A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. 261 249 259 252 255 256 258 254 Find unbiased estimates of the population mean and variance.

1. (a)A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. 261 249 259 252 255 256 258 254 Find unbiased estimates of the population mean and variance.[3]
2. (b)The supplier claims that the mean mass of boxes of cereal is 253g. A quality control officer suspects that the mean mass is actually more than 253 g. In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25 360 g.
3. (b)(i)Given that the population standard deviation of the masses is 3.5 g, test at the 5% significance level whether the population mean mass is more than 253 g.[5]
4. (b)(ii)An employee says, “This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.' Explain briefly whether this statement is true or not.[1]