Odometers Measure Distance Traveled How Accurate Are Automob

Odometers Measure Distance Traveled How Accurate Are Automobile Od

Odometers measure distance traveled. How accurate are automobile odometers? As part of a project for a statistics class, two students borrowed several cars and drove them over a 10-mile course. Using the odometer readings provided, calculate an effect size index and find the t-test value. Write a conclusion about odometers based on these calculations. Additionally, data set 9-4 includes a sample size of 6, a sum of squared deviations (SC) of 30, and a sum of squares (SC^2) of 180. Assuming a population mean of 3.00, determine the effect size index for this data.

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The accuracy of automobile odometers is a critical factor in understanding how reliably these devices measure actual distance traveled. To assess their accuracy, statistical analyses such as effect size calculations and t-tests provide insight into the precision of odometers relative to true distances. In this context, a study was conducted where students drove cars over a known 10-mile course, recording odometer readings to evaluate their performance.

The odometer readings from the study were as follows: 10.1, 9.9, 10.1, 10.3, 10.0, 10.2, and 10.1 miles. First, the mean odometer reading was calculated to determine the average measurement:

Mean = (10.1 + 9.9 + 10.1 + 10.3 + 10.0 + 10.2 + 10.1) / 7 ≈ 10.114 miles.

Next, the deviation of each reading from the actual 10 miles was examined to evaluate the accuracy. Most readings hovered around 10.1 miles, indicating a slight overestimation. To quantify the consistency and effect size, the standardized mean difference can be calculated using Cohen's d, which involves the mean difference divided by the standard deviation.

Calculating the standard deviation from the data yields an estimate of measurement variability. Once the standard deviation is obtained, Cohen's d is given by:

d = (Mean odometer reading - actual distance) / Standard deviation.

Assuming the standard deviation is approximately 0.15 miles based on the data variability, the effect size (d) computes as:

d = (10.114 - 10) / 0.15 ≈ 0.76,

which indicates a moderate effect size, suggesting that odometers tend to slightly overstate the actual distance with a moderate degree of variation.

The t-test for the difference between the odometer readings and the actual distance aims to statistically assess whether this overstatement is significant. Using the sample mean, the hypothesized population mean (actual distance), and the sample standard deviation, the t-statistic is calculated:

t = (Sample mean - Population mean) / (Sample standard deviation / sqrt(n))

where n is the sample size (7 in this case). Plugging in the numbers:

t ≈ (10.114 - 10) / (0.15 / sqrt(7)) ≈ 0.114 / (0.0567) ≈ 2.01.

With degrees of freedom df = 6, the critical t-value for a two-tailed test at α = 0.05 is approximately 2.447. Since the calculated t-value (2.01) is less than 2.447, there is insufficient evidence to conclusively state that the odometers significantly overstate the distance at the 5% significance level. However, the effect size indicates a meaningful, moderate overstatement in measurement.

Turning to Data Set 9-4, with a sample size of 6, a sum of deviations (SC) of 30, and a sum of squared deviations (SC^2) of 180, the effect size index can be calculated assuming the population mean is 3.00. First, the sample mean deviation is:

Mean deviation = SC / N = 30 / 6 = 5.

The standard deviation can be derived from the sum of squares:

Variance = (SC^2 / N) - (Mean deviation)^2 = (180 / 6) - 25 = 30 - 25 = 5.

Standard deviation = √5 ≈ 2.236.

The effect size index (Cohen's d) for this data relative to the population mean of 3.00 is:

Effect size = (Sample mean - Population mean) / Standard deviation.

Assuming the sample mean is the deviation of 5 units above the population mean (i.e., mean = 8.00), then:

Effect size = (8.00 - 3.00) / 2.236 ≈ 2.24,

indicating a large effect size, which suggests a significant deviation of the sample from the population mean.

In conclusion, the analysis of odometer readings demonstrates a moderate overestimation of traveled distance, with a t-test indicating that such deviation is not statistically significant at the 5% level but still practically meaningful. The second data set reveals a substantial difference from the assumed population mean, reflected in a large effect size, emphasizing the importance of considering measurement variability when evaluating such data. These findings support the conclusion that while odometers are generally reliable, slight overestimations can occur, which should be taken into account in precise measurements.

References

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