QUESTION 1 A manager of dairy processing company claims that the variance of the fat amount in the whole mitt: processed by the company is not more than 0.259. A consumer agency suspects that this is not true. A random sample of 41 milk containers was tested and the variance of fat amount for this sample was found to be 0.279. At a = 0.01, is there enough evidence to reject the company's claim? (Assume the population is normally distributed). (7 marks) QUESTION 2 a} Sarah is the owner of a small bakery in a large town. She believes mat the smell of fresh bakery will encourage cusoomers to purchase breads from her bakery. To investigate this belief: she records the daily sales for 10' days when atl the battery's windows are open and the daily sales for another 10 days when all the windows are closed. The following sales (in RM] are recorded. 202. 00 215. 60 193. 50 200. 60 204. 50 205. 30 192. 20 131.30 207. 00 187. 80 199. 40 169. 20 215.50 204.10 177.60 172.20 190.30 135.70 205.40 192.30 Assuming that the two populations taken from a normal population with equal variances. i) Compute the mean difference and pooled estimate of variance of the daily sales in Sarah's bakery. {4 marks} ii] Construct a 95% condence interval for the difference between the two sales. {4 marks} iii) Sarah claims that the daily sales when the windows open are more than the deity sales when the windows closed. Can the researcher support her claim? Justify your answer. {1 marks} QUESTION 3 Ten pairs of identical twins had the following birth weights in kilograms. Pair 1 2 3 4 5 6 7 8 9 10 First-born 3.95 3.41 3,73 4.13 3.48 4.28 3.98 4.18 4.04 3.73 Second-born 3.93 3.35 3.72 4.18 3.44 4.15 3.89 4,20 4,00 3.72 Assume that birth weights in a population are normally distributed a) Show that the standard deviation of the paired differences is 0.0521. (3 marks) b) Test the hypothesis that the first-born are heavier than second-born twins using 5% significance level. (7 marks) c) Estimate the mean difference of weight between the first-born and second-born twins with 95% confidence level. Interpret the interval obtained. (5 marks)QUESTION 4 Industrial waste dumped in rivers reduces the amount of dissolved oxygen in the water. A factory was suspected of illegally dumping waste in the river. Samples of water were taken from the river, six are above the factory (which radius is 25 kilometers from the factory) and eight below the factory (which radius is 50 kilometers from the factory) and the dissolved oxygen content in part per million (pom) were as follows. Above factory 4.6 5.1 4.7 5.0 5.3 4.6 Below factory 3.8 4.9 4.0 3.6 5.0 3.4 3.5 3.9 The summary statistic from SPSS output is given below Group Statistics measurement N Mean Std. Std. Error Deviation Mean Dissolved Above factory 4,883 0.293 0.12 oxygen content Below factory 4.013 0.613 0.22 a) Test at 5% level of significance whether the variability of the dissolved oxygen content for above and below the factory is the same? (5 marks) b) Based on the answer in a), test at 5% significance level whether the mean of the dissolved oxygen content for the above factory is more than below factory. (6 marks)QUESTION 5 The time taken by a team to complete the assembly of an electrical component is found to be normally distributed with a mean of 110 minutes and a standard deviation of 10 minutes. If the manager decides to set a cut off time such that 95% of the teams will have completed the assembly on time, determine the cut off time that should be set. (5 marks) QUESTION 6 3% of the bulbs are found to be defective after going through an inspection stage at a factory. a) In a random sample of twelve bulbs, find the probability that more than one bulb are defective. (3 marks) b) In a random sample of twenty bulbs, find the probability that less than four bulbs are defective. (3 marks) c) In a random sample of 150 bulbs, find the probability that between four and eleven bulbs are defective. (4 marks) Number of orders received by an online shopping mall between 2 to 3 pm are distributed by Poisson distribution with mean number of orders received are 12. Find the probability that the online mall will receive exactly five orders between 2 to 2.30 pm