Clear calculations please.

1 A statistical test is used to determine whether or not an anti-smoking campaign carried out 5 years ago has led to a significant reduction in the mean number of smoking related illnesses. If the probability value of the test statistic is 7%, determine the conclusion for a test of size: (1) 10% 5%. 2 A random sample, x1....*10, from a normal population gives the following values: 9.5 18.2 4.69 3.76 14.2 17.13 15.69 13.9 15.7 7.42 [x, =120.19 _x/ =1,693.6331 Test at the 5% level whether the mean of the whole population is 15 if the variance is: (a) unknown (b) 20. (ii) Test at the 5% level whether the population variance is 20. 3 A professional gambler has said: 'Flipping a coin into the air is fair, since the coin rotates about a horizontal axis, and it is equally likely to be either way up when it first clips the ground. So a flicked coin is equally likely to land showing heads or tails. However, spinning a coin on a table is not fair, since the coin rotates about a vertical axis, and there is a systematic bias causing it to tilt towards the side where the embossed pattern is heavier. In fact, when a new coin is spun, it is more than twice as likely to land showing tails as it is to land showing heads.' After hearing this, you carried out an experiment, spinning a new coin 25 times on a polished table, and found that it showed tails 18 times. Comment on whether the results of your experiment support the gambler's claims about the probabilities when a coin is spun. 4 The sample variances of two independent samples from normal populations A and B, which have the same population variance, are $ =12.4 and $; =25.8. If the sample sizes are nA = 10 and no =5 and the sample means are found to differ by 4.5, test whether the population means are equal. 5 Two populations X and Y are known to have the same variance, but the precise distributions are not known. A sample of 5 values from population X and 10 values from population Y had sample variances of s* =47.0 and sy =12.6. Apply a statistical test based on the F distribution to assess whether both populations can be considered to be normally distributed. 6 Determine the form of the best test of Hot = /p VS H1 :#= /, where > /0, assuming the distribution of the underlying population is A(/,o ), based on a sample of size n.7 The lengths of a random sample of 12 worms of a particular species have a mean of 8.54 cm and a style standard deviation of 2.97 cm. Let # denote the mean length of a worm of this species. It is required to test: Ho:#=7cm vs H1:/#7cm Assuming that the lengths of worms are normally distributed, calculate the probability-value of these sample results. [3] 8 A general insurance company is debating introducing a new screening programme to reduce the claim amounts that it needs to pay out. The programme consists of a much more detailed style application form that takes longer for the new client department to process. The screening is applied to a test group of clients as a trial whilst other clients continue to fill in the old application form. It can be assumed that claim payments follow a normal distribution. The claim payments data for samples of the two groups of clients are (in f100 per year): Without screening 24.5 21.7 45.2 15.9 23.7 34.2 29.3 21.1 23.5 28.3 With screening 22.4 21.2 36.3 15.7 21.5 7.3 12.8 21.2 23.9 18.4 (i) Test the hypothesis that the new screening programme reduces the mean claim amount. (5] (ii) Formally test the assumption of equal variances required in part (i). [3] [Total 8] 9 An environmentalist is investigating the possibility that oestrogenic chemicals are leading to a style particular type of deformity in a species of amphibians living in a lake. The usual proportion of deformed animals living in unpolluted water is 0.5%. In a sample of 1,000 animals examined, 15 were found to have deformities. (i) Test whether this provides evidence of the presence of harmful chemicals in the lake. [3] Following an extensive campaign to reduce these chemicals in the lake a further sample of 800 animals was examined and 10 were found to have deformities. (ii) Test whether there has been a significant reduction in the proportion of deformed animals in the lake. (3] [Total 6]