Week 7 Introduction to Statistical Inference

1. Introduction to Statistical Inference

  • Course: Biostatistics and Epidemiology (BIOE 211)

  • Institution: Our Lady of Fatima University College of Medical Laboratory Science

2. Understanding Statistical Inference

  • Definition: Process of generalizing conclusions about a target population based on results from a sample.

  • Method: Utilizes sample statistics to infer unknown parameters of the population.

3. Key Components of Statistical Inference

3.1 Parameters and Statistics

  • Parameters: Measures computed from the entire population; classified as constants.

  • Statistics: Measures computed from samples; considered variable.

3.2 Sampling Variation

  • Variability in statistics due to random selection of samples.

  • Recognizes that statistics can differ from one sample to another.

4. Statistical Notation Summary

Measure

Parameter

Statistic

Mean

𝝻

π‘₯Μ…

Variance

𝜎²

sΒ²

Standard Deviation

𝜎

s

Proportion

P

p

5. Methods of Statistical Inference

5.1 Estimation

  • Definition: Process of utilizing sample statistics to estimate population parameters.

  • Steps:

    1. Collect data from sample respondents.

    2. Calculate summary measures.

    3. Use statistics to estimate parameter values.

5.2 Hypothesis Testing

  • Definition: Evaluating the validity of a hypothesis regarding a population based on sample data.

  • Steps:

    1. Collect sample data.

    2. Calculate relevant statistics.

    3. Apply statistical tests.

    4. Make a decision regarding the hypothesis.

6. Properties of the Sampling Distribution of π‘₯Μ…

  1. Mean of the sampling distribution π‘₯Μ… (πœ‡ xΜ…) equals the population mean (πœ‡).

  2. Standard deviation of π‘₯Μ… (𝜎 xΜ…) equals population SD (𝜎) divided by the square root of n.

  3. The sampling distribution of π‘₯Μ… approximates a normal distribution:

    • Normally distributed samples lead to normally distributed π‘₯Μ… distribution.

    • Non-normally distributed populations approximate normality with a sufficiently large n (Central Limit Theorem).

7. Estimation of the Population Mean

7.1 Point Estimate

  • Definition: A single number representing the parameter estimate.

  • Application: Best estimate of population πœ‡ is the mean of the sample; requires a random sample.

  • Drawback: Subject to sampling error.

7.2 Interval Estimate

  • Definition: Estimate of the parameter within a specified range of values.

  • Procedure: Add and subtract an amount from π‘₯Μ… to create an interval that likely contains πœ‡.

  • Considerations: Select desired confidence level (e.g., 90%, 95%, 99%).

8. Interval Estimate Calculation

  • Using Standard Deviation:

    • Formula: π‘₯Μ… Β± Z(𝜎/√n)

  • Confidence Levels:

    • 99%: Z = 2.575

    • 95%: Z = 1.96

    • 90%: Z = 1.645

9. Example of Interval Estimate

  • Objective: Estimate the mean weight of school-aged children.

  • Sample Data: 70, 74, 75, 78, 74, 64, 70, 78, 81, 73, 82, 75, 71, 79, 73, 79, 85, 79, 71, 65, 70, 69, 76, 77, 66.

  • Mean (π‘₯Μ…): 74.16 lbs; Standard deviation: 5.37 lbs.

    • 95% CI: [72.05, 76.27] lbs.

10. Another Example of Interval Estimate

  • Scenario: Health services utilization affected by distance.

  • Sample Information: Mean distance travelled = 7 km; Standard deviation = 3.2 km.

  • Constructing 95% CI: (Details of calculations omitted).

11. Student's t Distribution Table

  • Displays critical values for various degrees of freedom and significance levels for one-tailed and two-tailed tests.

12. Conclusion

  • Reinforces the foundational concepts of statistical inference, the significance of estimations, and hypothesis testing in biostatistics.