Confidence Intervals and Hypothesis Testing in Biostatistics

Confidence Intervals

Formulas for Test 2

  • Point Estimate (PE) and Margin of Error (MOE)
    • Formula:
      \text{Point Estimate} + \text{Margin of Error} = \text{PE} + \text{MOE}
    • Lower Confidence Interval:
      \text{Lower C.I.} = \text{PE} - \text{MOE}
    • Upper Confidence Interval:
      \text{Upper C.I.} = \text{PE} + \text{MOE}

### The equations for estimating parameters

  • \text{PE} = \text{(mean or proportion)}
    • Margin of Error (MOE)
  • For proportions:
    \text{MOE} = \text{z} \times \sqrt{\frac{p(1-p)}{n}}
  • For means:
    \text{MOE} = t_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}}

Inference Using Single Sample and Two Samples

Dependent Samples

  • Confidence Interval:
    • Formula:
      \text{C.I.} = d \pm \frac{t{\alpha/2} \cdot Sd}{\sqrt{n}}
    • Where,
    • $d$ = difference of sample means
    • $S_d$ = standard deviation of differences
    • $n$ = sample size
    • Test Statistic:
      T = \frac{\bar{x} - d0}{Sd / \sqrt{n}}

Independent Samples

  • Non-Pooled is used when variances are unequal
    • Test Statistic:
    • T = \frac{\bar{x1} - \bar{x2}}{Sp \cdot \sqrt{\frac{1}{n1} + \frac{1}{n_2}}}
    • Where:
    • Sp = \sqrt{\frac{(n1 - 1)S1^2 + (n2 - 1)S2^2}{n1 + n_2 - 2}}

Bartlett’s F Test for Variances

  • Formula:
    • F = \frac{S1^2}{S2^2} where $S1^2 > S2^2$
  • Degrees of freedom
    • df{numerator} = n{1} - 1
    • df{denominator} = n{2} - 1

Quick TI cheats

  • Confidence Interval:
    • Use STAT
  • Single Sample Tests:
    • For mean: TINTERVAL for confidence interval
    • For proportion: 1-PROPZINT
  • Two Independent Samples:
    • Pooled: 2-SAMPTTEST
    • Non-Pooled: 2-SAMPTTEST

Principles of Biostatistics - Test 2 Instructions

  • Show all work for credit.
  • Using calculators:
    • Allowed for checking work; must show commands and round answers to 4 decimal places.

Hypothesis Testing Example: Comparing Average Salaries

Department A vs. Department B

  • Sample Size from A:
    • n_A = 40
  • Average Salary from A:
    • \bar{x}_A = 55000
  • Standard Deviation from A:
    • S_A = 5000
  • Sample Size from B:
    • n_B = 35
  • Average Salary from B:
    • \bar{x}_B = 53500
  • Standard Deviation from B:
    • S_B = 4800

Hypotheses

  • State the hypotheses:
    • Null hypothesis:
    • H0: \sigma{A}^2 = \sigma_{B}^2
    • Alternative hypothesis:
    • HA: \sigma{A}^2 > \sigma_{B}^2

Test Statistic

  • Calculate the test statistic using the F-test formula:
    • F = \frac{SA^2}{SB^2}

Decision and Interpretation

  • Determine p-value from the F-distribution.
  • Decision:
    • Reject $H_0$
    • Fail to reject $H_0$

Orange Yield Comparison Example: Florida vs. California

  • Sample Means from Table 1:
    • Florida Mean: \bar{x}_{FL} = 31.67
    • California Mean: \bar{x}_{CA} = 28.73
  • Hypothesis Test:
    • Test if yield in FL is more than in CA.
  • Statistical outputs:
    • p-value from Bartlett's Test:
    • p = 0.179

Hypotheses for Yield Comparison

  • State the hypotheses:
    • Null: H0: \mu{FL} \leq \mu_{CA}
    • Alternative: HA: \mu{FL} > \mu_{CA}

Interpretation of Results

  • Decision:
    • If p > 0.05, fail to reject $H_0$
    • If p \leq 0.05, reject $H_0$

Daily Step Count Analysis from Fitness Study

Study Details

  • Sample Size: 16 individuals
  • Mean Daily Step Count: 779.38 mg/dL

Data Input

  • Daily Step Count Before and After table.
  • Analyze possible shifts in average step count.

