An Engineering Research Center Claims Through Use Of
An Engineering Research Center Claims That Through The Use Of A New Co
An engineering research center claims that through the use of a new computer control system, automobiles should achieve, on average, an addition of 3 miles per gallon of gas. A random sample of 100 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 2.4, and the sample standard deviation was 1.8 miles per gallon. Test the hypothesis that the population mean is at least 3 miles per gallon. Find the p-value for this test, and interpret your findings.
Tests of Between-Subjects Effects
Dependent Variable: The grade level
Source | Type III Sum of Squares | df | Mean Square | F | Sig.
Corrected Model | 0.833a | 1 | 0.833 | 0.169 | 0.078
Intercept | 282.773 | 1 | 282.773 | ... | 0.000
Student Type | ... | 1 | ... | ... | 0.078
Error | 31 | ... | ... | ... | ...
Total | 314 | | | |
Corrected Total | 31 | | | |
Note: R Squared = 0.026 (Adjusted R Squared = 0.018)
Paper For Above instruction
The investigation of a new computer control system’s impact on automobile fuel efficiency involves hypothesis testing to determine whether the observed data supports the claimed improvement of at least 3 miles per gallon (mpg). This paper explores both the one-sample hypothesis test regarding the mean increase in miles per gallon and the analysis of the main effect of student type in a separate ANOVA study, interpreting the statistical significance and practical implications of the findings.
Hypothesis Test for the Mean Increase in Miles Per Gallon
Firstly, investigating whether the new computer control system produces an average fuel efficiency gain of at least three mpg involves setting up the null hypothesis (H0) and alternative hypothesis (Ha). The hypotheses are formalized as follows:
- H0: μ ≥ 3 mpg (The population mean increase is at least 3 mpg.)
- Ha: μ
Given the sample data—sample mean (x̄) of 2.4 mpg, standard deviation (s) of 1.8 mpg, and sample size (n) of 100—the test statistic is calculated using a one-sample t-test formula:
t = (x̄ - μ₀) / (s / √n)
where μ₀ = 3 mpg.
Plugging in the known figures:
t = (2.4 - 3) / (1.8 / √100) = (-0.6) / (1.8 / 10) = (-0.6) / 0.18 = -3.33
The degrees of freedom (df) for this test is n - 1 = 99.
Using a t-distribution table or statistical software, the p-value corresponding to t = -3.33 with df = 99 is approximately 0.0009. Because this is a one-sided test (we are testing if the mean is less than 3), the p-value indicates the probability of observing such a sample mean or less if the null hypothesis is true.
Since the p-value (~0.0009) is less than common significance levels (e.g., α = 0.05), we reject H0, suggesting that the true average increase in miles per gallon is statistically significantly less than 3 mpg. Therefore, the data does not support the claim that the new control system yields at least a 3 mpg increase.
Interpreting the p-value
The p-value of approximately 0.0009 quantifies the evidence against the null hypothesis. A very low p-value implies strong evidence that the true mean is less than 3 mpg. Practically, this suggests that, based on the sample, the control system's effectiveness in increasing miles per gallon may fall short of the claimed 3 mpg gain, raising questions about its practical impact.
Analysis of Main Effect of Student Type in ANOVA
The second part of the provided data pertains to a different statistical analysis: a between-subjects ANOVA examining whether the grade level differs significantly between student groups. The results show an F statistic of F(1, 118) = ?, with p = 0.078, indicating the p-value associated with the main effect of student type.
The F value is missing explicitly; however, the p-value of 0.078 suggests that the F statistic is close to the critical value for significance at the 0.05 level but does not reach it. In the context of ANOVA, a p-value of 0.078 exceeds the conventional significance threshold, indicating that there is not enough evidence to reject the null hypothesis that the mean grade levels are the same for the two student groups.
The R-squared value of 0.026 indicates that approximately 2.6% of the variance in grade levels is explained by student type. The adjusted R-squared of 0.018 further suggests a small effect size, emphasizing that differences in grade levels across student groups are minimal and statistically non-significant.
Therefore, the conclusion is that student type does not have a statistically significant main effect on grade level within this sample, implying the average grade is roughly comparable across the groups examined.
Conclusion
In sum, the hypothesis testing on the fuel efficiency data provides compelling evidence that the new control system does not produce the claimed average increase of 3 mpg, with the data indicating a statistically significant shortfall. Conversely, the ANOVA analysis reveals that the main effect of student type on grade level is not statistically significant, suggesting equivalence in averages among the groups. These findings underscore the importance of rigorous statistical evaluation when assessing technological claims and educational factors, ensuring decisions are grounded in empirical evidence.
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