3.1-3.2

CHAPTER 3

Estimation: How Large is the Effect?

SECTION 3.1

Statistical Inference: Confidence Intervals

Overview of Topics

  • So far studies have focused on:
    • Inference
    • Hypothesis Testing
    • Questions posed: Is a certain value of a parameter plausible?
  • New goal outlined is Estimation:
    • Questions asked have shifted to: What values of the parameter are plausible?
  • Statisticians typically use a confidence interval to address these questions.

Definition of Confidence Interval

  • A Confidence Interval (CI) is defined as:
    • A range of likely values for the parameter of interest.

Example from a Poll on Medical Marijuana

  • In February of 2014, a Star Tribune Minnesota poll was conducted:
    • Survey Question: “Do you support or oppose legalizing marijuana for medical purposes in Minnesota?”
    • Sample Size: 800 adults surveyed.
    • Respondents: 408 (51%) indicated support for legalization.
  • Objective of the Poll: Estimate the proportion of adults in Minnesota supporting medical marijuana legalization.

Analysis of Poll Data

  • Defining Components:
    • Population: All Minnesotans.
    • Parameter of Interest: Long run proportion of Minnesotans supporting legalization.
    • Sample: 800 Minnesota adults surveyed.
    • Observed Statistic of Interest:
    • ext{Support} = 408 \ ext{Total Sample} = 800 \ \text{Observed proportion} = p̂ = \frac{408}{800} = 0.51
    • This also represents point estimation.

Variation in Sampling

  • If a different sample of 800 Minnesota adults was taken, the point estimate might change, reinforcing the necessity of establishing a range for possible values.

Hypothesis Testing

  • Define Hypotheses:
    • Null hypothesis (H₀): The long run proportion of Minnesotans supporting legalization is equal to a specific value, denoted as _.
    • Alternative hypothesis (H₁): The long run proportion is different from _.
  • Set up the null and alternative hypotheses in symbols based on the parameters for the group.
  • Calculation of P-value:
    • Use the Theory Based Inference Applet to compute the p-value with:
    • Select test of significance; ensure it is two-sided.
    • Input: Sample size n=800 and proportion p=0.51.
    • Determine if you would Reject (R) or Fail to Reject (FTR) H₀ at a significance level of $ ext{α} = 0.05.

P-value Example Calculations

  • Various p-values can be generated by trying different hypothesized values for the parameter:
    • Sample of parameters:
    • For π = 0.40, P-value = 0.000
    • For π = 0.60, P-value = 0.5716
  • Trends Observed:
    • As the parameter increases, p-values decrease until π < 0.47 and later begin to increase after π > 0.55.
  • Interpretation:
    • A p-value greater than the significance level implies that the null hypothesis is plausible; conversely, a lower p-value suggests it is not plausible.

Confidence Interval

  • Interpretation of CI:
    • A Confidence Interval provides a range of values where the true population parameter may fall.
    • An interval estimate calculated from a sample statistic reflects the uncertainty due to sampling variability.
    • Formula for CI:
      ext{CI} = ext{Observed Statistic} ext{ ± Margin of Error}
  • Example of Margin of Error:
    • For the example, if the observed statistic is 51%, the reported margin of error is ext{3.5%} , leading to the interval representation ( 47.5% ext{ to } 54.5% .)
  • This coincides with results under repeated sampling indicated earlier.

Significance Levels (α)

  • The relationship between Significance Level and Confidence Level:
    • ext{Significance Level} + ext{Confidence Level} = 100 ext{%}
  • Common significance levels:
    • ext{α} = 0.10 ext{ (90 ext{%} CI)} ; Reject if $P ext{-value} < 0.10$
    • ext{α} = 0.05 ext{ (95 ext{%} CI)} ; Standard practice
    • ext{α} = 0.01 ext{ (99 ext{%} CI)} ; More stringent rejection

SECTION 3.2

Confidence Intervals for Proportions

  • Process Overview:
    • Confidence intervals are established as a range of likely values for the parameter of interest.
    • Multiple samples reveal varying confidence intervals that should collectively form a reliable estimate of the true parameter proportions.

General Formula for CI

  • For general computations involving sample proportions:
    ext{CI} = ext{p} ± ext{Margin of error}
  • Margin of error is crucial to narrow or widen intervals based on the given sample variability or confidence level.
  • Standard Error (SE):
    SE = rac{{ ext{p}(1- ext{p})}}{{n}}
  • This raises the need for proper calculation of the confidence interval with appropriate margins encased to obtain relative precision in estimates.

Calculation Example

  • For a given example:
    • Observed statistic: 51% lead to computing the confidence limits with ratios of - including determining the relevant lower and upper bounds.
  • Ultimately, it visibly highlights a range with interpretation linking back to confidence levels, and offering strength of hypothesis tests analysis.

Understanding Interval Width

  • Several factors impact the confidence interval width:
    1. Confidence Level: Increased certainty broadens the range.
    2. Sample Size: A larger sample leads to reduced variability and sharper precision.
    3. Standard Error: Variability in responses influences width.
Key Insights

Understanding these impacts enables effective deployment of statistical methods in research to ensure accurate reporting and feedback in inference and testing scenarios through data analytics.