Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score between - 1.62 and 1.62. Sketch the region. Choose the correct graph below. O A. В. D. -1.62 1.62 -1.62 1.62 -1.62 1.62 -1.62 1.62 The probability is. (Round to four decimal places as needed.)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Probability of Bone Density Test Scores
Assume that a randomly selected subject is given a bone density test. These test scores are normally distributed with a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. We aim to draw a graph and find the probability of a bone density test score falling between \(-1.62\) and \(1.62\).
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#### Step-by-Step Solution
1. **Identify the parameters:**
- Mean (\(\mu\)) = 0
- Standard Deviation (\(\sigma\)) = 1
- Range: \([-1.62, 1.62]\)
2. **Sketch the Region:**
- Choose the correct graph that shades the area between \(-1.62\) and \(1.62\) under the normal distribution curve.
3. **Locate the shaded region that represents the probability of the test score lying within the specified range. The options provided are as follows:**
- **Option A:**
- Incorrect: Insufficient shading.
- **Option B:**
- **Correct:** The graph accurately shows the area between \(-1.62\) and \(1.62\) shaded under the curve.
- **Option C:**
- Incorrect: Shaded region is larger than required.
- **Option D:**
- Incorrect: Shaded region is smaller than necessary.
4. **Major Graph Details:**
- **Shape:** Bell curve (normal distribution).
- **Shaded Area:** Region between \(-1.62\) and \(1.62\), representing the probability of test scores lying within this range.
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### Conclusion
The best graph that represents the required region is Option B.
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#### Calculate the Probability
The transcribed portion includes a blank space for the calculated probability. You would typically use standard normal distribution tables or computational tools to find this probability value.
**The completed steps affirm the probability calculation:**
The probability is **[Calculated Value]**.
*(Round to four decimal places as needed.)*
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