Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.3.7d

Chapter 3, Problem 3.3.7d

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.

d. Using the criteria summarized in Figure 3-6, is the commute time of 95 minutes significantly low, significantly high, or neither?

Verified step by step guidance

Step 1: Recall the formula for calculating a z-score: z = (X - μ) / σ, where X is the data value (in this case, 95.0 minutes), μ is the mean (42.6 minutes), and σ is the standard deviation (26.2 minutes).

Step 2: Substitute the given values into the formula. This will give you the z-score for the commute time of 95.0 minutes. Use the formula z = (95.0 - 42.6) / 26.2.

Step 3: Simplify the numerator by subtracting the mean (42.6) from the data value (95.0). Then divide the result by the standard deviation (26.2).

Step 4: Compare the calculated z-score to the criteria in Figure 3-6. Typically, a z-score greater than 2 or less than -2 is considered significantly high or significantly low, respectively. If the z-score falls between -2 and 2, it is considered neither significantly high nor low.

Step 5: Based on the z-score and the criteria, determine whether the commute time of 95.0 minutes is significantly low, significantly high, or neither. Provide reasoning based on the z-score's magnitude.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Z Scores

A z score, or standard score, indicates how many standard deviations an element is from the mean of a data set. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z scores allow for comparison between different data sets and help identify how unusual or typical a particular value is within its distribution.

Mean and Standard Deviation

The mean is the average of a data set, calculated by summing all values and dividing by the number of values. The standard deviation measures the dispersion or spread of the data points around the mean, indicating how much the values typically deviate from the average. Together, these statistics provide a foundation for understanding the distribution of data and calculating z scores.

Significance in Statistics

In statistics, a value is considered significantly high or low if it falls beyond a certain threshold, often determined by z scores. Typically, a z score greater than 1.96 or less than -1.96 indicates that a value is significantly different from the mean at a 95% confidence level. This concept helps in making inferences about data and determining whether observed values are typical or unusual.

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