H1 maths Estimator

11 A website states that the mean length of adults in a particular species of fish is 30 cm. A scientist claims that the true mean length is greater than 30 cm. To test this claim, a random sample of 100 adult fish of this species is captured from a lake. The lengths of the fish, x cm, are summarised by E(x -30) = 15 and E(x - 30) 2 = 82. (i) Find unbiased estimates of the population mean and variance. [3] (ii) Test at the 2.5% significance level whether the scientist's claim is supported by the data. [4] (iii) State, with a reason, whether it is necessary to assume that the lengths of these fish are distributed normally for the test to be valid. [1] A new random sample of 100 adult fish of this species is captured from a different lake. The mean length of the fish in this sample is mom and the population variance is 0.9 cm . A test at the 10% significance level supports the scientist's claim that the mean length of the fish is greater than 30 cm. (iv) Find the range of possible values of m. [3]