2007 HCI P2 Q10

Ben, a fruit seller, only sells mangoes form two orchards, \(A\) and \(B\). In each orchard the weights of the mangoes are normally distributed with a standard deviation of \(39\)g in A and an unknown standard deviation in B. The mean weights are \(200\)g in A and \(160\)g in B.

1. One day, Andy, a business partner of Ben, brought in a large batch of mangoes from orchard A. \(10\) mangoes were randomly selected from this batch and the weights (in g) of the mangoes were as follows:
   \(175 \qquad 182 \qquad 167 \qquad 160 \qquad 169 \qquad 189 \qquad 137 \qquad 164 \qquad x \qquad  207\)
   If a test at \(2.5\%\) significance level indicates there is a decrease in the mean weight of the mangoes form orchard A, find the greatest value of \(x\), correct to \(1\) decimal place.[4]
2. on another day, Ben examined \(20\) mangoes from orchard B which have mean weight \(140\)g and a standard deviation \(45\)g. He was worried that the mean weight of mangoes in orchard B might have changed. Andy helped by carrying out a statistical test to find out the truth.
   1. State the null and alternative hypotheses in carrying out such a test.[1]
   2. Andy has to reject the null hypothesis at \(k\%\) level of significance.
      Calculate the minimum value of \(k\), denoted by \({{k}_{0}}\), correct to \(2\) decimal places, in order to arrive at this decision.[2]
   3. If a mango from orchard B which weighs \(140\)g is added to the sample of \(20\). Would this affect Andy’s conclusion for the test at \({{k}_{0}}\%\) level of significance?[2]