Question

question 3 2 pts match the probability to the normal distribution of a z - score transformation. between 0 and +1 standard deviations above the mean choose between -1 standard deviation below the mean and +1 standard deviations above the mean choose more than +1 standard deviations choose between 0 and -2 standard deviations choose below the mean choose

Explanation:

Step1: Recall normal - distribution properties

The total area under the normal - distribution curve is 1. The normal distribution is symmetric about the mean ($z = 0$).

Step2: Analyze "Between 0 and +1 standard deviations above the mean"

The area between $z = 0$ and $z=1$ can be found using the standard normal table. The area to the left of $z = 0$ is 0.5 and the area to the left of $z = 1$ is approximately 0.8413. So the area between $z = 0$ and $z = 1$ is $0.8413−0.5=0.3413$.

Step3: Analyze "Between - 1 standard deviation below the mean and +1 standard deviations above the mean"

The area to the left of $z=-1$ is approximately 0.1587 and the area to the left of $z = 1$ is approximately 0.8413. So the area between $z=-1$ and $z = 1$ is $0.8413−0.1587 = 0.6826$.

Step4: Analyze "More than +1 standard deviations"

The area to the left of $z = 1$ is approximately 0.8413. So the area to the right of $z = 1$ is $1 - 0.8413=0.1587$.

Step5: Analyze "Between 0 and - 2 standard deviations"

The area to the left of $z = 0$ is 0.5 and the area to the left of $z=-2$ is approximately 0.0228. So the area between $z=-2$ and $z = 0$ is $0.5−0.0228 = 0.4772$.

Step6: Analyze "Below the mean"

Since the normal distribution is symmetric about the mean, the area below the mean ($z<0$) is 0.5.

Answer:

Between 0 and +1 standard deviations above the mean: 0.3413
Between - 1 standard deviation below the mean and +1 standard deviations above the mean: 0.6826
More than +1 standard deviations: 0.1587
Between 0 and - 2 standard deviations: 0.4772
Below the mean: 0.5