Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.


Find the bone density scores that are the quartiles Q1, Q2, and Q3.

Verified step by step guidance

1. Step 1: Understand the problem. The bone density test scores are normally distributed with a mean (μ) of 0 and a standard deviation (σ) of 1. Quartiles divide the data into four equal parts, and we are tasked with finding the scores corresponding to Q1 (25th percentile), Q2 (50th percentile, also the median), and Q3 (75th percentile).
2. Step 2: Recall the relationship between percentiles and z-scores in a standard normal distribution. Percentiles correspond to specific z-scores, which can be found using a z-score table or statistical software. For example, Q1 corresponds to the 25th percentile, Q2 to the 50th percentile, and Q3 to the 75th percentile.
3. Step 3: Use the z-score formula for a standard normal distribution: $Z=\; (X-\mu )/\sigma$. Here, μ = 0 and σ = 1, so the formula simplifies to $Z=X$. This means the z-scores directly correspond to the bone density scores.
4. Step 4: Find the z-scores for the quartiles using a z-score table or statistical software. For Q1 (25th percentile), the z-score is approximately -0.67. For Q2 (50th percentile), the z-score is 0 (the mean). For Q3 (75th percentile), the z-score is approximately 0.67.
5. Step 5: Interpret the results. The bone density scores corresponding to Q1, Q2, and Q3 are the same as the z-scores because the distribution is standard normal. Therefore, Q1 ≈ -0.67, Q2 = 0, and Q3 ≈ 0.67. Round these values to two decimal places as required.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the bone density scores are normally distributed with a mean of 0 and a standard deviation of 1, which allows for the application of statistical methods to find specific percentiles.

Quartiles

Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data. The first quartile (Q1) is the median of the lower half of the data, the second quartile (Q2) is the median of the dataset, and the third quartile (Q3) is the median of the upper half. In the context of the bone density scores, finding Q1, Q2, and Q3 involves determining the corresponding z-scores that represent these quartiles in the normal distribution.

Z-scores

A z-score indicates how many standard deviations an element is from the mean. In a standard normal distribution, z-scores can be used to find probabilities and percentiles. For the bone density scores, calculating the z-scores for Q1, Q2, and Q3 allows us to determine the specific scores that correspond to these quartiles, facilitating the interpretation of the distribution of bone density test results.