z Scores. In Exercises 5–8, express all z scores with two decimal places.

Diastolic Blood Pressure of Females For the diastolic blood pressure measurements of females listed in Data Set 1 “Body Data” in Appendix B, the highest measurement is 98 mm Hg. The 147 diastolic blood pressure measurements of females have a mean of 70.2 mm Hg and a standard deviation of 11.2 mm Hg.

b. How many standard deviations is that [the difference found in part (a)]?

검증된 단계별 안내

Step 1: Recall the formula for calculating a z-score: z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation.

Step 2: Identify the given values from the problem: X = 98 mm Hg (the highest measurement), μ = 70.2 mm Hg (the mean), and σ = 11.2 mm Hg (the standard deviation).

Step 3: Substitute the given values into the z-score formula. This will look like: z = (98 - 70.2) / 11.2.

Step 4: Simplify the numerator by subtracting the mean (70.2) from the value of interest (98). This gives the difference, which represents how far the value is from the mean.

Step 5: Divide the difference obtained in Step 4 by the standard deviation (11.2) to calculate the z-score, which represents how many standard deviations the value is from the mean.

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주요 개념

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

Z-Scores

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. This standardization allows for comparison across different datasets by indicating how many standard deviations a data point is from the mean.

Mean

The mean, or average, is a measure of central tendency that is calculated by summing all values in a dataset and dividing by the number of values. In the context of the diastolic blood pressure measurements, the mean provides a reference point to understand the typical blood pressure level among the sampled females, which is essential for calculating z-scores.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this question, the standard deviation is crucial for determining how far a specific measurement is from the mean in terms of standard deviations.