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The hydrocarbon emissions are known to have decreased dramatically during the 1980 s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), for automobiles of 1980 and \(1990 .\) Twenty cars of each year model were randomly selected and their hydrocarbon emission leycls were recorded. The data are as follows 1980 models: \(\begin{array}{llllllllll}141 & 359 & 247 & 940 & 882 & 494 & 306 & 210 & 105 & 880 \\ 200 & 223 & 188 & 940 & 241 & 190 & 300 & 435 & 241 & 380\end{array}\) 1990 models: \(\begin{array}{rrrrrrrrrr}140 & 160 & 20 & 20 & 223 & 60 & 20 & 95 & 360 & 70 \\\ 220 & 400 & 217 & 58 & 235 & 380 & 200 & 175 & 85 & 65\end{array}\) Test the hypothesis that \(\sigma_{1}=\sigma_{2}\) against the alternative that \(\sigma_{1} \neq \sigma_{2}\). Assume both populations are normal. Use a P-value.

Step by step solution

01

Calculating variances

Calculate the variance for both 1980 and 1990 car models' data sets. Use the formula for variance: \(s^2 = \frac{\Sigma (x_i - \overline{x})^2}{n-1}\) where \(\overline{x}\) is the sample mean, \(x_i\) is each value from the data set, and \(n\) is the number of data items in the sample.

02

Finding the F-statistic

Calculate the F-statistic using the formula: \(F = \frac{s1^2}{s2^2}\) where \(s1^2\) and \(s2^2\) are the computed variances of the 1980 and 1990 car models respectively. Make sure to keep the larger variance in the numerator to ensure the F value is greater than 1.

03

Computing P-value

Use an F-distribution table or a statistical calculator/tool to find the P-value corresponding to the calculated F-statistic and with degrees of freedom \(df1 = n1 - 1\) and \(df2 = n2 - 1\), where \(n1\) and \(n2\) are the number of samples from 1980 and 1990 car models respectively.

04

Making decision

The decision, based on P-value, is dependent on the predetermined alpha level, typically 0.05. If the P-value is less than 0.05, then there is evidence to reject the null hypothesis, meaning variances are not equal. If the P-value is more than or equal to 0.05, there is not enough evidence to reject the null hypothesis, meaning variances can be considered equal.

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Key Concepts

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

F-test

The F-test is a statistical method used to compare the variances of two populations. It helps in determining if the populations have different variances. In our exercise, we're investigating the variances of hydrocarbon emissions from cars manufactured in 1980 and 1990. By using an F-test, we can conclude if the emission variance from one year significantly differs from the other. The formula for the F-statistic is simple: compute it by dividing the larger variance by the smaller variance to ensure the outcome is greater than 1. This is crucial because an F value below 1 isn't valid for this test. The calculated F-value is then compared against a critical value from the F-distribution table, or used to find the P-value, to test the hypothesis that the variances are equal.

Variance Calculation

Variance calculation is essential for determining the spread of data points in a dataset. It tells us how much the values deviate from the mean of the dataset. To calculate variance, use the formula \[ s^2 = \frac{\Sigma (x_i - \overline{x})^2}{n-1} \]where \(x_i\) represents each data point, \(\overline{x}\) is the sample mean, and \(n\) is the number of data items in the sample.

• First, find the mean of the dataset.
• Subtract each data point by this mean and square the result.
• Sum up all squared results and divide by \(n-1\) to find the variance.

For our study, calculate the variance for the 1980 and 1990 car models separately. This forms the foundation for conducting the F-test.

Normal Distribution

Normal distribution, often called the bell curve, is a common pattern in statistics. It assumes that data points are symmetrically distributed around the mean. This distribution is important as many statistical tests—including the F-test—rely on the assumption that data follows a normal distribution. In our scenario, we assume that the hydrocarbon emissions data from 1980 and 1990 models are normally distributed. This assumption allows us to use the F-test confidently. By ensuring data follows a normal pattern, we can apply various statistical analyses more accurately. The normal distribution is advantageous because:

• It enables proportion estimation within specified ranges.
• It simplifies the application of statistical models.

Being aware of this groundwork ensures that our hypothesis testing remains valid and trustworthy.

P-value

The P-value is a probability measure that helps determine the strength of evidence against the null hypothesis. A lower P-value indicates stronger evidence to reject the null hypothesis. It gives us insight into whether a statistically significant difference exists between the variances of two populations. After calculating the F-statistic from our variance data, find the corresponding P-value using the F-distribution. This value is compared against an alpha level, commonly set at 0.05, which signifies a 5% risk of concluding that a difference exists, when there isn't one.

• If the P-value is less than 0.05, reject the null hypothesis indicating variances are different.
• If the P-value is greater than or equal to 0.05, there isn't enough evidence to reject the null hypothesis, hence the variances can be viewed as equal.

Relying on P-values aids in making informed decisions based on statistical results.