find the area and the normal probability under the normal curve for the interval 0 < z < 2+answer

To find the area under the normal curve for the interval 0 < z < 2, we need to use the standard normal distribution table or a calculator with a normal distribution function.

The standard normal distribution table gives the probability that a standard normal random variable Z is less than or equal to a given value z.

Step 1: Identify the z-score In this case, the z-score is 2.

Step 2: Look up the z-score in the standard normal distribution table The value corresponding to z=2 in the standard normal distribution table is approximately 0.9772. This means that the probability that Z is less than or equal to 2 is 0.9772.

Step 3: Subtract the probability at 0 Since we are looking for the probability for the interval 0 < z < 2, we need to subtract the probability at z=0 from the probability at z=2. The value corresponding to z=0 in the standard normal distribution table is 0.5.

So, the area under the normal curve for the interval 0 < z < 2 is approximately 0.9772 - 0.5 = 0.4772.

The normal probability under the normal curve for the interval 0 < z < 2 is also 0.4772, because the total area under a probability density function (including a normal curve) represents the total probability, which is 1. So, the area under the curve corresponds to the probability for that interval.