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

New York City Commute Time New York City commute times (minutes) are listed in Data Set 31 “Commute Times” in Appendix B. The 1000 times have a mean of 42.6 minutes and a standard deviation of 26.2 minutes. Consider the commute time of 95.0 minutes.

c. Convert the commute time of 95.0 minutes to a z score.

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Step 1: Recall the formula for calculating a z-score:

z = (x - μ) / σ

, where

x

is the data value,

μ

is the mean, and

σ

is the standard deviation.

Step 2: Identify the given values from the problem. The data value

x

is 95.0 minutes, the mean

μ

is 42.6 minutes, and the standard deviation

σ

is 26.2 minutes.

Step 3: Substitute the given values into the z-score formula:

z = (95.0 - 42.6) / 26.2

Step 4: Perform the subtraction in the numerator:

95.0 - 42.6

Step 5: Divide the result of the subtraction by the standard deviation

26.2

to calculate the z-score. Round the z-score to two decimal places.

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주요 개념

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

Z Score

A z score, or standard score, indicates how many standard deviations a data point is from the mean of a dataset. It is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. Z scores allow for the comparison of different data points within the same distribution, providing insight into their relative position.

Mean

The mean is the average value of a dataset, calculated by summing all the data points and dividing by the number of points. In the context of the question, the mean commute time of 42.6 minutes serves as a reference point for determining how far a specific commute time, like 95.0 minutes, deviates from the average. Understanding the mean is crucial for interpreting z scores.

Standard Deviation

Standard deviation is a measure of 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 a wider spread. In this question, the standard deviation of 26.2 minutes is essential for calculating the z score, as it quantifies how much the commute times vary from the average.