Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

c. For a randomly selected subject, find the probability of a bone density test score between -0.67 and 1.29.

검증된 단계별 안내

Step 1: Understand the problem. The z-scores are normally distributed with a mean (μ) of 0 and a standard deviation (σ) of 1. We are tasked with finding the probability that a randomly selected z-score lies between -0.67 and 1.29.

Step 2: Recall that the probability of a z-score falling between two values can be found using the cumulative distribution function (CDF) of the standard normal distribution. The formula for this is P(a ≤ Z ≤ b) = P(Z ≤ b) - P(Z ≤ a), where Z is the standard normal variable.

Step 3: Use the standard normal table (or a statistical software) to find the cumulative probabilities for the z-scores -0.67 and 1.29. Specifically, find P(Z ≤ 1.29) and P(Z ≤ -0.67).

Step 4: Subtract the cumulative probability of the lower z-score from the cumulative probability of the higher z-score. This gives P(-0.67 ≤ Z ≤ 1.29) = P(Z ≤ 1.29) - P(Z ≤ -0.67).

Step 5: Interpret the result. The value obtained represents the probability that a randomly selected subject has a bone density test score between -0.67 and 1.29. This probability can be expressed as a percentage if needed.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Z Score

A z score indicates how many standard deviations an element is from the mean of a distribution. In the context of a bone density test, a z score of 0 represents the average bone density, while positive and negative values indicate above or below average densities, respectively. Understanding z scores is crucial for interpreting test results and assessing the likelihood of certain outcomes.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. In this case, the z scores of bone density tests follow a normal distribution with a mean of 0 and a standard deviation of 1, which allows for the application of statistical methods to calculate probabilities and make inferences about the population.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. For the bone density test, calculating the probability of a score falling between -0.67 and 1.29 requires using the properties of the normal distribution, specifically the cumulative distribution function (CDF), to find the area under the curve between these two z scores.