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Chapter 3: Problem 82

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of a. 41 to 47 months b. 38 to 50 months c. 35 to 53 months

Short Answer

Expert verified
a. 68% of batteries will last between 41 to 47 months.\n b. 95% of batteries will last between 38 to 50 months.\n c. 99.7% of batteries will last between 35 to 53 months.

Step by step solution

01

Identify the Intervals

First, let's identify the intervals in terms of standard deviations from the mean. 41 to 47 months represents an interval of one standard deviation from the mean (44 months), as 44 - 3 = 41 and 44 + 3 =47. 38 to 50 months represents an interval of two standard deviations from the mean, as 44 - 2*3 = 38 and 44 + 2*3 = 50. 35 to 53 months represents an interval of three standard deviations from the mean, as 44 - 3*3 = 35 and 44 + 3*3 = 53.
02

Application of Empirical Rule

Applying the Empirical Rule, for the interval of 41 to 47 months which is one standard deviation from the mean, 68% of batteries will last this long. For the interval of 38 to 50 months which is two standard deviations from the mean, 95% of batteries will last this long. For the interval of 35 to 53 months which is three standard deviations from the mean, 99.7% of batteries will last this long.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Standard Deviation
Standard deviation is a statistical measurement that shows how data points are spread out around the mean, or average, value of a dataset. In simpler terms, it tells us how much variation or "dispersion" exists from the average. If data points are close to the mean, the standard deviation is low. If data extends far from the mean, the standard deviation is high.

In the case of the auto batteries, we are given a standard deviation of 3 months. This means most battery lives are within 3 months of the mean life span of 44 months. Standard deviation helps understand the range of data within a normal, bell-shaped distribution, which is crucial for applying the Empirical Rule reliably.
Recognizing Bell-Shaped Distribution
A bell-shaped distribution, also known as a normal distribution, is one of the most common distributions found in statistics. It is characterized by its symmetric, bell-like shape, where most of the data points are concentrated around the mean, and probabilities tail off equally in both directions as you move away from the mean.

This distribution is important because it allows for the use of the Empirical Rule, which estimates the percentages of data within certain standard deviations from the mean. For our auto batteries example, the bell-shaped distribution indicates that most battery lifespans cluster around 44 months, with fewer batteries expected to last significantly less or more time than this average.
Calculating Mean Life
Mean life refers to the average lifespan of a set of items, in this case, auto batteries. The mean is a measure of central tendency and is calculated by adding all data points together and dividing by the number of data points.

For the auto batteries, the mean life is reported to be 44 months. This serves as a central point from which we can measure the typical deviation or variation of all batteries using standard deviation. By understanding the mean life, we recognize the expected lifespan around which most batteries will last, thanks to the underlying normal distribution. Utilizing the mean and standard deviation together allows effective application of the Empirical Rule to predict the percentage of battery lifespans falling within given ranges.

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Most popular questions from this chapter

According to an article in the Washington Post ( Washington Post, January 5, 2009), the average employee share of health insurance premiums at large U.S. companies is expected to be \(\$ 3423\) in \(2009 .\) Suppose that the current annual payments by all such employees toward health insurance premiums have a bell-shaped distribution with a mean of \(\$ 3423\) and a standard deviation of \(\$ 520\). Using the empirical rule, find the approximate percentage of employees whose annual payments toward such premiums are between a. \(\$ 1863\) and \(\$ 4983\) b. \(\$ 2903\) and \(\$ 3943\) c. \(\$ 2383\) and \(\$ 4463\)

On a 300 -mile auto trip, Lisa averaged 52 mph for the first 100 miles, 65 mph for the second 100 miles, and 58 mph for the last 100 miles. a. How long did the 300 -mile trip take? b. Could you find Lisa's average speed for the 300 -mile trip by calculating \((52+65+58) / 3 ?\) If not, find the correct average speed for the trip.

The heights of five starting players on a basketball team have a mean of 76 inches, a median of 78 inches, and a range of 11 inches. a. If the tallest of these five players is replaced by a substitute who is 2 inches taller, find the new mean, median, and range. b. If the tallest player is replaced by a substitute who is 4 inches shorter, which of the new values (mean, median, range) could you determine, and what would their new values be?

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The following data give the numbers of driving citations received by 12 drivers. \(\begin{array}{llllllllllll}4 & 8 & 0 & 3 & 1 & 17 & 4 & 1 & 48 & 1 & 37 & 9\end{array}\) a. Find the mean, median, and mode for these data. b. Calculate the range, variance, and standard deviation. c. Are the values of the summary measures in parts a and b population parameters or sample statistics?
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