A random sample of 8 cereal boxes from one supplier was selected. After weighing each box, the recorded masses in grams were: $261, 249, 259, 252, 255, 256, 258, 254$. Determine unbiased estimates for the population mean and variance.
The supplier says that the mean mass of cereal boxes is $253\text{ g}$. A quality control officer thinks the mean mass is actually greater than $253\text{ g}$. To examine this claim, he weighs a random sample of $100$ cereal boxes and gets a total mass of $25\,360\text{ g}$. If the population standard deviation of the masses is $3.5\text{ g}$, test at the $5\%$ significance level whether the population mean mass exceeds $253\text{ g}$.
Explain briefly whether the statement ‘This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.’ is true or false.