A Particular Brand Of Tires Claims Its Deluxe Tire Average

Question

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly inconsistent with the claim? The assignment involves testing the claim of a tire manufacturer that their deluxe tires last at least 50,000 miles on average. Using the data collected from a survey of 28 tires, with a sample mean of 46,500 miles and a known standard deviation of 8,000 miles, we are to perform a hypothesis test at the 0.05 significance level. The goal is to determine whether the observed data is highly inconsistent with the manufacturer's claim that the average lifespan of the tires is at least 50,000 miles.

Dr. Jack HW Helper · Accepted Answer

Introduction The validity of manufacturing claims in the automotive industry often relies on statistical evidence. In this case, we evaluate whether a particular brand of tires truly meets its advertised standard of longevity. The company's claim is that each tire lasts an average of at least 50,000 miles, supported by previous studies. Our goal is to assess this claim with recent survey data using hypothesis testing. Hypotheses Formulation The null hypothesis (H₀) states that the population mean lifespan of the tires is at least 50,000 miles: H₀: μ ≥ 50,000 miles. The alternative hypothesis (H₁) indicates that the true mean lifespan is less than 50,000 miles: H₁: μ This is a one-tailed test because we are assessing whether the current data significantly contradicts the manufacturer's claim by being lower than the claimed mean. Data Summary From the survey: Sample size (n): 28 tires Sample mean (x̄): 46,500 miles Standard deviation (σ): 8,000 miles (known) Given the large sample size relative to the population standard deviation, we will perform a z-test. Test Statistic Calculation The z-score formula when the population standard deviation is known is: $$ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} $$ where: - $\bar{x} = 46500$ - $\mu_0 = 50000$ - $\sigma = 8000$ - $n = 28$ Calculating: $$ z = \frac{46500 - 50000}{8000 / \sqrt{28}} = \frac{-3500}{8000 / 5.2915} = \frac{-3500}{1512.9} \approx -2.31 $$ Decision Rule At a significance level ($\alpha$) of 0.05, the critical z-value for a one-tailed test is approximately -1.645. Since the calculated z-value (-2.31) is less than -1.645, we reject the null hypothesis. Conclusion The sample data provides sufficient evidence at the 0.05 significance level to conclude that the mean lifespan of the tires is significantly less than 50,000 miles. Therefore, the data is highly inconsistent with the manufacturer's claim, indicating potential issues with tire durability or exaggeration of performance. Discussion This an

A Particular Brand Of Tires Claims Its Deluxe Tire Average ✓ Solved

Sample Paper For Above instruction

Introduction

Hypotheses Formulation

Data Summary

Test Statistic Calculation

Decision Rule

Conclusion

Discussion

References