Question 1. (i) A researcher is interested in whether the average age of a population in a region is 40 years. A random sample of 51 people from this population has a mean of 43 years and standard deviation of 10 years. Assuming that the ages are normally distributed, test the researcher's claim using an appropriate statistical test at the 5% significance level. [25] (ii) Weights (kg) of 12 animals are measured before and after a new diet. The data and SPSS output for a paired t-test are presented below. Animal 1 2 3 4 5 6 7 8 9 10 11 12 Before 12.3 14.5 10.1 13.4 12.3 13.6 16.5 14.3 12.6 14.1 16.8 16.7 After 14.3 15.4 11.2 12.7 13.4 15.6 17.2 15.6 14.5 14.3 17.9 17.6 Paired Samples Test Paired Differences Df Sig. (2- Mean Std. Std. Error 95% Confidence Interval tailed) Deviation Mean of the Difference Lower Upper Pair Before - -1.041 .77161 .22274 -1.53192 -.55141 -4.677 11 .001 After (a) State the hypotheses for the paired t-test. [5] (b) Using the above results, what can you conclude about the new diet? Assume a 5% level of significance. [5] (c) Does the 95% confidence interval support your answer in part (b)? [5] (d) One assumption of the paired t-test is that the differences are normally distributed. Describe how this assumption could be verified for the data in this question. [10] (Total 50 marks)