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Chapter 3: Problem 43

In December \(2004,\) the average price of regular unleaded gasoline excluding taxes in the United States was \(1.37\) per gallon according to the Energy Information Administration. Assume that the standard deviation price per gallon is \(0.05\) per gallon to answer the following. (a) What percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between \(\$ 1.27\) and \(\$ 1.47 ?\)

Short Answer

Expert verified
a) 99.7%. b) 98.76% and prices within \(1.245 to \)1.495. c) 95.44%.

Step by step solution

01

Understanding the Problem

Given the average price (\text{mean}) \( \bar{x} = 1.37 \ \text{USD per gallon} \) and the standard deviation (sd) \( \text{sd} = 0.05 \ \text{USD per gallon} \). We need to find percentages for different standard deviations from the mean using standard normal distribution (Z-scores).
02

Finding percentage within 3 standard deviations

Using the empirical rule (68-95-99.7 rule) for normal distribution: Within \( \pm 3 \) standard deviations of the mean, approximately \( 99.7\text{\text{%}} \) of data falls. Thus, \( 99.7\text{\text{%}} \) of gasoline stations had prices within 3 standard deviations from the mean.
03

Finding percentage within 2.5 standard deviations

Using Z-scores and a standard normal distribution table, \( P(-2.5 \leq z \leq 2.5) = 0.9876 \text{\text{%}} \). This means \( 98.76 \text{\text{%}} \) of gasoline stations had prices within 2.5 standard deviations of the mean.
04

Calculating gasoline prices within 2.5 standard deviations

Within 2.5 standard deviations from the mean: \( \bar{x} \pm 2.5 \times \text{sd} \ = 1.37 \pm 2.5(0.05) \ = 1.37 \pm 0.125 \). Thus gasoline prices are within the range \( 1.245 \ \text{USD to} \ 1.495 \ \text{USD} \).
05

Finding the percentage for prices between \(1.27 and \)1.47

First convert to Z-scores: \( Z_{1.27} = \frac{1.27-1.37}{0.05} = -2 \) and \( Z_{1.47} = \frac{1.47-1.37}{0.05} = 2 \). Using the empirical rule or standard normal table, \( P(-2 \leq z \leq 2) = 0.9544 \text{\text{%}} \). Therefore, the minimum percentage of gasoline stations that had prices between \ \(1.27 \ and \ \)1.47 \ is \ 95.44 \text{\text{%}} \.

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Key Concepts

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

normal distribution
A normal distribution is a probability distribution that is symmetric about the mean. This means that most values cluster around a central point, and the probabilities of values farther from the mean taper off equally in both directions. The shape of a normal distribution is often referred to as a 'bell curve' because of this symmetry.
z-scores
Z-scores are a way to describe a value's position relative to the mean of a dataset. They are measured in terms of standard deviations. For example, a Z-score of 1 means the value is 1 standard deviation above the mean, while a Z-score of -1 means the value is 1 standard deviation below the mean.
empirical rule
The empirical rule, or 68-95-99.7 rule, is a statistical rule for normal distributions. It states that approximately 68% of data falls within 1 standard deviation of the mean, around 95% falls within 2 standard deviations, and about 99.7% falls within 3 standard deviations. This rule helps in understanding the spread and probability of data points in a normal distribution.
price analysis
In the context of gasoline prices, price analysis involves examining the average price and its variation. By using the normal distribution and concepts like Z-scores and the empirical rule, you can determine the percentages of gas stations with prices within certain ranges. This helps in understanding the distribution of prices across different stations.

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