Imagine That Car Middle East: A Car Magazine Comparison
Imagine That Car Middle East A Car Magazine Is Comparing That Repair
Imagine that Car Middle East, a car magazine is comparing the repair costs incurred during the first four years on two types of family cars, the Camry and Altima. Random samples of 60 Camry cars and 66 Altima cars are selected. The mean repair cost is $1,679 for Camry and $1,485 for Altima, with sample standard deviations of $267 and $326, respectively. Assume that the distribution is normally distributed with equal variances.
a. Find the pooled standard deviation.
b. Based on the sample information, determine the 90% confidence interval estimate for the difference between the two population means.
Paper For Above instruction
Introduction
The comparison of repair costs between different vehicle models provides valuable insights for consumers, manufacturers, and industry analysts. In this context, the focus is on evaluating whether there is a significant difference in the first four-year repair costs between Toyota Camry and Nissan Altima using statistical methods. This paper describes the procedures for calculating the pooled standard deviation and constructing a 90% confidence interval for the difference in mean repair costs, based on sample data obtained from random samples of each vehicle type.
Data Overview and Assumptions
The sample statistics provided include:
- Sample size for Camry (n₁) = 60
- Sample mean for Camry (x̄₁) = $1,679
- Sample standard deviation for Camry (s₁) = $267
- Sample size for Altima (n₂) = 66
- Sample mean for Altima (x̄₂) = $1,485
- Sample standard deviation for Altima (s₂) = $326
The assumption of normally distributed populations and equal variances underpins the application of pooled variance methods and confidence interval calculations for the difference between means.
Part A: Calculation of Pooled Standard Deviation
The pooled standard deviation synthesizes individual sample standard deviations into a single estimate of the common population standard deviation. It is calculated as follows:
\[
s_p = \sqrt{\frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2}}
\]
Substituting the values:
\[
s_p = \sqrt{\frac{(60 - 1) \times 267^2 + (66 - 1) \times 326^2}{60 + 66 - 2}}
\]
\[
s_p = \sqrt{\frac{59 \times 71,289 + 65 \times 106,276}{124}}
\]
\[
s_p = \sqrt{\frac{4,204,851 + 6,907,940}{124}} = \sqrt{\frac{11,112,791}{124}} \approx \sqrt{89,644.58} \approx 299.41
\]
Thus, the pooled standard deviation is approximately $299.41.
Part B: Constructing the 90% Confidence Interval for the Difference of Means
To estimate the range within which the true difference between the population means lies, the confidence interval formula is used:
\[
(x̄_1 - x̄_2) \pm t_{\alpha/2, df} \times s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}
\]
where:
- \((x̄_1 - x̄_2) = 1,679 - 1,485 = 194\)
- \(s_p \approx 299.41\) (from Part A)
- Degrees of freedom \(df = n_1 + n_2 - 2 = 124\)
Using a t-distribution table or calculator, for a 90% confidence level and 124 degrees of freedom:
\[
t_{0.05, 124} \approx 1.98
\]
Calculating the standard error (SE):
\[
SE = s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = 299.41 \times \sqrt{\frac{1}{60} + \frac{1}{66}} \approx 299.41 \times \sqrt{0.01667 + 0.01515} \approx 299.41 \times \sqrt{0.03182}
\]
\[
SE \approx 299.41 \times 0.1784 \approx 53.43
\]
Constructing the margin of error (ME):
\[
ME = t_{0.05, 124} \times SE = 1.98 \times 53.43 \approx 105.76
\]
Finally, the confidence interval:
\[
194 \pm 105.76 \Rightarrow \text{Lower bound: } 88.24; \quad \text{Upper bound: } 299.76
\]
Interpretation: At the 90% confidence level, the true difference in average repair costs between the Camry and Altima can be estimated to be between approximately $88.24 and $299.76, with the Camry incurring higher repair costs on average.
Discussion
The statistical analysis indicates a significant difference in the first four-year repair costs between the Toyota Camry and Nissan Altima. The confidence interval suggests that, on average, the Camry’s repair expenses are between $88 and $300 higher than those of the Altima within the specified period. This difference might influence consumer choice, with some consumers possibly preferring the Altima based on lower expected repair costs.
The assumption of equal variances is supported by the similar magnitude of sample standard deviations, and the use of pooled variance is statistically justified given the normality assumption and sample sizes. However, for more precise analysis, tests for equality of variances such as Levene’s test could further validate this assumption.
Conclusion
This study successfully calculated the pooled standard deviation and constructed a 90% confidence interval for the difference in mean repair costs between the Camry and Altima. The findings suggest that the Camry tends to have higher repair expenses within the first four years of ownership, which could be a factor in consumer decision-making and warranty considerations.
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