What is the probability that a random selected observation from the standard normal distribution is greater than -1.04
The correct answer and explanation is:
To find the probability that a randomly selected observation from the standard normal distribution is greater than -1.04, we first need to understand the concept of the standard normal distribution.
A standard normal distribution has a mean of 0 and a standard deviation of 1. The area under the curve represents the total probability, which is 1. The value -1.04 is a z-score, which indicates how many standard deviations an observation is from the mean.
To calculate the probability that a random observation is greater than -1.04, we first need to find the cumulative probability up to -1.04, which corresponds to the area to the left of -1.04. This cumulative probability can be found using a z-table or a statistical calculator.
From the z-table, the cumulative probability at -1.04 is approximately 0.1492. This means that about 14.92% of the observations fall below -1.04. Since the total area under the curve is 1, the probability of selecting a value greater than -1.04 is the complement of the cumulative probability.
Thus, the probability is: P(X>−1.04)=1−P(X≤−1.04)=1−0.1492=0.8508P(X > -1.04) = 1 – P(X \leq -1.04) = 1 – 0.1492 = 0.8508
Therefore, the probability that a randomly selected observation is greater than -1.04 is approximately 0.8508, or 85.08%.
This result makes sense because -1.04 is relatively close to the left tail of the standard normal distribution, and since the distribution is symmetric, the majority of the data (about 85%) lies to the right of this value.