A viticulturist is investigating the survival of a new hybrid vine graft. The probability that a graft is successful is denoted by ppp. A sample of 100 grafts is taken, and the random variable S S\,S represents the number of successful grafts, where S∼B(100,p)S \sim \text{B}(100, p)S∼B(100,p).
Using a normal approximation, the probability that S S\,S is at least 53 is 0.0054 to 4 decimal places.
Show that p p\,p satisfies 426.01p2−446.01p+110.25=0426.01p^2 - 446.01p + 110.25 = 0426.01p2−446.01p+110.25=0 when normal probability tables are used.
Hence find the value of ppp, justifying your answer.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.