Confidence Interval Calculation

  • Construct a 95% confidence interval:
    • Compute t_{\frac{\alpha}{2}} & proposals for CI
    • Results: Calculate and round as necessary.

Evaluation of New Study Results

  • Findings suggest:
  • Confidence Interval for weight loss:
    • Reported as (0.51, 0.69)
  • Calculate:
    • Point Estimate (P.E.)
    • Margin of Error (M.O.E.)
    • \text{P.E.} = \frac{0.51 + 0.69}{2} = 0.601
    • \text{M.O.E.} = \frac{0.69 - 0.51}{2} = 0.09

Interpretation of the Results

  • Interpretation of Confidence Interval:

    • We are 90% confident that the true, unknown population mean of weight loss μ is within the interval (0.51, 0.69).
    • The results suggest a reasonable expectation about weight loss under the new program

  • Overall conclusion of tests must adhere to statistical significance criteria, coupled with clear articulation of conditions and methodology utilized during analyses.

Hypothesis Testing Example: Comparing Average Salaries

Department A vs. Department B
  • Sample Size from A:
    n_A = 40
  • Average Salary from A:
    \bar{x}_A = 55000
  • Standard Deviation from A:
    S_A = 5000
  • Sample Size from B:
    n_B = 35
  • Average Salary from B:
    \bar{x}_B = 53500
  • Standard Deviation from B:
    S_B = 4800
Hypotheses
  • State the hypotheses:
    • Null hypothesis:
      H0: \sigmaA^2 = \sigma_B^2
    • Alternative hypothesis:
      HA: \sigmaA^2 > \sigma_B^2
Test Statistic
  • Calculate the test statistic using the F-test formula:
    • Formula:
      F = \frac{SA^2}{SB^2}
    • Substitute the values:
      F = \frac{(5000)^2}{(4800)^2}
    • Calculation step:
      F = \frac{25000000}{23040000} \approx 1.087
Decision and Interpretation
  • Determine the p-value from the F-distribution based on the calculated F statistic and degrees of freedom.
  • Decision:
    • If p-value < significance level (e.g., 0.05), reject $H_0$.
    • If p-value ≥ significance level, fail to reject $H_0$.

Orange Yield Comparison Example: Florida vs. California

  • Sample Means from Table 1:
  • Florida Mean:
    \bar{x}_{FL} = 31.67
  • California Mean:
    \bar{x}_{CA} = 28.73
  • Hypothesis Test:
    • Test if yield in FL is more than in CA.
Hypotheses for Yield Comparison
  • State the hypotheses:
    • Null:
      H0: \mu{FL} \leq \mu_{CA}
    • Alternative:
      HA: \mu{FL} > \mu_{CA}
Interpretation of Results
  • Decision:
    • If p > 0.05, fail to reject $H_0$
    • If p \leq 0.05, reject $H_0$

Daily Step Count Analysis from Fitness Study

Study Details
  • Sample Size: 16 individuals
  • Mean Daily Step Count: 779.38 mg/dL
Data Input
  • Daily Step Count Before and After table.
  • Analyze possible shifts in average step count.
Confidence Interval Calculation
  • Construct a 95% confidence interval:
    • Compute t_{\frac{\alpha}{2}} & proposals for CI.
    • Results: Calculate and round as necessary.
Evaluation of New Study Results
  • Findings suggest:
    • Confidence Interval for weight loss:
    • Reported as (0.51, 0.69)
    • Calculate:
    • Point Estimate (P.E.)
    • Margin of Error (M.O.E.)
    • \text{P.E.} = \frac{0.51 + 0.69}{2} = 0.601
    • \text{M.O.E.} = \frac{0.69 - 0.51}{2} = 0.09
Interpretation of the Results
  • Interpretation of Confidence Interval:
    • We are 90% confident that the true, unknown population mean of weight loss μ is within the interval (0.51, 0.69).
    • The results suggest a reasonable expectation about weight loss under the new program.
    • Overall conclusion of tests must adhere to statistical significance criteria, coupled with clear articulation of conditions and methodology utilized during analyses.