To determine what percentage of the data will fall between the mean and 2 standard deviations above the mean in a Standard Normal Model, let's break down the steps involved.
1. Understanding the Standard Normal Model:
The Standard Normal Distribution, or Z-distribution, is a special case of the normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1.
2. Setting the Bounds:
- The mean (μ) is 0.
- The value that is 2 standard deviations above the mean is calculated as:
[tex]\[
\text{Upper Bound} = \mu + 2\sigma = 0 + 2 \times 1 = 2
\][/tex]
- Therefore, we are interested in the interval [0, 2].
3. Converting to Z-scores:
- The Z-score for the lower bound (mean) is:
[tex]\[
Z_{\text{lower}} = \frac{0 - 0}{1} = 0.0
\][/tex]
- The Z-score for the upper bound (2 standard deviations above mean) is:
[tex]\[
Z_{\text{upper}} = \frac{2 - 0}{1} = 2.0
\][/tex]
4. Using the Cumulative Distribution Function (CDF):
The CDF of the standard normal distribution gives the probability that a value will fall to the left of a given Z-score.
- The probability of Z being less than or equal to 2 (using the CDF) is:
[tex]\[
P(Z \leq 2.0) \approx 0.9772
\][/tex]
- The probability of Z being less than or equal to 0 (using the CDF) is:
[tex]\[
P(Z \leq 0.0) = 0.5
\][/tex]
5. Calculating the Probability Between Z = 0 and Z = 2:
To find the probability that a value lies between the mean and 2 standard deviations above the mean, subtract the CDF value of the lower bound from the CDF value of the upper bound.
[tex]\[
P(0 \leq Z \leq 2) = P(Z \leq 2.0) - P(Z \leq 0.0) = 0.9772 - 0.5 = 0.4772
\][/tex]
6. Converting the Probability to Percentage:
Convert the probability to a percentage by multiplying by 100.
[tex]\[
\text{Percentage} = 0.4772 \times 100 \approx 47.72\%
\][/tex]
Therefore, approximately 47.72% of the data in a Standard Normal Distribution will fall between the mean and 2 standard deviations above the mean.
