1. Experience has shown that 20 % of a manufactured product is of top quality. In one day's production of 400 articles, only 50 are of top quality. Show that either the production of the day chosen was not a representative sample or the hypothesis of 20% was wrong. Based on the particular day's production, find also 95% confidence limits for the percentage of top quality product. 2. The mean value of a random sample of 60 items was found to be 145 with a S.D. f 40. Find the 95% confidence limits for the populations mean. What size of the sample is required to estimate the population mean within five of its actual value with 95% or more confidence, using the sample mean? 3. Hydrocarbon emissions from cars are known to have decreased dramatically during the 1980s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million, for automobiles from 1980 and 1990. Twenty cars of each model year were randomly selected, and their hydrocarbon emission levels were recoded. The data are as follows: 1980 models 1990 models 141 494 200 190 140 60 220 380 359 306 223 300 160 20 400 200 247 210 188 435 20 95 217 175 940 105 940 241 20 360 58 85 882 880 241 380 223 70 235 65 Test the hypothesis that the variances are equal and also the same normal population. 4. A coin is thrown until a head occur and the number X of tosses recorded. After repeating the experiment 256 times, we obtained the following results: X 1 2 3 4 5 6 7 8 f 136 60 34 12 9 1 3 1 Test the hypothesis, at 0.05 level of significance, that the observed distribution of X may be fitted by the binomial distribution B (x; 1/2), x =1, 2, 3, ..., 8. 5. A company wishes to purchase one of five different machines A, B, C, D and E. In an experiment designed to determine whether there is a performance difference between the machines, five experienced operators each work on the machines for equal times, the table shows the number of units produced by each machine. Test the hypothesis that there is no difference between the machines at 5% level of significance. A 68 72 77 42 53 B 72 53 63 53 C 60 82 64 75 72 D 48 61 57 64 50 E 64 65 70 68 53 6. A car rental agency, which uses 5 different brands of tyres in the process of deciding the brand of tyre to purchase as standard equipment for its fleet, finds that each of 5 tyres of each brand last the following number of kilometers (in thousands). Test at 5% level of significance. Test the hypothesis that the five tyre brands have almost the same average lives. ABCDE 36 46 35 45 41 37 39 42 36 39 42 35 37 39 37 38 37 43 35 35 47 43 38 32 38