A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 414 gram setting. It is believed that the machine is underfilling the bags. A 8 bag sample had a mean of 407 grams with a standard deviation of 18 . A level of significance of 0.025 will be used. Assume the population distribution is approximately normal. Is there sufficient evidence to support the claim that the bags are underfilled?
There is not sufficient evidence to support the claim that the bags are underfilled.
There is sufficient evidence to support the claim that the bags are underfilled.

Step 1 ：The problem is asking us to determine if there is sufficient evidence to support the claim that the bags of banana chips are underfilled. The manufacturer believes that the bag filling machine, set at 414 grams, is underfilling the bags. A sample of 8 bags was taken, with a mean weight of 407 grams and a standard deviation of 18 grams. We are to use a level of significance of 0.025. The population distribution is assumed to be approximately normal.

Step 2 ：We will perform a one-sample t-test. The null hypothesis (\(H_0\)) is that the mean weight of the bags is 414 grams. The alternative hypothesis (\(H_1\)) is that the mean weight is less than 414 grams.

Step 3 ：The sample mean (\(\bar{x}\)) is 407 grams, the population mean (\(\mu\)) is 414 grams, the sample standard deviation (\(s\)) is 18 grams, and the sample size (\(n\)) is 8.

Step 4 ：We calculate the t-statistic using the formula \(t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\). Substituting the given values, we get a t-statistic of approximately -1.10.

Step 5 ：Using a significance level (\(\alpha\)) of 0.025, we compare the calculated t-statistic with the critical t-value from the t-distribution table. If the calculated t-statistic is less than the critical t-value, we reject the null hypothesis in favor of the alternative hypothesis.

Step 6 ：Since the calculated t-statistic is less than the critical t-value, we reject the null hypothesis. Therefore, there is sufficient evidence to support the claim that the bags are underfilled.

Step 7 ：\(\boxed{\text{Final Answer: There is sufficient evidence to support the claim that the bags are underfilled.}}\)