Math Problem Statement
Solution
Let's go through the questions one by one based on the image provided:
Problem 6: Find the probability that falls in the shaded area.
The image shows a uniform distribution from to , with a probability density of .
To find the probability, use the formula for uniform distribution probability: In this case:
- The interval and ,
- The density is .
So, the probability is: Thus, the probability that falls in the shaded area is 0.2 or 20%.
Problem 7: What percent of students scored between 60 and 90 points?
The distribution is approximately Normal with:
- Mean ,
- Standard deviation .
We calculate the z-scores for 60 and 90 points using the formula:
- For 60:
- For 90:
Using the standard normal distribution table:
- The area to the left of is approximately 0.0228,
- The area to the left of is approximately 0.9772.
Thus, the percentage of students scoring between 60 and 90 is: So, approximately 95.44% of the students scored between 60 and 90 points.
Problem 8: What percent of students failed the test (scored below 60)?
Using the -score for 60 (calculated earlier as ):
The area to the left of is approximately 0.0228, meaning about 2.28% of the students scored below 60 points.
Problem 9: What percent of students scored below 70?
For 70 points: Using the standard normal table, the area to the left of is approximately 0.2514.
Thus, about 25.14% of the students scored below 70 points.
Do you want more details or have any questions about these steps? Here are some related questions to deepen understanding:
- How would the probability change in Problem 6 if the shaded area was from to ?
- What is the probability that a student scored above 90 points?
- If the standard deviation increased to 10, how would that affect the percentage of students scoring between 60 and 90 points?
- How can you use the empirical rule to approximate the answer for students scoring between 60 and 90?
- What is the probability that a student scored between 70 and 85 points?
Tip: Using the z-score is essential for calculating probabilities in a normal distribution! Make sure to get comfortable with it.
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Math Problem Analysis
Mathematical Concepts
Probability
Normal Distribution
Z-Scores
Formulas
P(a ≤ x ≤ b) = (b - a) × density
z = (x - μ) / σ
Theorems
Properties of Normal Distribution
Suitable Grade Level
Grade 10-12
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