An automobile manufacturer claims that their car has a 59.1 miles per gallon (MPG) rating. An independent testing firm tested 51 cars and found a mean MPG of 59.3 with a standard deviation of 1.4 MPG. Is there sufficient evidence at the 0.1 significance level to conclude that the manufacturer's MPG rating is incorrect?

Using a one-sample z-test, the calculated test statistic of 1.0194 does not exceed the critical Z-value of ±1.645 for a 0.1 level of significance. Therefore, there is not sufficient evidence to claim that the manufacturer's MPG rating is incorrect.

Explanation:

To determine if there is sufficient evidence that the cars have an incorrect manufacturer's MPG rating, we can conduct a hypothesis test. The null hypothesis (H0) is that the true mean MPG is equal to the manufacturer's claim of 59.1 MPG, while the alternative hypothesis (H1) is that the true mean MPG is not equal to 59.1 MPG.

We need to calculate the test statistic using the given sample mean, the population mean under the null hypothesis, the sample standard deviation, and the number of observations. The formula for the test statistic in a one-sample z-test is:

Z = (sample mean - population mean) / (standard deviation / sqrt(n))

Using the data provided:

Plugging these into the formula gives us:

Z = (59.3 - 59.1) / (1.4 / sqrt(51))
= 0.2 / (1.4 / sqrt(51))
= 0.2 / (1.4 / 7.1418)
= 0.2 / 0.1962
= 1.0194

Next, we compare the calculated Z-value with the critical Z-value(s) for alpha = 0.1 (two-tailed test), which are approximately ±1.645. Since the calculated Z-value of 1.0194 does not exceed the critical value, we do not reject the null hypothesis. This means that there is not sufficient evidence at the 0.1 level to claim that the manufacturer's MPG rating is incorrect.