PLEASE GIVE CORRECT WORKING AND EXPLANATION

1)The sample size in Problem 1 is reduced to 50. Determine the type I error if the defect rate remains at 7% and the type II error when the defect rate rises to 9%. The decision is now to stop the machine for adjustment if a sample contains 4 or more defective screws. The sample size in Problem 1 is now reduced to 25. Determine the type I error if the defect rate remains at 7%, and the type II error when the defect rate rises to 10%. The decision is now to stop the machine for adjustment if a sample contains 3 or more defective screws

2) A random sample of 100 components is drawn from the output of a machine whose defect rate is 3%. Determine the type I error if the decision rule is to stop production when the sample contains: (a) 4 or more defective components, (b) 5 or more defective components, and (c) 6 or more defective components. [(a) 35.3% (b) 18.5% (c) 8.4%] 4. If there are 4 or more defective components in a sample drawn from the machine given in problem 3 above, determine the type II error when the actual defect rate is: (a) 5% (b) 6% (c) 7%. [(a) 26.5% (b) 15.1% (c) 8.18%]

3) A batch of cables produced by a manufacturer have a mean breaking strength of 2000 kN and a standard deviation of 100 kN. A sample of 50 cables is found to have a mean breaking strength of 2050 kN. Test the hypothesis that the breaking strength of the sample is greater than the breaking strength of the population from which it is drawn at a level of significance of 0.01. z (sample) = 3.54,z = 2.58, hence hypothesis is rejected, where z is the z-value corresponding to a level of significance of

4) Nine estimations of the percentage of copper in a bronze alloy have a mean of 80.8% and standard deviation of 1.2%. Assuming that the percentage of copper in samples is normally distributed, test the null hypothesis that the true percentage of copper is 80% against an alternative hypothesis that it exceeds 80%, at a level of significance of 0.1. t0.95, 8 = 1.86, |t| = 1.88, hence null hypothesis rejected

5) The internal diameter of a pipe has a mean diameter of 3.0000 cm with a standard deviation of 0.015 cm. A random sample of 30 measurements are taken and the mean of the samples is 3.0078 cm. Test the hypothesis that the mean diameter of the pipe is 3.0000 cm at a level of significance of 0.01. z (sample) = 2.85,z = 2.58, hence hypothesis is rejected 4. A fishing line has a mean breaking strength of 10.25 kN. Following a special treatment on the line, the following results are obtained for 20 specimens taken from the lineBreaking strength Frequency (kN) 9.8 1 10 1 10.1 4 10.2 5 10.5 3 10.7 2 10.8 2 10.9 1 11.0 1 Test the hypothesis that the special treatment has improved the breaking strength at a level of significance of 0.1. x = 10.38,s = 0.33, t0.9519 = 1.73, |t| = 1.72, hence hypothesis is accepted

6)A machine produces ball bearings having a mean diameter of 0.50 cm. A sample of 10 ball bearings is drawn at random and the sample mean is 0.53 cm with a standard deviation of 0.03 cm. Test the hypothesis that the mean diameter is 0.50 cm at a level of significance of (a) 0.05 and (b) 0.01. |t| = 3.00, (a) t0.9759 = 2.26, hence hypothesis rejected, (b) t0.9959 = 3.25, hence hypothesis is accepted

7) Six similar switches are tested to destruction at an overload of 20% of their normal maximum current rating. The mean number of operations before failure is 8200 with a standard deviation of 145. The manufacturer of the switches claims that they can be operated at least 8000 times at a 20% overload current. Can the manufacturer's claim be supported at a level of significance of (a) 0.1 and (b) 0.2? |t| = 3.08, (a) t0.955 = 2.02, hence claim supported, (b) t0.995 = 3.36, hence claim not supported

8). A comparison is being made between batteries used in calculators. Batteries of have a mean lifetime of 24 hours with a standard deviation of 4 hours, this data being calculated from a sample of 100 of the batteries. A sample of 80 of thE batteries has a mean lifetime of 40 hours with a standard deviation of 6 hours. Test the hypothesis that the type B batteries have a mean lifetime of at least 15 hours more than those of , at a level of significance of 0.05. Take x as 24 + 15, i.e. 39 hours, z = 1.28,z0.05, one-tailed test = 1.645, hence hypothesis is accepted 2. Two randomly selected groups of 50 operatives in a factory are timed during an assembly operation. The first group take a mean time of 112 minutes with a standard deviation of 12 minutes. The second group take a mean time of 117 minutes with a standard deviation of 9 minutes. Test the hypothesis that the mean time for the assembly operation is the same for both groups of employees at a level of significance of 0.05. z = 2.357,z0.05, two-tailed test = 1.96, hence hypothesis is rejected 3. Capacitors having a nominal capacitance of 24F but produced by two different companies are tested. The values of actual capacitance are: Company 1 21.4 23.6 24.8 22.4 26.3 Company 2 22.4 27.7 23.5 29.1 25.8 Test the hypothesis that the mean capacitance of capacitors produced by company 2 are higher than those produced by company 1 at 606 STATISTICS AND PROBABILITY a level of significance of 0.01. Bessel's correction is 2 = s2N N 1 . x1 = 23.7,s1 = 1.73, 1 = 1.93, x2

9)25.7, s2 = 2.50, 2 = 2.80, |t| = 1.62, t0.9958 = 3.36, hence hypothesis is accepted 4. A sample of 100 relays produced by manufacturer operated on average 1190 times before failure occurred, with a standard deviation of 90.75. Relays produced by manufacturer B, operated on average 1220 times before failure with a standard deviation of 120. Determine if the number of operations before failure are significantly different for the two manufacturers at a level of significance of (a) 0.05 and (b) 0.1. z (sample) = 1.99, (a) z0.05, two-tailed test = 1.96, no significance, (b) z0.1, two-tailed test = 1.645, significant difference

10) A sample of 12 car engines produced by manufacturer A showed that the mean petrol consumption over a measured distance was 4.8 litres with a standard deviation of 0.40 litres. Twelve similar engines for manufacturer B were tested over the same distance and the mean petrol consumption was 5.1 litres with a standard deviation of 0.36 litres. Test the hypothesis that the engines produced by manufacturer A are more economical than those produced by manufacturer B at a level of significance of (a) 0.01 and (b) 0.1. Assuming null hypothesis of no difference, = 0.397, |t| = 1.85, (a) t0.995, 22 = 2.82, hypothesis rejected, (b) t0.95, 22 = 1.72, hypothesis accepted 6. Four-star and unleaded petrol is tested in 5 similar cars under identical conditions. For four-star petrol, the cars covered a mean distance of 21.4 kilometres with a standard deviation of 0.54 kilometres for a given mass of petrol. For the same mass of unleaded petrol, the mean distance covered was 22.6 kilometres with a standard deviation of 0.48 kilometres. Test the hypothesis that unleaded petrol gives more kilometres per litre than four-star petrol at a level of significance of 0.