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Assume that a randomly selected subject is given a bone density test. Those tests follow a standard normal distribution. Find the probability that the bone density score for this subject is between -1.93 and 2.37. Student’s answer: We first need to find the probability for each of these z-scores using Excel. For -1.93 the probability from the left is 0.0268, and for 2.37 the probability from the left is 0.9911. Continue the solution: Finish the problem giving step-by-step instructions and explanations. We need to subtract the two probabilities to get the required probability. The probability that the bone density score for this subject is between -1.93 and 2.37 is 0.9643.

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If a subject’s bone density scores are normally distributed with mean 0 and standard deviation 1, then to find the probability that the score lies between -1.93 and 2.37, we can utilize the properties of the standard normal distribution, particularly the cumulative distribution function (CDF), which can be accessed through statistical tools like Excel, scientific calculators, or statistical software. The approach involves calculating the cumulative probability for each z-score and then subtracting these probabilities to determine the probability of the scores lying within the specified interval.

Firstly, we convert the raw scores to their corresponding z-scores, which are already provided: -1.93 and 2.37. Using Excel, the CDF (represented by the function NORM.S.DIST in Excel or standard normal distribution tables) provides the probability of a randomly selected score falling below a particular z-score. For z = -1.93, the cumulative probability from the left end is approximately 0.0268. This indicates that about 2.68% of the scores fall below -1.93.

Similarly, for z = 2.37, the cumulative probability from the left is approximately 0.9911, meaning about 99.11% of the scores are below 2.37. To find the probability that a score falls between -1.93 and 2.37, we subtract the smaller cumulative probability from the larger one: 0.9911 - 0.0268 = 0.9643. This result indicates that approximately 96.43% of the bone density scores are within this interval.

This method demonstrates the straightforward process of deriving probabilities from the standard normal distribution using known cumulative probabilities. The key steps include converting raw scores to z-scores, retrieving the cumulative probabilities associated with these z-scores, and then calculating the difference to find the probability within the given range. This approach is applicable across various contexts where normal distribution assumptions hold, including biological measurements, test scores, and natural phenomena.

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