Assume You Are Working At The Consumer Protection Agency ✓ Solved

Assume You Are Working At The Consumer Protection Agency Recently Yo

Assume you are working at the Consumer Protection Agency. Recently, you have been getting complaints about the highway gas mileage of a new minivan. The car company agrees to allow you to select randomly 41 of its new minivans to test their highway mileage. The company claims that its minivans get 28 miles per gallon on the highway. Your test results show a sample mean of 26.7 and a sample standard deviation of 4.2.

Part 1 (Confidence Interval):

• Calculate a 95% confidence interval around your sample mean.
• Is the claimed mean inside your confidence interval?
• What does your result mean, in terms of the company's claim?

Part 2 (Two-tail test):

• List the null and alternative hypotheses for the appropriate test.
• Use alpha = 0.05. Find the critical value(s) and calculate the observed value of the test statistic.
• Is the observed test statistic in the critical (rejection) region?
• Will the p-value be higher or lower than your alpha? What does this result mean, in terms of the company's claim?

Part 3 (One-tail test):

• List the null and alternative hypotheses for the test.
• Use alpha = 0.05.
• Find the critical value and calculate the observed value of the test statistic.
• Is the observed test statistic in the critical region?
• Will the p-value be higher or lower than your alpha?
• What does this result mean, in terms of the company's claim?

Part 4 (Conclusion):

• What conclusions did you reach?
• What did you learn from each method of checking the claim for means?
• Were there important differences between methods? Which method would you prefer?
• Which carries a higher risk of a type I error?
• Based on this experience, why do you think it’s important to decide on the method before conducting the test?
• Based on your results, do you support the company's claim?
• What action, if any, should the company take?

Sample Paper For Above instruction

The evaluation of the company's claim regarding the highway gas mileage of its new minivans requires a combination of statistical methods, primarily confidence intervals and hypothesis testing. These methods help determine whether the sample data supports or contradicts the manufacturer's assertion that the minivans average 28 miles per gallon (mpg). In this paper, we analyze the data obtained from testing 41 minivans, which yielded a sample mean of 26.7 mpg and a standard deviation of 4.2 mpg. We perform a 95% confidence interval calculation, conduct both two-tailed and one-tailed hypothesis tests, and interpret the findings to draw meaningful conclusions about the company's claim.

Part 1: Confidence Interval Analysis

To calculate the 95% confidence interval (CI) for the true mean of the minivan's highway mileage, we employ the standard formula for a t-distribution, given the sample size and standard deviation:

CI = x̄ ± t_α/2 * (s / √n)

Where:

• x̄ = 26.7 mpg (sample mean)
• s = 4.2 mpg (sample standard deviation)
• n = 41 (sample size)
• t_α/2 = t-value for 95% confidence and 40 degrees of freedom

Using a t-distribution table, for 40 degrees of freedom at α/2=0.025, the t-value is approximately 2.021.

Calculating the standard error:

SE = 4.2 / √41 ≈ 4.2 / 6.403 ≈ 0.656

Therefore, the confidence interval is:

CI = 26.7 ± 2.021 * 0.656 ≈ 26.7 ± 1.324

Constructed CI: [25.376 mpg, 28.024 mpg]

The company's claimed mean of 28 mpg lies at the upper bound of the confidence interval but is technically within the interval boundary, suggesting that, based on this sample, the claim cannot be rejected at 95% confidence.

Part 2: Two-Tailed Hypothesis Test

The hypotheses are formulated as:

• Null hypothesis (H₀): μ = 28 mpg
• Alternative hypothesis (H_a): μ ≠ 28 mpg

Significance level (α): 0.05.

Calculating the test statistic (t):

t = (x̄ – μ₀) / (s / √n) = (26.7 – 28) / 0.656 ≈ -1.3 / 0.656 ≈ -1.98

The critical t-values for a two-tailed test at α=0.05 and df=40 are approximately ±2.021.

Since |t| = 1.98

The p-value associated with t = -1.98 and df=40 is slightly above 0.05, approximately 0.054.

Therefore, the p-value exceeds the significance level, indicating the evidence is insufficient to reject the null hypothesis. Thus, statistically, there's not enough evidence to conclude that the true mean differs from 28 mpg, aligning with the company's claim.

Part 3: One-Tail Hypothesis Test

The hypotheses are formatted as:

• Null hypothesis (H₀): μ ≥ 28 mpg
• Alternative hypothesis (H_a): μ

The critical t-value for a one-tailed test at α=0.05 with 40 degrees of freedom is approximately –1.684.

The observed t-value remains –1.98.

Since –1.98

The p-value for t = –1.98 in a one-tailed test is about 0.027, which is less than 0.05.

This indicates statistically significant evidence at the 5% level that the true mean is less than 28 mpg, suggesting the company's claim may be overstated.

Part 4: Conclusions and Implications

Based on the confidence interval and hypothesis testing, the findings are somewhat nuanced. The 95% confidence interval includes 28 mpg, implying that we cannot definitively reject the claim with high confidence. The two-tailed test yields a p-value just above 0.05, indicating weak evidence against the null hypothesis. However, the one-tailed test strongly suggests that the true mean may be less than 28 mpg, aligning with consumer complaints.

This discrepancy between methods highlights the importance of selecting the appropriate hypothesis test based on the context. The one-tailed test is more powerful for detecting if the actual mileage is less than claimed, but it also carries a higher risk of a Type I error if misapplied. Conversely, the two-tailed test is more conservative, suitable when deviations in either direction are of concern.

From this analysis, the company should consider that consumer complaints are supported by statistical evidence that the actual highway mileage may be below the claimed 28 mpg. A prudent course of action includes further testing with larger samples and transparency in reporting real-world performance data, which can enhance consumer trust and avoid potential legal and reputational risks.

References

• Agresti, A., & Finlay, B. (2009). Statistical methods for the social sciences (4th ed.). Pearson.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W.H. Freeman.
• Rosenbaum, P. R. (2002). Observational Studies. Springer.
• Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods (8th ed.). Iowa State University Press.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences (9th ed.). Pearson.
• Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.
• Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge.
• Gibbons, J. D., & Chakraborti, S. (2011). Nonparametric Statistical Inference (5th ed.). CRC Press.