4. A car manusacturer wants to test a new engine to determine whether it meets new air pollution standard (true mean emission must be less than 20 parts per million of carbon). Ten engines manusactured for testing purposes yield the following emission levels: \[ 15.9,16.2,21.5,19.5,16.4,18.4,19.8,17.8,12.7 \text {, } \] 14.8 Assume that the normal distribution assumption is satisfied for this sample. Do the data supple sufficient evidence to allow the manufacturer to conclude that this type of engine meets the pollution standard? use $a=0.01$

Step 1 ：We are given a set of emission levels from ten engines: \(15.9, 16.2, 21.5, 19.5, 16.4, 18.4, 19.8, 17.8, 12.7, 14.8\). We want to test whether the true mean emission level is less than 20 parts per million of carbon.

Step 2 ：We first calculate the sample mean and standard deviation of the emission levels. The sample mean is \(17.3\) and the standard deviation is approximately \(2.62\).

Step 3 ：We then perform a hypothesis test. The null hypothesis is that the true mean emission level is equal to or greater than 20 parts per million, and the alternative hypothesis is that the true mean emission level is less than 20 parts per million. We use a significance level of \(0.01\).

Step 4 ：We calculate the test statistic, which follows a t-distribution with \(n-1\) degrees of freedom under the null hypothesis. The test statistic is approximately \(-3.26\).

Step 5 ：We calculate the p-value, which is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. The p-value is approximately \(0.0049\).

Step 6 ：Since the p-value is less than the significance level of \(0.01\), we reject the null hypothesis. This means that the data provides sufficient evidence to conclude that the true mean emission level of the new engine is less than 20 parts per million of carbon.

Step 7 ：Final Answer: The data provides sufficient evidence to conclude that the true mean emission level of the new engine is less than 20 parts per million of carbon. Therefore, the manufacturer can conclude that this type of engine meets the pollution standard. The test statistic is approximately \(-3.26\) and the p-value is approximately \(0.0049\). \(\boxed{Reject \: the \: null \: hypothesis}\).