It is known from past experience that for students who studied AMA1110 for at least 100 hours, their chance of passing the exam is 0.95, while for those who studied AMA1110 for less than 100 hours, their chance of passing the exam is 0.1. It is also known that the passing rate is 90%. If a student is randomly chosen and is found to have passed, what is the probability that he/she has studied for less than 100 hours? [8 marks] 2. The following table lists the probability distribution of the number of breakdowns per month for a machine based on past data: Breakdowns per week Probability 0 1. 2. 3 c 0.2 0.4 0.3 a Find the value of c. b Find the mean μ and variance σ2 for this probability distribution. [8 marks] 3. In a quality control unit of a newspaper printing, a newspaper is selected randomly each day from the production line to check the number of misprints in the newspaper. It is known from past experience that the average number of misprints is 2.5. By policy, if 3 or more misprints are detected in a randomly selected newspaper, the printing process will be stoped. (a) What is the probability that the printing process is stopped in a randomly chosen day? [5 marks] (b) Suppose that the quality control is conducted for 7 consecutive days. Find the probability that the printing process is stopped on exactly 5 out of these 7 days. [5 marks] 4. A manufacturer fills jars with coffee. The weight of coffee, W grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams. (i) Find P (W 227)? [4 marks] (ii) One hundred jars of coffee are selected at random. Find the probability that more than 23 jars contains less than 227 grams of coffee. [6 marks] 5. The weights (in kg) of the students in a school is normally distributed with standard deviation 15. (a) If we know that the population mean is 67. Find the probability that the mean weight of a random sample of 31 students is over 70kg. [4 marks] (b) Assume that the population mean is unknown and we want to estimate it by taking a random sample of 31 students. It is found that the mean of the weights in the sample is 65kg. Find an approximate 90% confidence interval for the mean of the weights of students in the school.

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