When dealing with large sample sizes, the distribution of sample means will be approximately normal regardless of the shape of the population distribution.
Standard Normal Curve or Z-distribution
Z-scores: Utilized to determine the position of a score in relation to the mean of a distribution.
The area under the standard normal curve represents probabilities.
Finding Areas under the Curve
To find the area under the curve between two z-scores, integrate or use z-tables.
Example: For a normal distribution, z-scores of -1.96 and 1.96 encompass approximately 95% of the data, reflecting a common confidence level.
Z-Scores and Confidence Intervals
Confidence Intervals: Reflect the range within which we can expect a population parameter to lie, based on sample statistics.
Critical Values: The z-score multipliers used to calculate confidence intervals.
As the confidence level increases, the width of the confidence interval increases.
Important Z-Scores for Confidence Intervals
90% Confidence Interval: Z-score multiplier around 1.645
95% Confidence Interval: Z-score multiplier around 1.96
98% Confidence Interval: Z-score multiplier around 2.33
99% Confidence Interval: Z-score multiplier around 2.576
These critical values are essential for determining the width of the confidence intervals.
Key Points
The higher the level of confidence, the wider the confidence interval.
Critical values are essential in calculating accurate confidence intervals for various confidence levels.