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 3.3.7b

Chapter 3, Problem 3.3.7b

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.

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

Verified step by step guidance

Step 1: Recall the formula for calculating a z-score: \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Step 2: Identify the given values from the problem: \( x = 95.0 \), \( \mu = 42.6 \), and \( \sigma = 26.2 \).

Step 3: Substitute the values into the z-score formula: \( z = \frac{95.0 - 42.6}{26.2} \).

Step 4: Perform the subtraction in the numerator: \( 95.0 - 42.6 \). This gives the difference between the data value and the mean.

Step 5: Divide the result from Step 4 by the standard deviation \( \sigma = 26.2 \) to find the z-score, which represents how many standard deviations the data value is from the mean.

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Key Concepts

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

Z-Score

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 commute times, the mean provides a baseline to understand the typical commute duration, which is essential for calculating Z-scores and assessing individual commute times relative to this average.

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 a wider spread. In this question, the standard deviation is crucial for determining how far the specific commute time of 95.0 minutes is from the mean in terms of standard deviations.

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Textbook Question

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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Textbook Question

Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47.

a. Find the missing value.

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Textbook Question

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)]?

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Textbook Question

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

a. What is the difference between the commute time of 95.0 minutes and the mean commute time?

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Textbook Question

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

a. What is the difference between the highest diastolic blood pressure and the mean of the diastolic blood pressures for females?

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Textbook Question

Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

b. After listing the nine different possible samples of two values selected with replacement, find the sample variance (which includes division by ) for each of them; then find the mean of the nine sample variances s2.

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