A snack company produces potato chips. The company claims that the mean weight of a bag is $50$ grams. To monitor quality, a random sample of $16$ bags is selected, and the mean weight is found to be $51$ grams with a standard deviation of $0.4$ grams. Assume the weights are approximately normally distributed. Using a significance level of $\alpha =0.05$ (two-tailed), are the bags meeting the claimed mean weight? (Critical $t$-values for $df=15$ is approximately $\pm 2.131$)