Math Problem Statement
Solution
The problem involves a normally distributed variable with a mean of 0 and a standard deviation of 1, and it asks to find the probability of a bone density test score being greater than -1.76.
The correct region to sketch on the normal curve for a probability of a score greater than -1.76 would be the right tail of the distribution, starting at -1.76 and extending to infinity. Option A shows this correctly, where the shaded region represents the area to the right of -1.76, corresponding to the probability of a score greater than -1.76.
Now, let's calculate the probability:
- We need to find the z-score corresponding to -1.76. This is given directly as the z-score, since the distribution has a mean of 0 and a standard deviation of 1.
- Using a standard normal table or a calculator, the cumulative probability up to -1.76 is approximately 0.0392.
- Since we are interested in the probability greater than -1.76, we subtract this from 1:
Thus, the probability is 0.9608.
Would you like further clarification, or do you have any additional questions?
Here are five related questions to explore:
- What is the probability of a score greater than 1.76 in a normal distribution?
- How do you calculate probabilities for values between two z-scores, for example, between -1.76 and 1.76?
- How does the area under the normal curve change with different standard deviations?
- What is the total area under the normal distribution curve, and why is it significant?
- Can you explain the symmetry of the normal distribution and how it affects probability calculations?
Tip: When solving normal distribution problems, it is always useful to check if you're dealing with a "greater than" or "less than" probability to apply the correct cumulative probability subtraction.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Probability
Z-Scores
Formulas
Cumulative probability formula for Z-scores
P(X > z) = 1 - P(X ≤ z)
Theorems
Empirical Rule (68-95-99.7 Rule)
Properties of the Standard Normal Distribution
Suitable Grade Level
College or Advanced High School Statistics
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