Overview
This lecture discusses how to determine if a daily cash receipt is unusually low using the concepts of mean, standard deviation, z-score, and the empirical rule.
Graphing the Distribution
- The mean daily cash receipt is $9,200, placed at the center of the graph.
- Each tick mark on the graph represents an increase or decrease of $400, the standard deviation.
- Adding standard deviations above the mean: $9,600, $10,000, $10,400.
- Subtracting standard deviations below the mean: $8,800, $8,400, $8,000.
Analyzing the Observation
- The cash receipt on the day a new employee was hired was $8,300.
- The question is whether $8,300 is an unusually low value.
Calculating the Z-Score
- To determine if a value is unusual, calculate the z-score: ( \text{z} = \frac{X - \bar{X}}{S} ).
- Here, ( X = $8,300 ), ( \bar{X} = $9,200 ), ( S = $400 ).
- The difference ( $8,300 - $9,200 = -$900 ) shows the observation is $900 below the mean.
- The z-score is ( \frac{-900}{400} = -2.25 ).
Interpreting with the Empirical Rule
- The empirical rule states that 95% of data lies within two standard deviations of the mean.
- Values beyond two standard deviations (z < -2 or z > 2) are considered unusual.
- The z-score of -2.25 is between -2 and -3, indicating the receipt is unusually low.
Key Terms & Definitions
- Mean ((\bar{X})) — The average value of a data set.
- Standard Deviation (S) — A measure of the spread or variability of a data set.
- Z-score — The number of standard deviations an observation is from the mean, calculated by ( (X - \bar{X}) / S ).
- Empirical Rule — States that about 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean.
Action Items / Next Steps
- Practice calculating z-scores for other values.
- Review and memorize the empirical rule.