6.3 Confidence intervals and p-values

Confidence intervals and p-values are crucial tools in statistical inference. They help us understand the reliability of our estimates and the strength of evidence against null hypotheses. These concepts build on the probability foundations covered earlier in the chapter.

By learning to construct and interpret confidence intervals and p-values, you'll be better equipped to make informed decisions based on data. These tools are essential for drawing meaningful conclusions from statistical analyses across various fields of study.

Confidence Intervals for Population Parameters

Constructing Confidence Intervals

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• Confidence intervals provide a range of values likely to contain the true population parameter with a specified level of confidence (typically 95%)
• Constructing a confidence interval requires the sample mean, sample size, standard deviation (or standard error), and desired confidence level
• The general formula for a confidence interval is: $sample mean ± (critical value × standard error)$
  • The critical value is determined by the desired confidence level and sample size, and can be found using a t-distribution table or calculator
  • The standard error is the standard deviation of the sampling distribution, calculated as $sample standard deviation ÷ \sqrt{sample size}$
• Confidence intervals can be one-sided (upper or lower bound) or two-sided (both upper and lower bounds)
• The width of the confidence interval is influenced by sample size, data variability, and desired confidence level
  • Larger sample sizes lead to narrower intervals
  • Less variability in the data leads to narrower intervals
  • Lower confidence levels (90% vs 95%) lead to narrower intervals

Properties and Limitations of Confidence Intervals

• Confidence intervals provide a range of plausible values for the population parameter rather than a single point estimate
• The confidence level (95%) represents the proportion of intervals that would contain the true population parameter if the sampling process were repeated many times
  • A 95% confidence interval does not mean there is a 95% probability that the true population parameter lies within the interval for a single sample
  • If the sampling process were repeated many times, 95% of the resulting intervals would contain the true population parameter
• The width of the confidence interval indicates the precision of the estimate
  • Narrower intervals suggest more precise estimates
  • Wider intervals suggest less precision
• Confidence intervals can determine if there is a statistically significant difference between two groups or treatments by checking for overlap
  • Non-overlapping intervals suggest a significant difference
  • Overlapping intervals do not necessarily imply a non-significant difference (further testing may be required)

Interpretation of Confidence Intervals

Understanding the Meaning of Confidence Intervals

• A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence
• The level of confidence (95%) represents the proportion of intervals that would contain the true population parameter if the sampling process were repeated many times
  • Example: If 100 different samples were taken and a 95% confidence interval was calculated for each, approximately 95 of those intervals would contain the true population parameter
• The confidence level is not the probability that the true population parameter lies within the interval for a single sample
  • Example: A 95% confidence interval of (0.2, 0.4) does not mean there is a 95% probability that the true population parameter is between 0.2 and 0.4 for that specific sample
• Confidence intervals provide a range of plausible values for the population parameter, accounting for sampling variability

Implications and Applications of Confidence Intervals

• The width of the confidence interval indicates the precision of the estimate
  • Narrower intervals suggest more precise estimates and less uncertainty
  • Wider intervals suggest less precision and more uncertainty
  • Example: A 95% confidence interval of (0.2, 0.4) is more precise than (0.1, 0.5)
• Confidence intervals can be used to determine if there is a statistically significant difference between two groups or treatments
  • Non-overlapping intervals suggest a significant difference
  • Overlapping intervals do not necessarily imply a non-significant difference (further testing may be required)
  • Example: If the 95% confidence interval for the mean height of men is (170 cm, 180 cm) and for women is (160 cm, 170 cm), there is evidence of a significant difference in height between the two groups
• Confidence intervals can be used to estimate population parameters in various fields (medicine, social sciences, business, etc.)
  • Example: A 95% confidence interval for the proportion of voters supporting a candidate can inform campaign strategies
  • Example: A 95% confidence interval for the mean effectiveness of a new drug can guide treatment decisions

Concept and Interpretation of p-values

Understanding p-values

• A p-value is the probability of obtaining a result as extreme as, or more extreme than, the observed result, assuming the null hypothesis is true
• The null hypothesis (H₀) typically represents no effect or no difference, while the alternative hypothesis (H₁) represents the presence of an effect or difference
• P-values are calculated using the sampling distribution of the test statistic under the null hypothesis
  • Example: In a t-test comparing the means of two groups, the p-value is calculated using the t-distribution
• A smaller p-value indicates stronger evidence against the null hypothesis and in favor of the alternative hypothesis
  • Example: A p-value of 0.01 provides stronger evidence against the null hypothesis than a p-value of 0.1
• P-values do not provide information about the size or importance of an effect, only the likelihood of observing the data if the null hypothesis is true

Interpreting p-values

• The interpretation of a p-value depends on the context of the research question and the chosen significance level (α)
• A p-value less than or equal to the significance level (p ≤ α) is considered statistically significant, and the null hypothesis is rejected in favor of the alternative hypothesis
  • Example: If α = 0.05 and p = 0.02, the result is statistically significant, and the null hypothesis is rejected
• A p-value greater than the significance level (p > α) is not considered statistically significant, and there is insufficient evidence to reject the null hypothesis
  • Example: If α = 0.05 and p = 0.1, the result is not statistically significant, and the null hypothesis is not rejected
• P-values should be interpreted in the context of the study design, sample size, and practical significance
  • Example: A statistically significant result with a small effect size may not be practically meaningful
• P-values are affected by sample size; larger sample sizes can lead to smaller p-values even for small effects
  • Example: A study with a large sample size (n = 1000) may find a statistically significant result for a small difference, while the same difference in a smaller sample (n = 50) may not be significant

Statistical Significance vs p-values

Determining Statistical Significance

• Statistical significance is determined by comparing the p-value to a pre-specified significance level (α), often set at 0.05
• If p ≤ α, the result is considered statistically significant, and the null hypothesis is rejected in favor of the alternative hypothesis
  • Example: If α = 0.05 and p = 0.02, the result is statistically significant, and the null hypothesis is rejected
• If p > α, the result is not considered statistically significant, and there is insufficient evidence to reject the null hypothesis
  • Example: If α = 0.05 and p = 0.1, the result is not statistically significant, and the null hypothesis is not rejected
• The choice of significance level (α) is somewhat arbitrary and depends on the field of study and the consequences of making a Type I error (rejecting a true null hypothesis) or a Type II error (failing to reject a false null hypothesis)
  • Example: In medical research, a lower significance level (α = 0.01) may be used to reduce the risk of Type I errors, as the consequences of falsely concluding a treatment is effective can be severe

Limitations and Considerations

• Statistical significance does not necessarily imply practical or clinical significance
  • Example: A study may find a statistically significant difference in blood pressure between two groups, but the difference may be too small to have any meaningful impact on health outcomes
• The size and importance of the effect should be considered alongside the p-value when interpreting results
  • Example: A study with a large sample size may find a statistically significant result for a small effect size, while a study with a smaller sample size may not find a significant result for a larger effect size
• Multiple comparisons and testing of multiple hypotheses can inflate the Type I error rate
  • Example: If 20 hypotheses are tested at α = 0.05, the probability of making at least one Type I error is 1 - (1 - 0.05)^20 ≈ 0.64
• Adjustments to the significance level, such as the Bonferroni correction, may be necessary to maintain the desired overall significance level when conducting multiple tests
  • Example: If testing 5 hypotheses, the Bonferroni-corrected significance level would be α/5 = 0.01 for each individual test to maintain an overall significance level of 0.05
• P-values should be reported alongside effect sizes, confidence intervals, and other relevant statistics to provide a more comprehensive understanding of the results