20.6. Example Problems

Example 1:

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

Solution: • Given: 𝜇 = 70, 𝜎 = 10, 𝑋 = 85 • Convert to Z: 85− 70 𝑍 = = 1.5 10 • From Z-table: 𝑃(𝑍 < 1.5) = 0.9332 Answer: Probability = 0.9332 or 93.32%

In this example, we begin by identifying the known values: the mean (𝜇) is 70, the standard deviation (𝜎) is 10, and we want to find out the probability that a student scores less than 85. To convert the raw score (85) to a Z-score, we subtract the mean from the score and divide by the standard deviation:

Z = (X - μ) / σ = (85 - 70) / 10 = 1.5.

Next, we use the Z-table to find the probability corresponding to a Z-score of 1.5, which gives us a probability of 0.9332. This means that there's a 93.32% chance that a student scores less than 85.

Imagine you're in a classroom where the students' test scores follow a bell curve. If the average score is 70, and one student scores 85, we can find out how many students likely scored less than this. By calculating a Z-score, we can understand how exceptional that score is within the context of the class. If 93 out of 100 students are expected to score lower than this student, it underscores the student's strong performance.

Example 2:

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

Solution: • 𝜇 = 70, 𝜎 = 10 • Convert both values to Z: 60−70 o For 60: 𝑍 = = −1 1 10 80−70 o For 80: 𝑍 = = 1 2 10 From Z-table: • 𝑃(𝑍 < 1) = 0.8413 • 𝑃(𝑍 < −1) = 0.1587 𝑃(60 < 𝑋 < 80)= 0.8413−0.1587 = 0.6826 Answer: 68.26%

In this example, we are tasked with finding the probability that a student's score lies between 60 and 80. We start by noting the mean (𝜇) is still 70 and the standard deviation (𝜎) is 10. We convert both scores to Z-scores. For 60: Z = (60-70) / 10 = -1, and for 80: Z = (80-70) / 10 = 1.

Now, we reference the Z-table to determine the probabilities for these Z-scores: P(Z < 1) gives us 0.8413 and P(Z < -1) gives us 0.1587. To find the probability between these two scores, we subtract the smaller probability from the larger one: 0.8413 - 0.1587 = 0.6826, which indicates a 68.26% probability that a student's score will fall between 60 and 80.

Think of a game where scoring between certain points matters. If the average score is set at 70, we want to know the chances of a player scoring between 60 and 80 points. Calculating Z-scores provides insights into how likely it is for players to achieve scores within that range, giving teams a better strategy to adapt their plays. If about 68 out of 100 players can be expected to score within this range, it suggests a competitive atmosphere but with room for strong performances.

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