lecture recording on 06 March 2025 at 14.23.17 PM

Introduction to Confidence Intervals and Estimations

• Understand how to calculate confidence intervals using different scores and data sets.

Z Scores and Standard Errors

• When given a percent (e.g., 95%), convert it to the appropriate z-score.

  • For 95%, the z-score is 1.96.

• To calculate the standard error (SE):

  • Use the formula: SE = sigma / sqrt(n) where sigma = standard deviation and n = sample size.

  • In one example: SE = 9 / sqrt(42).

Confidence Interval Formula

• The confidence interval (CI) uses:

  • CI = x̄ ± (z-score × SE)

  • For example: x̄ = 86, z-score = 1.96, use SE calculated previously.

Switching Between Z and T Scores

• When population standard deviation (sigma) is unknown:

  • Use the t-score instead of the z-score.

  • Degrees of freedom for t-score: n - 1.

• Ensure to estimate population mean using the sample standard deviation.

Importance of Data Sets

• For calculations involving sample data:

  • Download necessary data sets (e.g., for parts of statistical problems).

  • Use Excel or other tools to perform calculations.

Estimating Population Proportions

• For categorical data, we estimate proportions (p) instead of means:

  • Example: Preference between vanilla and chocolate ice cream.

  • Important to restrict categories to two for accurate calculation.

Calculating Proportions

• Formula for sample proportion:

  • p = x / n where x is the number of successes and n is the total sample size.

  • Calculate one minus p for proportion of non-successes.

• Calculate standard error for proportion:

  • SE = sqrt(p(1-p)/n).

Critical Values and Margin of Error

• For confidence intervals:

  • Critical z-score for given confidence level (e.g., 95% is 1.96).

  • Margin of error (ME) calculated as: ME = z-score × SE.

  • Confidence limits derived from p ± ME.

Example Calculation

• Given a sample of 300 where 99 prefer vanilla:

  • p = 99 / 300 = 0.33.

  • 1 - p = 0.67.

  • Calculate SE and ME, leading to confidence interval ranging between calculated values (e.g., approximately 27.7% to 38.3%).

Small Sample Size Considerations

• Small sample sizes lead to larger margins of error.

  • Suggestion: To decrease margin of error, increase the sample size.

• Remind to keep rounding to a minimum until the final answer in calculations to avoid errors.

Communicating Results

• Confidence intervals indicate a range of where the population parameter is likely to lie (e.g., 95% confident that actual proportion of preference is within calculated range).

• It is crucial not to overestimate certainty and always clarify confidence levels used in reporting findings.