Jan 27

Sampling Distribution

  • The sampling distribution refers to the distribution of sample means over a range of samples drawn from a population.

  • For this case, it is given that the population mean is 40%.

Standard Deviation of the Statistic

  • The formula for standard deviation of the statistic is:

    • Standard Deviation = √(pq/n)

  • Where:

    • p = population proportion (0.4 in this example)

    • q = 1 - p (0.6 in this example)

    • n = sample size

  • In this instance, the standard deviation calculated is 15.5%, assuming n = 100.

  • If the sample size (n) is increased to 20, the standard deviation decreases.

Maximum Standard Deviation

  • The maximum standard deviation possible occurs when p = 0.5:

    • Maximum Standard Deviation = 1.5 * √(0.5 * 0.5)

  • This results in a maximum of 0.5, indicating that the standard deviation can only go down from this point.

Parameters and Descriptions

  • p stands for a certain parameter that is being described:

    • Examples: The average height of men aged 18-24 in the U.S. or the proportion of girls with blue eyes.

  • It's important to define what mean or proportion you are working with.

Assumptions for Testing

  • Important assumptions must be met prior to performing statistical tests:

    • Independence: Refers to whether individuals in a sample are independent from each other.

      • Example: A sample must be a simple random sample and should not exceed 10% of the population.

    • Sample Size: Should be sufficiently large enough to assume normality using the Central Limit Theorem.

      • For proportions, check that np ≥ 10 and nq ≥ 10.

  • For the sampling distribution to be normal, an adequate sample size is essential.

Test Statistic and P-Value

  • The test statistic often involves calculating a z-score:

    • Formula:

      • Z = (p hat - p) / √(pq/n)

  • Obtain the p-value to determine the probability of observing a sample statistic as extreme as what was observed.

    • This indicates the probability that the test statistic is greater than, less than, or not equal to a certain value.

Example: Helsinki Heart Study

  • The study assesses whether the anti-cholesterol drug Gimfimrazole reduces heart attacks. Sample included:

    • 2,000 men receiving the drug and 2,000 men receiving a placebo.

    • Probability of heart attack in participants = 4% (0.04).

  • Calculation: What is the probability of at least 75 heart attacks in the treatment group?

    • The hypothesis involves checking if the sample is from a simple random sample and ensuring independence.

    • Since it’s a proportion problem, the absence of a standard deviation indicates the need for further calculations.

Conclusion

  • As we proceed through statistical tests and theories, it’s crucial to continuously validate assumptions and recalculate as necessary to ensure accurate results.

  • Lastly, maintaining an organized approach in assessments, including set reminders and familiar formulas, can simplify the testing process.