question 3 2 pts match the probability to the normal distribution of a z - score transformation. between 0…

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: Probability between 0 and + 1 standard deviation

The area between $z = 0$ and $z=1$ can be found using the standard normal table. The area between $z = 0$ and $z = 1$ is approximately 0.3413.

Step3: Probability between - 1 and + 1 standard deviation

Since the normal distribution is symmetric, the area between $z=-1$ and $z = 1$ is $2\times0.3413=0.6826$.

Step4: Probability more than + 1 standard deviation

The area to the right of $z = 1$ is $\frac{1 - 0.6826}{2}=0.1587$.

Step5: Probability between 0 and - 2 standard deviation

The area between $z = 0$ and $z=-2$ can be found from the standard - normal table. The area between $z=-2$ and $z = 0$ is the same as the area between $z = 0$ and $z = 2$. The area between $z = 0$ and $z = 2$ is approximately 0.4772.

Step6: Probability 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 above the mean: 0.1587 Between 0 and - 2 standard deviations: 0.4772 Below the mean: 0.5