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.

a. For a randomly selected subject, find the probability of a bone density test score greater than -1.37.

검증된 단계별 안내

Step 1: Understand the problem. The z-score represents the number of standard deviations a data point is from the mean. Here, the z-score is -1.37, and we are tasked with finding the probability that a randomly selected subject has a z-score greater than -1.37 in a standard normal distribution (mean = 0, standard deviation = 1).

Step 2: Recall that the cumulative distribution function (CDF) of the standard normal distribution gives the probability that a z-score is less than or equal to a given value. Denote this as P(Z ≤ -1.37).

Step 3: Use the complement rule to find the probability of a z-score greater than -1.37. This is given by P(Z > -1.37) = 1 - P(Z ≤ -1.37).

Step 4: Look up the cumulative probability P(Z ≤ -1.37) in a standard normal distribution table or use statistical software to find its value. This value represents the area under the curve to the left of z = -1.37.

Step 5: Subtract the cumulative probability P(Z ≤ -1.37) from 1 to find P(Z > -1.37). This result represents the probability of a bone density test score greater than -1.37.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Z Score

A Z score represents the number of standard deviations a data point is from the mean of a distribution. In the context of a bone density test, a Z score of 0 indicates 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 a subject's score relative to the population.

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. This property allows for the use of standard statistical methods to calculate probabilities and make inferences about the population.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. For the bone density test, calculating the probability of a score greater than -1.37 requires using the properties of the normal distribution, specifically the cumulative distribution function (CDF), to find the area under the curve to the right of the given Z score.