Statistics Lecture Notes

Overview of Section 6

• Section 6 contains complex formulas and mathematics, particularly focused on statistics and data analysis.

• The importance of understanding how to work with formulas in Excel to calculate necessary statistics was highlighted.

Transition from Simulations to Theoretical Distributions

• Transitioning from bootstrap and randomization distributions in Sections 3-5 to using theoretical normal distributions for confidence intervals and hypothesis tests.

• This shift simplifies calculations when the underlying statistics are normally distributed.

Standard Error and Its Importance

• Definition: Standard error quantifies how far a sample statistic (like a sample mean) is expected to be from the population parameter it's estimating.

• Larger sample sizes lead to smaller standard errors—this is crucial for accurate estimations.

Calculating Standard Error

• Formula: Standard error can be computed with the formula:
  SE = \sqrt{\frac{p(1-p)}{n}}

• Where:

  • p is the population proportion.

  • n is the sample size.

• Example given where for a population proportion of 0.651 and sample size of 100, calculations showed that:
  SE = \sqrt{\frac{0.651 \times (1-0.651)}{100}}

• This leads to a calculated standard error of approximately 0.048 .

Reliability of Sample Statistics

• Sample statistics vary based on specific samples taken, but tend to cluster around the true population parameters as the number of samples increases.

• Key Point: Increased sample size reduces variability, improving precision in estimating population parameters.

Normal Distribution and Sample Size

• For the sampling distribution to be normally distributed:

  • A sufficient sample size is required.

  • The sample proportion should not be too close to the extremes (0 or 1) which can distort the distribution.

• Check to ensure:
  n \times p \geq 10 and n \times (1-p) \geq 10

Using Confidence Intervals

• Confidence intervals utilize the standard error to estimate possible ranges for population parameters:

  • Formula discussed emphasizes the use of sample proportion \hat{p} in estimation contexts.

• Example with a calculated confidence interval using sample size of 1000, where 320 favored sin taxes:

  • Calculate \hat{p} = \frac{320}{1000}

  • Check validity using 10-count checks for both favorable and unfavorable reactions.

• Margin of Error: Calculated using the standard error and z-values derived from confidence levels (1.96 for 95% confidence).

Conclusion and Study Tips

• Understanding of statistical concepts is crucial, including recognition of how formulas apply in practical settings.

• Recommended to practice past exam papers and simulation tests to become familiar with calculations and concepts in statistics.

• Use Excel as a tool to assist in calculations where allowed, validating proficiency in theory as applied in practical situations.