Final answer:
To calculate the probabilities for a Standard Normal Distribution, one uses a Z-table, calculator, or software to find the area under the curve. Z-scores are used to find these probabilities and are calculated based on the mean and standard deviation of the distribution.
Step-by-step explanation:
The Standard Normal Distribution is a critical concept in statistics represented by Z~N(0,1), where the mean is 0 and the standard deviation is 1. To find probabilities related to the standard normal distribution, one can use a Z-table, a calculator, or statistical software. Here's how to find the probabilities for the given values:
- P(Z < 2.51): Look up the value of 2.51 in the Z-table to find the area to the left of Z which gives us the probability.
- P(Z > -1.48): This time, look up the value of -1.48 in the Z-table. Since tables typically give the area to the left, subtract this value from 1 to find the area to the right.
- The question appears incomplete for parts c) P(-1.16), d) P(-0.94), and e) P(1.11), but the methodology remains the same as steps 1 and 2 assuming the full Z-values are provided.
A Z-score can be calculated using the formula Z = (X - μ) / σ, where μ is the mean, σ is the standard deviation, and X is the value in question. For example, if we have a mean (μ) of 5 and standard deviation (σ) of 2, a score of 11 would be 3 standard deviations above the mean, as calculated by (11 - 5) / 2 = 3. This is how we obtain the Z-score.