I am working on a Normal Distribution problem and am a bit stuck. As the image shows below these are the probabilities associated with values corresponding to their distance from the mean (one standard deviation, two standard deviations, etc.).
Say that we are given (for simplicity) that the mean $\mu$ = 0 as is shown in the picture and that the standard deviation $\sigma$ = 1.
We are then asked which value is larger:
The probability that X falls between $-{1\over2}$ and ${1\over2}$ or two other options:
X falls in between 1 and 2
X is greater than 1
What I have done to solve this is the following:
We know for sure that the probability that X falls between 1 and 2 to be .14 percent. For X greater than 1 we know it to be .16 percent. We therefore need to compare these values to X between $-{1\over2}$ and ${1\over2}$. However, this is where I am stuck. We know that since more density falls closer towards the middle (mean) that the probability in-between $-{1\over2}$ and ${1\over2}$ is larger than .17 + .17 = .34. Can we conclude then from this that the probability in between $-{1\over2}$ and ${1\over2}$ is larger than both of the other two? Even though we cannot arrive at this value explicitly?