Math Problem Statement
Solution
This problem involves a normal distribution, where scores on a standardized exam are normally distributed with a mean and a standard deviation . The goal is to calculate the probability (shaded area) that a score lies between 355 and 687.
Step-by-step solution:
-
Convert the raw scores to z-scores using the formula: where:
- is the value from the distribution,
- is the mean,
- is the standard deviation.
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Calculate the z-score for 355:
-
Calculate the z-score for 687:
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Find the cumulative probability associated with these z-scores:
- From the standard normal distribution table, the cumulative probability for is approximately .
- The cumulative probability for is approximately .
-
Calculate the area between the two z-scores:
Thus, the shaded area under the curve, which represents the probability of a score falling between 355 and 687, is 0.95 (rounded to two decimal places).
Final answer:
Do you need any additional details or further explanation?
Here are some relative questions for further understanding:
- How does changing the standard deviation affect the width of the normal curve?
- What is the significance of a z-score in a normal distribution?
- How do you interpret probabilities in the context of a normal distribution?
- Can you explain why the area under the normal curve totals to 1?
- How do different values of the mean affect the center of the normal distribution?
Tip: When using a standard normal distribution table, make sure to interpret the values correctly depending on whether the z-score is positive or negative.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Z-scores
Probability
Formulas
Z = (X - μ) / σ
Cumulative probability from the Z-table
Theorems
Normal distribution properties
Suitable Grade Level
Grades 9-12
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