Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.5.49b

Chapter 2, Problem 2.5.49b

Life Spans of Tires A brand of automobile tire has a mean life span of 35,000 miles, with a standard deviation of 2250 miles. Assume the life spans of the tires have a bell-shaped distribution.

b. The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.

Verified step by step guidance

Step 1: Recall the Empirical Rule, which states that for a bell-shaped (normal) distribution: approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ), 95% within 2σ, and 99.7% within 3σ.

Step 2: Calculate the z-score for each tire's life span using the formula: z = (X - μ) / σ, where X is the observed value, μ is the mean (35,000 miles), and σ is the standard deviation (2250 miles).

Step 3: For each z-score, determine how many standard deviations the value is from the mean. For example, if z = -2, the value is 2 standard deviations below the mean.

Step 4: Use the Empirical Rule to estimate the percentile. For example, if a value is 1 standard deviation below the mean, it corresponds to approximately the 16th percentile (since 68% of the data is within 1σ, leaving 16% below).

Step 5: Repeat this process for each tire's life span (30,500 miles, 37,250 miles, and 35,000 miles) to find the corresponding percentiles based on their z-scores and the Empirical Rule.

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Key Concepts

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

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in understanding how data is spread around the mean and is essential for calculating percentiles in normally distributed data.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this context, a standard deviation of 2250 miles indicates how much individual tire life spans deviate from the mean of 35,000 miles. Understanding standard deviation is crucial for applying the Empirical Rule and determining how far a specific value is from the mean.

Percentiles

A percentile is a measure used to indicate the value below which a given percentage of observations fall. For example, if a tire's life span is at the 25th percentile, it means that 25% of the tires have a life span less than that value. Calculating percentiles using the Empirical Rule allows us to understand the relative standing of specific tire life spans within the overall distribution.

Related Practice

Textbook Question

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

ii.

iii.

Textbook Question

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40

37 49 34 38 43 47 35

c. Each employee in the sample takes a pay cut of \$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

Textbook Question

Using and Interpreting Concepts

Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

(b) find the interquartile range

56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

Textbook Question

Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula

MAD = Σ |x − x̄| / n.

b. Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16.

Textbook Question

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

b. Compare the four measures of central tendency, including the midrange.

Textbook Question

Extending Concepts

Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360

c. Compare the measures you found in part (b) with those found in part (a). What do you notice?

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