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 3.3.7c

Chapter 3, Problem 3.3.7c

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.

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,

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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Key Concepts

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

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.

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Related Practice

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.

c. Convert the highest diastolic blood pressure to a z score.

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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.)

c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n); then find the mean of those nine population variances.

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Textbook Question

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

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.

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)]?

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Textbook Question

Percentile Use the weights from Exercise 1 to find the percentile for 3647 mg.

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Textbook Question

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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