The probability P(-1.96 ≤ Z ≤ 1.96) is approximately 0.95. (rounded to 2 decimal places)
Finding the probability P(-1.96 ≤ Z ≤ 1.96) in the standard normal distribution:
Recognize the problem as calculating area under the curve
We are interested in the probability of a standard normal variable Z falling within the interval -1.96 to 1.96. This can be visualized as the shaded area under the standard normal curve between these two z-scores:
Image of standard normal distribution curve with the shaded area between 1.96 and 1.96Opens in a new window
standard normal distribution curve with the shaded area between 1.96 and 1.96
Use the standard normal cumulative distribution function (CDF)
The CDF of the standard normal distribution, denoted by Φ(z), gives the probability that a standard normal variable will be less than or equal to a certain value z.
Therefore, to find the probability P(-1.96 ≤ Z ≤ 1.96), we can calculate the area between -1.96 and 1.96 using the CDF:
P(-1.96 ≤ Z ≤ 1.96) = Φ(1.96) - Φ(-1.96)
Look up Φ(1.96) and Φ(-1.96) in a standard normal table or use a calculator/software
Φ(1.96) is approximately 0.9750.
Φ(-1.96) is approximately 0.0250.
Calculate the final probability
P(-1.96 ≤ Z ≤ 1.96) = 0.9750 - 0.0250 = 0.95
The probability P(-1.96 ≤ Z ≤ 1.96) is approximately 0.95. (rounded to 2 decimal places)