If Z represents the standard normal variate, then the probability P (−2 ≤Z ≤1) would be approximately equal to:

Correct option is A

Introduction:

• In environmental studies, the Standard Normal Variate ($Z$) is a dimensionless quantity that represents the number of standard deviations a data point is from the mean ($$\mu$$​) of a distribution where $$\mu = 0 and \sigma = 1$$​
• This transformation ($$Z = \frac{x - \mu}{\sigma}$$​ is essential for comparing different environmental datasets, such as comparing the air quality index (AQI) of two different cities or determining the probability of a pollutant concentration falling within a specific risk range.
• The area under the normal curve represents the total probability (equal to 1), and specific intervals provide the likelihood of an event occurring.

Information Booster:

• Step-by-Step Calculation:
• We need to find the area under the curve between $$Z = -2 and Z = 1$$​. This can be split into two parts from the mean (Z = 0):
• ​$$P(-2 \le Z \le 1) = P(-2 \le Z \le 0) + P(0 \le Z \le 1)$$​​

1. Probability from $$Z = -2 to 0$$​: Due to the symmetry of the normal curve, this is equal to $$P(0 \le Z \le 2)$$​.

   From standard Z-tables or the Empirical Rule, the area within $2$ standard deviations is $$\approx 95.4\%$$​ Half of that area (from 0 to 2) is approximately 0.4772.

2. Probability from $$Z = 0 to 1$$​: From the Empirical Rule, the area within 1 standard deviation is $$\approx 68.2\%$$​ Half of that area (from 0 to 1) is approximately 0.3413.

3. Total Area:

   ​$$0.4772 + 0.3413 = \mathbf{0.8185}$$​​

   When rounded to two decimal places, this equals 0.82.

• Key Rule to Remember: $$P(Z \le 1) \approx 0.8413 and P(Z \le -2) \approx 0.0228$$​​
• $$P(-2 \le Z \le 1) = 0.8413 - 0.0228 = 0.8185 \approx 0.82.$$

Additional knowledge:

• This value is significantly higher than the actual area.
• A student might arrive at this if they mistakenly used the area for $$P(-2 \le Z \le 2), which is 0.95$$​, and then subtracted a smaller value incorrectly, or if they looked at the wrong row/column in a cumulative Z-table (e.g., confused $$Z=1.3 with Z=1.0$$​​
• This corresponds approximately to the area for $$P(-1.1 \le Z \le 1.1)$$​. It is too small because it fails to capture the significant probability mass between $$Z = -1 and Z = -2$$​ (which is about 13.6%).
• This value is often confused with the probability $$P(Z \le 0.7) or P(-1.2 \le Z \le 1.2)$$.