Questions: Truck Pollution: In an experiment to determine the effect of ambient temperature on the emissions of oxides of nitrogen (NOX) of diesel trucks, ten trucks were run at temperatures of 40 degrees F and 80 degrees F. The emissions, in parts per billion, are presented in the following table. Truck 40 degrees F 80 degrees F 1 838.7 817.8 2 752.8 764.9 3 856.9 843.0 4 901.0 796.1 5 784.1 763.2 6 863.2 821.3 7 881.3 783.4 8 739.4 695.5 9 749.5 771.6 10 848.6 795.7

Solution Steps

The mean emissions at \(40^{\circ} \mathrm{F}\) is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{8215.5}{10} = 821.55 \text{ ppb} \]

The mean emissions at \(80^{\circ} \mathrm{F}\) is calculated similarly: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{7852.5}{10} = 785.25 \text{ ppb} \]

To compare the means of the two temperatures, we perform a paired sample t-test. The standard error (SE) of the mean difference is calculated as: \[ SE = \frac{s_d}{\sqrt{n}} = \frac{41.6258}{\sqrt{10}} = 13.1632 \]

The test statistic \(t\) is calculated using the formula: \[ t = \frac{\bar{d}}{SE} = \frac{36.3}{13.1632} = 2.7577 \]

For a two-tailed test at \(\alpha = 0.05\), the critical value is found using the t-distribution: \[ t_{\alpha/2, df} = t_{(0.025, 9)} = 2.2622 \]

The p-value is calculated as: \[ P = 2 \times (1 - T(|t|)) = 2 \times (1 - T(2.7577)) = 0.0222 \]

The results of the paired sample t-test are summarized as follows:

• \(t\)-statistic: \(2.7577\)
• p-value: \(0.0222\)
• Critical value: \(2.2622\)
• Mean difference: \(36.3\)
• Standard deviation of differences: \(41.6258\)
• Standard error of mean difference: \(13.1632\)

Final Answer