To compute the P-value for the test involving $\mu_d = \mu_1 - \mu_2$ based on the provided data, we will perform the following steps:

1. Calculate the differences ($d_i = x_{40°F} - x_{80°F}$).
2. Compute the mean of the differences ($\bar{d}$).
3. Compute the standard deviation of the differences ($s_d$).
4. Perform a t-test to find the test statistic ($t$).
5. Use the t-distribution to calculate the P-value.

Let me calculate it for you.import numpy as np from scipy.stats import t

Data for 40°F and 80°F emissions

emissions_40 = np.array([851.4, 775.8, 872.8, 937.2, 832.5, 880.2, 944.5, 806.9, 763.0, 933.5]) emissions_80 = np.array([839.7, 795.8, 867.9, 837.0, 817.8, 844.1, 830.6, 750.1, 796.1, 882.1])

Calculate differences

differences = emissions_40 - emissions_80

Compute mean and standard deviation of differences

mean_diff = np.mean(differences) std_diff = np.std(differences, ddof=1) # Sample standard deviation n = len(differences) # Number of paired observations

Test statistic (t-value)

t_stat = mean_diff / (std_diff / np.sqrt(n))

Degrees of freedom

df = n - 1

Compute the two-tailed p-value

p_value = 2 * (1 - t.cdf(abs(t_stat), df))

mean_diff, std_diff, t_stat, p_value(33.65999999999999, 47.98993412975036, 2.218012338868391, 0.053738082618467686)(33.65999999999999, 47.98993412975036, 2.218012338868391, 0.053738082618467686)successThe results are as follows:

• Mean difference ($\bar{d}$): 33.6600
• Standard deviation of differences ($s_d$): 47.9900
• Test statistic ($t$): 2.2180
• P-value: 0.0537

Would you like to delve deeper into interpreting the results?

Related Questions:

1. What is the null hypothesis in this test?
2. How do we determine statistical significance for this P-value?
3. Can we conclude that temperature impacts NOx emissions significantly?
4. Should we conduct a one-tailed test instead?
5. How does sample size affect the outcome of this test?

Tip: Ensure your significance level ($\alpha$) is predetermined before interpreting the P-value.