1. The random variable X has a normal distribution with mean 5 and standard deviation 2. Find: a) P(X<6) b) P(X<3) c) P(3<X<6) 2. The random variable X has a normal distribution with mean 6 and standard deviation 3. Find: a) P(X>10) b) P( X-6 <2) c) The constant a such that P(X<a)=0.15 [ 3] 3. The weight of the popcorn contained in a standard container sold in a cinema is normally distributed with mean 250g and variance 20g 2. Experience suggests that a customer will complain if s/he receives less than 240g of popcorn in a standard container. a) Find the probability that a randomly selected customer who buys a standard container of popcorn will complain. b) Two customers both buy a standard container of popcorn. Find the probability that neither will complain. pg. 1
4. The weights (in grammes) of jars of jam are normally distributed with mean 250 and variance 25. a) Find the probability a randomly selected jar i) weighs less than 242g ii) weighs between 255g and 265g b) A customer buys 3 jars of jam. He will complain if any of them weigh less than 242g. Find the probability he complains. 5. A machine in a sweet factory fills bags of jellybeans. The weight of jellybeans placed in each bag can be modelled by a normal distribution with mean 50g and standard deviation 2g. After the machine has filled the bags, they are weighed. Bags are rejected if they weigh less than 46g. a) From 1000 filled bags, find the expected number that will be rejected. A new law requires that at least 99% of bags of jellybeans must contain at least the weight stated on the outside of the bag. The weight stated on the outside of a filled bag is 50g. To comply with this law, the owner of the sweet factory decides to change the setting of the machine. This alters the mean, but does not affect the standard deviation. b) Find the smallest possible setting for the new mean. c) Comment on the suitability of the normal distribution as a model. 6. X~N(µ, σ 2 ) a) P( X-µ <98) = 0.95. Show that σ=50 Given µ = 230, find b) The probability X is greater than 300 or less than 100 c) The value that X exceeds with a probability of 0.5% [5] pg. 2
7. X~N(µ,25) a) P(X< 0) = 0.05. Find µ. b) Find P(X>10) c) P(a<X<15)=0.9. Find a. [6] 8. The continuous random variable X follows a normal distribution with mean 60 and variance σ 2. a) Given that P(X>68) = 0.0228, find σ. b) Find P(X>70 X>68) 9. The lengths of a certain species of worm follow a normal distribution. Thirty percent of the worms are at least 16cm long, and 15% of the worms are less than 10cm long. Find, to 2 decimal places, the standard deviation of the lengths of the worms. [8] pg. 3
10. The examination results of 11-year-old children on a particular test can be assumed to have a normal distribution with mean 100 and standard deviation σ. a) Give two reasons why this can only be an assumption. b) If only 2.5% of children get a score of 139 or above, find the value of σ correct to 1 decimal place The children are to be given grades A, B, or C on their test results. c) The maximum mark for a C is 85. Find, to the nearest whole number, the percentage of children that get this grade. d) 15% of children get an A grade. Find the lowest possible mark for an A grade. e) If three children are selected at random, find, correct to 2 decimal places, the probability that they all obtained different grades. 11. A class were conducting a statistics investigation into the speed of cars (in miles per hour) passing a certain point. They decided to model it using the random variable X~N(µ, 225) a) i) Using this model, find P(X-µ<-25) ii) Explain what is meant by this probability. b) A pupil suggested that µ was 25 mph. Use your answer to a) to explain why this would not be a good choice for µ. c) The class found that only 1% of cars were travelling at over 80 mph. Use this information to find µ. The speed limit on this particular road was 50 mph. d) Find the percentage of cars that were within the speed limit, giving your answer to the nearest whole number. e) Find, to 3 decimal places the probability that out of the next 10 cars to pass along the road, exactly three will be within the speed limit. pg. 4
12. Rownbury s chocolate bars are labelled as weighing 100g. Their weights are normally distributed with variance 0.4. a) A bar is substandard if it weighs 98g or less. Assuming a mean weight of 100g for each bar, calculate, to the nearest whole number, the expected number of substandard bars in a consignment of 1000. Rownbury s charge shopkeepers A pence per bar. They must pay shopkeepers 2A pence per substandard bar. b) i) Using your answer to a), write down an expression, in terms of A, for Rownbury s expected income per consignment of 1000 bars. ii) If this must be at least 100, and A must be a whole number, find the minimum acceptable value of A To comply with new legislation, Rownbury s have to change their manufacturing process so that no more than 1% of their chocolate bars weigh less than the weight on their label. c) Assuming the variance is unchanged, find, to 2 decimal places, the new mean that must be used to comply with this requirement. pg. 5