The probabilities on the second set of branches represent the conditional probabilities of B given that A has, or has not, happened. Given that events A and B are independent and that P(A) = x and P(B) = y, find, in terms of x and y: a P(A ∩ B)  The random variable X ~ N(20, 12). a Find the value of a and the value of b such that: ii P(X . b) = 0.6915 i P(X , a) = 0.40 b Find P(b , X , a). 4 The random variable Y ~ N(100, 152). a Find the value of a and the value of b such that: ii P(Y , b) = 0.10 i P(Y . a) = 0.975 b Find P(a , Y , b). 5 The random variable X ~ N(80, 16). a Find the value of a and the value of b such that: ii P(X , b) = 0.5636 i P(X . a) = 0.40 b Find P(b , X , a). P Each of 10 cows was given an additive (x) every day for four weeks to see if it would improve the milk yield ( y). At the beginning, the average milk yield per day was 4 gallons. The milk yield of each cow was measured on the last day of the four weeks. The data collected is shown in the table. Cow The masses, M kg, of a population of badgers are modelled as M ~ N(4.5, 0.62). For this population, find: a the lower quartile b the 80th percentile c Explain without calculation why Q2 = 4.5 kg.

There are seven outcomes where one of the dice lands on a 3. The yellow circles show the restricted sample space. Two of these outcomes have a total of 5. a Given that 60% of plates are less than x cm, find x. b Find the interquartile range of the plate diameters. It is thought that the relationship between E and t is of the form E = aT b. a By plotting an appropriate scatter diagram, verify that this relationship is valid for the data given. b By drawing a suitable line on your scatter diagram and finding its equation, estimate the values of a and b. c Give a reason why the model will not predict the efficiency of the engine when the temperature is 0 °C. On a randomly chosen day the probabilities that Bill travels to work by car, by bus or by train are 0.1, 0.6 and 0.3 respectively. The probabilities of being late when using these methods of travel are 0.55, 0.3 and 0.05 respectively. a Draw a tree diagram to represent this information.  The probability that a plate made using a particular production process is faulty is given as 0.16. A sample of 20 plates is taken. Find: Biologists use the normal distribution to model the distributions of physical characteristics, such as height and mass, in large populations. → Exercise 3E, Q13 If two events, A and B, are mutually exclusive, then their intersection is the empty set, ​∅​. You can write A ∩ B = ​∅​.

This is the Agreement for accessing CGP Online Editions (the Service). The Service provides online access to a range of titles published by Coordination Group Publications Ltd. (CGP). This Agreement covers access to the Service regardless of the device or network you access it through. By using the Service you agree to be bound by this Agreement. The table shows some data collected on the temperature, t °C, of a colony of insect larvae and the growth rate, g, of the population. Temp, t (°C) Growth rate, g These histograms show the distribution of heights of adult males in a particular city. As the class width reduces, the distribution gets smoother. It is believed that there is a relationship between the age and maintenance cost of these machines. b Using a 5% level of significance and quoting from the table of critical values, interpret your correlation coefficient. Use a two-tailed test and state clearly your null and alternative hypotheses.  (3 marks) Ea P(C ∩ D) =  P(C|D) × P(D) = 0.3 × 0.6 = 0.18 P(D ∩ C) b P(D|C) = ​​ _________ ​​ P(C) 0.18 = _____ ​ ​ ​​ = 0.9 0.2 c P(C ∪ D) =  P(C) + P(D) − P(C ∩ D) = 0.2 + 0.6 − 0.18 = 0.62

The percentage scores of a group of students in a test, S, are modelled as a normal distribution with mean 45 and standard deviation 15. Find: a P(S > 45) b P(30 < S < 60) c P(15 < S < 75) Alexia states that since it is impossible to score above 100%, this is not a suitable model. d State, with a reason, whether Alexia is correct. Data on the daily maximum temperature and the daily total sunshine is taken from the large data set for Leuchars in May and June 1987. A meteorologist finds that the product moment correlation coefficient for these data is 0.715. Given that the researcher tests for positive correlation at the 2.5% level of significance, and concludes that the value is significant, find the smallest possible sample size. The percentage scores, X, of a group of learner drivers in a theory test is modelled as a normal distribution with X ~ N(72, 62). a Find the value of a such that P(X , a) = 0.6. b Find the interquartile range of the scores.

The product moment correlation coefficient between weight and age for these babies was found to be 0.972. By testing for positive correlation at the 5% significance level interpret this value.  (3 marks) E The distribution becomes bell-shaped and is symmetrical about the mean. You can model the heights of adult males in this city using a normal distribution, with mean 175 cm and standard deviation 12 cm. has parameters µ, the population mean and σ2, the population variance The gradient of the line is 1.8. This corresponds to the value of n in the non-linear relationship. The y-intercept is at (0, −1). This corresponds to log a hence a = 10−1 = 0.1, as expected. Use the Inverse Normal function on your calculator to calculate values which satisfy given probability statements for the normal distribution. A Code must be redeemed before the title it is linked to can be accessed. An unredeemed Code may be transferred to another person or organisation who can then redeem it.

The probability that team A scores first and wins the match is 0.48. b Find the probability that team A scores first and does not win the match.  Jean always goes to work by bus or takes a taxi. If one day she goes to work by bus, the probability she goes to work by taxi the next day is 0.4. If one day she goes to work by taxi, the probability she goes to work by bus the next day is 0.7. Given that Jean takes the bus to work on Monday, find the probability that she takes a taxi to work on Wednesday. In the normal distribution N(40, 9) the second parameter is the variance. The standard __ deviation in this normal distribution is √​​ 9 ​​ = 3. random variable, you write X ~ N(µ, σ 2) where µ is the population mean and σ 2 is the population variance. Anna and Bella are sometimes late for school. The events A and B are defined as follows: A is the event that Anna is late for school B is the event that Bella is late for school P(A) = 0.3, P(B) = 0.7 and P(A9 ∩ B9) = 0.1. On a randomly selected day, find the probability that: a both Anna and Bella are late to school 

a P(D < x) = 0.6 ⇒ x = 20.38 cm b P(D < Q1) = 0.25 ⇒ Q1 = 18.99 cm P(D < Q3) = 0.75 ⇒ Q3 = 21.01 cm The interquartile range is 21.01 − 18.99 = 2.02 cm (2 d.p.)