To determine the fraction of a Gaussian population within given intervals, we can refer to the standard normal distribution table (Z-table). This table provides the area under the curve, which represents the probability of a value falling within a specific range in a standard normal distribution.
Intervals Explained:
- a: The area under the curve from negative infinity to value ‘a’.
- b’s width typically represents standard deviations from the mean.
- 2: This likely indicates the range of two standard deviations.
- c to: Refers to the interval starting from ‘c’ to infinity.
- d to: Represents the interval from ‘d’ to infinity.
- 0: Indicates the mean value of the Gaussian distribution.
- 4: This might represent four standard deviations, indicating how far out from the mean.
- e to: Suggests an upper limit interval.
To find the fractions corresponding to each of these intervals, one would typically look up the Z-scores for each endpoint in the Z-table. The Z-score is calculated as:
Z = (X - μ) / σ
Where ‘X’ is the value from the population, ‘μ’ is the mean, and ‘σ’ is the standard deviation. After finding the Z-score, you can consult the Z-table to get the corresponding area (or probability) for that Z-score.
For example, for 2 standard deviations (which corresponds to approximately 95% of the data in a normal distribution), you’d look up the Z-score of +2 and -2. Typically, this indicates that about 95% of the values lie between -2 and +2 standard deviations from the mean.
Similarly, for values of ‘c to + d’ and ‘0 to 4’, you would compute the relevant Z-scores and find the probabilities from the standard normal distribution table.
In summary, the fractions of the Gaussian population within the specified intervals can be extracted via their corresponding Z-scores from the standard normal distribution table, revealing the probabilities associated with each interval.