An agricultural scientist is monitoring the yield of a specific variety of wheat. The yield per hectare is normally distributed with a standard deviation of 4.8 kg4.8\text{ kg}4.8 kg. Traditionally, the mean yield, μ\muμ, is known to be 190 kg190\text{ kg}190 kg.
The scientist suspects that a new fertilizer has increased the mean yield. She collects data from a random sample of 666 hectares and performs a hypothesis test at the 10%10\%10% significance level.
She tests H0:μ=190H_0: \mu = 190H0:μ=190 against H1:μ>190H_1: \mu > 190H1:μ>190.
When reviewing her notes, she finds that she has lost one of the recordings. The five available recordings are:
192.5,188.2,197.6,191.4,195.3 192.5, \quad 188.2, \quad 197.6, \quad 191.4, \quad 195.3 192.5,188.2,197.6,191.4,195.3Given that the result of the hypothesis test was to reject the null hypothesis H0H_0H0 at the 10%10\%10% level of significance, calculate the minimum possible value of the missing measurement, xxx. Give your answer correct to 1 decimal place.
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.