluate hypothesis test results.

Statistical Significance vs. Practical Significance

• Statistical significance means the result is unlikely due to random sampling error.
  
• Practical significance considers whether the effect is meaningful in context.
  
• p-values are affected by sample size—large samples can make small effects statistically significant.
  
• Effect size (e.g., Cohen’s \(d\)) helps assess practical importance.

In R

• Critical value: qt(probability, df)
  • For a two-sided test, use \(\alpha/2\) and \(1 - \alpha/2\).
    
  • For a one-sided test, use \(\alpha\) or \(1 - \alpha\).
• p-value: pt(t_statistic, df, lower.tail = TRUE/FALSE)
  • Multiply by 2 for a two-sided p-value.

Complete Analysis Format

1. State the problem clearly.
   
2. Check that the assumptions are satisfied.
   
3. Perform the appropriate test (using the six steps).
   
4. Write a conclusion in plain language, including:
   • The p-value
     
   • The confidence interval
     
   • The scope of inference