Question

An automobile manufacturer claims that their car has a 59.4 miles/gallon (mpg) rating. An independent testing firm has been contracted to test the mpg for this car. After testing 751 cars, they found a mean mpg of 59.3 with a standard deviation of 2 mpg. Is there sufficient evidence at the 0.05 level that the cars have an incorrect manufacturer's mpg rating?

Answer 1

At the 0.05 significance level, there is insufficient evidence to conclude that the cars have an incorrect manufacturer's mpg rating. The calculated mean mpg of 59.3 falls within the acceptable range considering the manufacturer's claim of 59.4 mpg, and the slight difference is likely due to sampling variability.

Step-by-step explanation:

To assess whether there is sufficient evidence to dispute the manufacturer's mpg rating claim, we perform a hypothesis test. The null hypothesis
`(\(H_0\))` posits that the mean mpg is equal to the claimed value (59.4 mpg), while the alternative hypothesis
`(\(H_1\))` suggests that the mean mpg differs from the claimed value. The test is conducted at a 0.05 significance level.

Mathematically, this can be expressed as:

`\[ H_0: \mu = 59.4 \, \text{mpg} \]`

`\[ H_1: \mu \\eq 59.4 \, \text{mpg} \]`

Using the provided data, specifically the sample mean
`(\(\bar{x}\) = 59.3)` and standard deviation s = 2, we can conduct a t-test to compare the sample mean to the claimed value. The calculated t-statistic is then compared to the critical t-value at a 0.05 significance level with 750 degrees of freedom (751 cars - 1). If the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis, indicating insufficient evidence to dispute the manufacturer's claim.

In conclusion, the calculated mean mpg of 59.3 is within the acceptable range, and the hypothesis test does not provide enough evidence to reject the manufacturer's claim at the 0.05 significance level. Any observed difference is likely attributable to random sampling variability rather than a significant discrepancy in the mpg rating.