Stats 4/14 notes

Point Estimation and Interval Estimation

• Point Estimation: Refers to the process of providing a single value as an estimate of an unknown population parameter.
  • Examples include:
  • Sample Mean ($ar{x}$): The average of sample data.
  • Sample Proportion ($rac{x}{n}$): The ratio of the number of successes to the total number of trials.
• Interval Estimation: Used when point estimates do not exactly represent the population parameters, thus we create intervals (confidence intervals) within which we believe the parameters lie.
  • Margin of Error: Represents the extent of uncertainty in the estimation, calculated based on confidence levels.

Sample Statistics vs Population Statistics

• Sample Variance ($s^2$): Measures the spread of sample data points around the sample mean.
• Sample Standard Deviation ($s$): The square root of the sample variance, providing a measure of dispersion.
• Population Variance ($ au^2$): Variance of a whole population, often unknown and inferred from sample statistics.
• Population Standard Deviation ($ au$): The square root of the population variance, reflecting how data points in a population are distributed.
• Common terms include:
  • Sample Mean ($ar{x}$)
  • Population Mean ($ ext{μ}$)
  • Sample Proportion ($rac{x}{n}$)
  • Population Proportion ($rac{X}{N}$)

Chi-Square Distribution

• Used for variance estimates when dealing with normally distributed populations.
• Degrees of Freedom (df): Calculated as $n - 1$ where $n$ is the sample size. For example, with a sample size of 10, df = 10 - 1 = 9.

Confidence Intervals

• To construct a confidence interval, we utilize:
  • Critical Values: From distribution tables based on the desired confidence level (e.g., 90%, 95%). For a 90% confidence interval, 1 - rac{ ext{alpha}}{2} = 0.95 and alpha is set at 0.10.
  • Example for 90% Confidence Interval
  • If the sample mean is $M$ and the margin of error is $E$, the interval is given by: (M - E, M + E).
• Example calculations include:
  • Determining critical values for $ ext{chi-square}$ distribution to find the interval estimate.

Sample Problems and Calculations

• To find a sample variance or standard deviation:
  • Given Standard Deviation ($1.7$): If $N = 10$, compute the square of standard deviation for variance calculation.
• Example: Find Variance for Salary Data:
  • Average salary of web designers = $5300/month$ and standard deviation given. Calculate sample variance and apply confidence levels to find intervals accordingly.

Critical Values Exploration

• For critical values corresponding to degrees of freedom calculated, refer to a chi-square table for values at $ ext{df} = 29$ and locate left/right critical values based on the desired confidence level.

Future Chapters and Continued Learning

• Aim to complete all calculations and practice with examples; plan to move onto chapter 8 in the following sessions, ensuring a comprehensive understanding of statistical inference, confidence intervals, and hypothesis testing.