Example Problems

This section illustrates example problems involving the Normal Distribution, demonstrating the calculations of probabilities using Z-scores.

In this section, we cover two example problems related to the Normal Distribution, detailing the processes for calculating probabilities. These examples illustrate conversions to Z-scores and the use of Z-tables.

Example Problems

This section explores two practical examples to solidify the understanding of the Normal Distribution and associated calculations.

Understanding the Problems

Both examples focus on the fundamental steps involved in solving problems related to Normal Distribution: identifying the mean and standard deviation, calculating Z-scores, and referring to Z-tables to find probabilities.

Example 1:

Problem: The marks of students in a class are normally distributed with a mean of 70 and a standard deviation of 10. What is the probability that a student scores less than 85?
Solution:

1. Identifying Variables: Mean (μ) = 70, Standard Deviation (σ) = 10, X = 85.
2. Convert to Z-score:
   $Z = \frac{X - μ}{σ} = \frac{85 - 70}{10} = 1.5$
3. Using the Z-table:
   $P(Z < 1.5) = 0.9332$ Final Answer: Probability = 0.9332 or 93.32%.

Example 2:

Problem: Find the probability that a value lies between 60 and 80.
Solution:

1. Identifying Variables: Mean (μ) = 70, Standard Deviation (σ) = 10.
2. Convert both values to Z-scores:
   • For 60: $Z = \frac{60 - 70}{10} = -1$
   • For 80: $Z = \frac{80 - 70}{10} = 1$
3. Using the Z-table:
   • $P(Z < 1) = 0.8413$
   • $P(Z < -1) = 0.1587$
   • $P(60 < X < 80) = 0.8413 - 0.1587 = 0.6826$ Final Answer: 68.26%

These examples illustrate how to apply the steps systematically to solve problems relating to the Normal Distribution, highlighting the importance of Z-scores and tables.