Problem Description
In this Statistical Analysis assignment, we aim to determine whether the average vehicle price in a town area deviates significantly from the 80th percentile of the dataset. To do this, we need to conduct a hypothesis test. The choice between a z-test and a t-test hinges on two essential factors: the sample size and our knowledge of the population standard deviation.
Given a relatively small sample size (n = 10) and an unknown population standard deviation, it is appropriate to utilize a t-test. The t-test is designed to accommodate smaller sample sizes and employs the sample standard deviation to estimate the population standard deviation, making it a more suitable choice for cases where the population standard deviation is unknown. Therefore, for this scenario, a t-test is the appropriate statistical method for conducting the hypothesis test.
Data Summary:
- Sample Mean (x̅): $85,623
- Sample Standard Deviation (s): $59,931.02
- Population Mean (μ – 80th percentile): $114,592
- Sample Size (n): 10
Hypothesis:
- Null Hypothesis (Ho): The average vehicle price in the town area is equal to the 80th percentile ($114,592). μ = $114,592
- Alternative Hypothesis (Ha): The average vehicle price in the town area is not equal to the 80th percentile ($114,592). μ ≠ $114,592
T-Test Statistic: To analyze our hypothesis, we'll perform a t-test, considering our small sample size (n = 10) and an unknown population standard deviation. The t-test accommodates smaller sample sizes and uses the sample standard deviation to estimate the population standard deviation. This robust approach is suitable for small samples where the population standard deviation is not known.
T-Test Statistic (TS) = (Sample Mean - Population Mean) / (Sample Standard Deviation / √Sample Size) TS = ($85,623 - $114,592) / ($59,931.02 / √10) = -1.5286
P-Value: The p-value, which measures the probability of obtaining results as extreme as the ones observed in our sample, is a crucial part of the hypothesis test.
P-value ≈ 0.1607
Conclusion: After a thorough analysis of the hypothesis test results, it becomes clear that the data does not provide substantial support for the claim that the average vehicle price in the town area significantly differs from the 80th percentile of the dataset.
The calculated p-value is approximately 0.1607, which exceeds our chosen significance level (α = 0.05). Therefore, we are unable to reject the null hypothesis (Ho). This implies that, within the scope of this analysis, there is no strong statistical evidence to suggest a meaningful difference between the average vehicle price and the 80th percentile value.