Determine unbiased estimates of the mean and variance of the loss in weight achieved over the month by people using this diet.

(i)     State suitable hypotheses for testing whether or not this diet causes a mean loss in weight.

(ii)     Determine the value of a suitable statistic for testing your hypotheses.

(iii)     Find the 1 % critical value for your statistic and state your conclusion.

the weight losses are

2.2\(\,\,\,\,\,\)3.5\(\,\,\,\,\,\)4.3\(\,\,\,\,\,\)–0.5\(\,\,\,\,\,\)4.2\(\,\,\,\,\,\)–0.2\(\,\,\,\,\,\)2.5\(\,\,\,\,\,\)2.7\(\,\,\,\,\,\)0.1\(\,\,\,\,\,\)–0.7     (M1)(A1)

\(\sum {x = 18.1} \), \(\sum {{x^2} = 67.55} \)

UE of mean = 1.81     A1

UE of variance \( = \frac{{67.55}}{9} - \frac{{{{18.1}^2}}}{{90}} = 3.87\)     (M1)A1

Note: Accept weight losses as positive or negative. Accept unbiased estimate of mean as positive or negative.

Note: Award M1A0 for 1.97 as UE of variance.

[5 marks]

(i)     \({H_0}:{\mu _d} = 0\) versus \({H_1}:{\mu _d} > 0\)     A1

Note: Accept any symbol for \({\mu _d}\)

(ii)     using t test     (M1)

\(t = \frac{{1.81}}{{\sqrt {\frac{{3.87}}{{10}}} }} = 2.91\)     A1

(iii)     DF = 9     (A1)

Note: Award this (A1) if the p-value is given as 0.00864

1% critical value = 2.82     A1

accept \({H_1}\)     R1

Note: Allow FT on final R1.

[6 marks]

In (a), most candidates gave a correct estimate for the mean but the variance estimate was often incorrect. Some candidates who use their GDC seem to be unable to obtain the unbiased variance estimate from the numbers on the screen. The way to proceed, of course, is to realise that the larger of the two ‘standard deviations’ on offer is the square root of the unbiased estimate so that its square gives the required result. In (b), most candidates realised that the t-distribution should be used although many were awarded an arithmetic penalty for giving either t = 2.911 or the critical value = 2.821. Some candidates who used the p-value method to reach a conclusion lost a mark by omitting to give the critical value. Many candidates found part (c) difficult and although they were able to obtain t = 2.49…, they were then unable to continue to obtain the confidence interval.

In (a), most candidates gave a correct estimate for the mean but the variance estimate was often incorrect. Some candidates who use their GDC seem to be unable to obtain the unbiased variance estimate from the numbers on the screen. The way to proceed, of course, is to realise that the larger of the two ‘standard deviations’ on offer is the square root of the unbiased estimate so that its square gives the required result. In (b), most candidates realised that the t-distribution should be used although many were awarded an arithmetic penalty for giving either t = 2.911 or the critical value = 2.821. Some candidates who used the p-value method to reach a conclusion lost a mark by omitting to give the critical value. Many candidates found part (c) difficult and although they were able to obtain t = 2.49…, they were then unable to continue to obtain the confidence interval.