RI Hypothesis Testing Tutorial Q2

A hand sanitiser company, Cleanser, claims that the mean volume of each bottle of their hand sanitiser is \(15\) ml. The consumer association took a random sample of \(80\) bottles and measured the volume of each bottle. It was found that the sample mean and the sample variance were \(\bar{x}\) and \(2\) ml\({^2}\) respectively. A test was conducted at the \(5\)% level of significance to test if Cleanser has overstated the mean volume of its hand sanitiser.

1. Explain, in the context of the question, the meaning of a random sample.[1]
2. Explain, in the context of the question, the meaning of ‘at the \(5\)% level of significance’.[1]
3. Find the unbiased estimate of the population variance.[1]

It was found that there was insufficient evidence to conclude that Cleanser had overstated the mean volume of their hand sanitiser.

4. By defining any parameters used and stating the hypotheses clearly, find the range of values of \(\bar{x}\), correct to \(2\) decimal places.[5]

Another random sample of \(80\) bottles of hand sanitiser was taken from another hand sanitiser company, Germsfree. The distribution of the volume of the \(80\) bottles of Germsfree sanitisers were given in the table below.

5. Find an unbiased estimate for the population mean and variance of volume of a bottle of Germsfree sanitiser.[2]
6. The same test was performed on the sample from Germsfree to see if Germsfree had also overstated the mean volume of its hand sanitiser.[2]
   It was found that both samples from Cleanser and Germsfree had the same mean. Let \({{p}_{1}}\) and \({{p}_{2}}\) be the \(p\)-value from the test on the sample from Cleanser and Germsfree respectively. Without doing any further calculation, state with justification an inequality relating \({{p}_{1}}\) and \({{p}_{2}}\).[1]