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The EPA has set a maximum noise level for heavy trucks at 83 decibels (dB). The manner in which this limit is applied will greatly affect the trucking industry and the public. One way to apply the limit is to require all trucks to conform to the noise limit. A second but less satisfactory method is to require the truck fleet's mean noise level to be less than the limit. If the latter rule is adopted, variation in the noise level from truck to truck becomes important because a large value of \(\sigma^{2}\) would imply that many trucks exceed the limit, even if the mean fleet level were 83 dB. A random sample of six heavy trucks produced the following noise levels (in decibels): \(\begin{array}{llllll}85.4 & 86.8 & 86.1 & 85.3 & 84.8 & 86.0 .\end{array}\) Use these data to construct a \(90 \%\) confidence interval for \(\sigma^{2}\), the variance of the truck noiseemission readings. Interpret your results.

Step by step solution

01

Understand the Problem

We need to construct a 90% confidence interval for the variance \( \sigma^2 \) of the truck noise emission based on a sample of six trucks' noise levels. This will help us understand the variability in noise levels among the trucks.

02

Calculate the Sample Variance

First, find the mean of the noise levels: \( \bar{x} = \frac{85.4 + 86.8 + 86.1 + 85.3 + 84.8 + 86.0}{6} = 85.73 \). Then calculate the sample variance \( s^2 \) by using the formula \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \), where \( x_i \) are the sample noise levels and \( n = 6 \). Substitute to find \( s^2 \).

03

Use the Chi-Square Distribution

When constructing a confidence interval for \( \sigma^2 \), we use the chi-square distribution. The formula for the endpoints of the confidence interval of \( \sigma^2 \) are \( \left( \frac{(n-1)s^2}{\chi^2_{\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} \right) \), where \( \alpha = 0.10 \).

04

Determine Chi-Square Values

For the given confidence level of \( 90\% \), the degrees of freedom \( df = n - 1 = 5 \). From chi-square distribution tables, find \( \chi^2_{0.05,5} \approx 1.145 \) and \( \chi^2_{0.95,5} \approx 11.070 \).

05

Calculate the Confidence Interval

Substitute \( s^2 \) and the chi-square values into the formula from Step 3. Calculate the values to obtain the confidence interval for \( \sigma^2 \).

06

Interpret the Results

A confidence interval means that we can be 90% confident the true variance \( \sigma^2 \) of the noise emissions falls within this interval. If the interval is wide, it indicates higher uncertainty about the true variance.

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Key Concepts

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

Understanding Variance

Variance is a measure of how far a set of numbers are spread out from their average value. In the context of noise pollution from trucks, variance helps us understand

• the level of inconsistency in the noise levels produced by different trucks.
• how much individual trucks deviate from the average noise level.

Low variance means noise levels are close to each other, whereas high variance indicates more disparity. When applied to compliance with noise pollution limits, high variance in truck noise levels could mean that even if the average level is within limits, many trucks might still produce noise above the acceptable threshold. Thus, calculating variance is crucial in assessing whether it is the rule for individual trucks or the fleet average that ought to be applied.

Chi-Square Distribution and Its Role

The chi-square distribution is a statistical tool used to assess the variability of a sample. It is especially useful when calculating confidence intervals for the variance of a normally distributed population. Given the sample of truck noise levels, the chi-square distribution helps us determine

• the range of values within which the true variance of the truck noise levels likely falls.
• the degree of certainty we have about this assessment.

For our problem, the degrees of freedom (df) are calculated as the sample size minus one. In this case, with six trucks, the df equals five. Values from chi-square tables for different percentiles corresponding to the degrees of freedom allow us to construct the confidence interval for the variance.

Calculating Sample Variance

Sample variance provides an estimate of the variance of an entire population based on a sample. To calculate sample variance: 1. Find the mean (average) of your sample data. 2. Subtract the mean from each data point and square the result. 3. Add up all the squared differences. 4. Divide by the number of data points minus one (for a sample variance, not population). For instance, using the truck noise levels, the sample variance gives us a snapshot of how noise varies across these trucks, providing essential data for constructing the confidence interval for the true variance.

Noise Pollution Limits and Their Implications

Noise pollution limits are regulations set to control the noise levels in a particular environment. For the trucking industry, this means

• setting maximum allowable noise levels for trucks to reduce environmental and human health impacts.
• ensuring trucks operate under conditions that minimize excessive noise pollution.

The 83 dB limit specified requires regulatory enforcement strategies. One requires each truck to meet the limit, promoting uniform compliance, while another allows for the fleet's average to meet the limit. The latter introduces variability concerns, as high variance could result in numerous trucks exceeding the noise threshold, even if the mean is below it. Understanding these implications is critical when defining standards on individual or fleet-based compliance.