Z is a standard normal variable. Determine the probability: The probability that z is between 1.18 and 1.97 P(1.18 < z < 1.97) = Round to four decimal places.
The Correct Answer and Explanation is:
Correct Answer:
P(1.18 < z < 1.97) = 0.1291
Explanation:
In statistics, a standard normal variable (Z) refers to a random variable that follows the standard normal distribution — a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. This distribution is symmetric and bell-shaped, and the probabilities associated with it can be found using standard normal distribution (Z) tables or statistical software.
We are asked to find the probability that the Z value lies between 1.18 and 1.97, which is denoted as:P(1.18<Z<1.97)P(1.18 < Z < 1.97)P(1.18<Z<1.97)
To solve this, we use the cumulative distribution function (CDF) of the standard normal distribution, denoted as Φ(z). The cumulative probability up to a certain z-value gives the area under the curve to the left of that z-value.
We calculate this probability by subtracting the cumulative probability up to 1.18 from the cumulative probability up to 1.97:P(1.18<Z<1.97)=P(Z<1.97)−P(Z<1.18)P(1.18 < Z < 1.97) = P(Z < 1.97) – P(Z < 1.18)P(1.18<Z<1.97)=P(Z<1.97)−P(Z<1.18)
Using a standard normal table or calculator:
- Φ(1.97) ≈ 0.9756
- Φ(1.18) ≈ 0.8465
P(1.18<Z<1.97)=0.9756−0.8465=0.1291P(1.18 < Z < 1.97) = 0.9756 – 0.8465 = 0.1291P(1.18<Z<1.97)=0.9756−0.8465=0.1291
This means there is a 12.91% chance that a randomly selected value from the standard normal distribution will fall between 1.18 and 1.97.
The area under the normal curve between two z-scores represents the probability of the variable falling within that interval. Because the normal distribution is continuous and symmetric, this process applies universally for any z-values using the same subtraction method.
