To determine if the new engine is more economical than the old engine, we can use a statistical test. Since we are comparing the means of two independent samples, the two-sample t-test is appropriate. Here's the step-by-step plan:

1. Hypotheses:

   • Null Hypothesis ($H_0$): There is no difference in the fuel economy of the new and old engines ($\mu_{\text{new}} = \mu_{\text{old}}$).
   • Alternative Hypothesis ($H_1$): The new engine is more economical than the old engine ($\mu_{\text{new}} > \mu_{\text{old}}$).

2. Level of significance: Typically $\alpha = 0.05$.

3. Data:

   • Old engine: $8, 9, 7.5, 8.5, 6, 9, 9, 10, 7, 8.5, 6, 10, 9, 8, 9, 5, 9.5, 10, 8$
   • New engine: $10, 9, 9, 6, 9, 11, 11, 8, 9, 6.5, 7, 9, 10, 8, 9, 10, 9, 12, 11.5, 10, 7, 10, 8.5$

4. Calculate:

   • Mean and standard deviation for each sample.
   • Perform the two-sample t-test.

Let me calculate this for you. numpy as np from scipy.stats import ttest_ind

Data

old_engine = [8, 9, 7.5, 8.5, 6, 9, 9, 10, 7, 8.5, 6, 10, 9, 8, 9, 5, 9.5, 10, 8] new_engine = [10, 9, 9, 6, 9, 11, 11, 8, 9, 6.5, 7, 9, 10, 8, 9, 10, 9, 12, 11.5, 10, 7, 10, 8.5]

Calculate means and standard deviations

old_mean = np.mean(old_engine) new_mean = np.mean(new_engine) old_std = np.std(old_engine, ddof=1) new_std = np.std(new_engine, ddof=1)

Perform two-sample t-test (one-tailed)

t_stat, p_value = ttest_ind(new_engine, old_engine, alternative='greater')

Summarize results

results = { "Old Engine Mean": old_mean, "New Engine Mean": new_mean, "Old Engine Std Dev": old_std, "New Engine Std Dev": new_std, "T-Statistic": t_stat, "P-Value": p_value }

import pandas as pd results_df = pd.DataFrame([results]) import ace_tools as tools; tools.display_dataframe_to_user(name="Engine Comparison Results", dataframe=results_df)The analysis results have been presented in a table for your review. Based on the p-value (0.038), there is sufficient evidence at a 0.05 significance level to conclude that the new engine is more economical than the old engine.

Would you like more detailed explanations about the calculations or have further questions? Here are some related questions to consider:

1. How is the t-statistic calculated in a two-sample t-test?
2. Why is a one-tailed test used instead of a two-tailed test in this scenario?
3. What are the assumptions of a two-sample t-test, and were they satisfied here?
4. How does the sample size affect the reliability of this test?
5. What other tests could be used to compare these samples?

Tip: Always check the assumptions of a statistical test (e.g., normality and equal variances) to ensure valid conclusions.