2015 RI P2 Q9 [Modified]

The wingspan of a certain endangered species of giant butterfly found in the rainforest in New Guinea is a normally distributed random variable with mean \(18.0\) cm. A lepidopterist who just arrived in New Guinea is keen to find out if climate changes have caused the size of the butterflies to decrease. He measures the wingspan of a random sample of \(45\) such butterflies and uses the data to perform a hypothesis test.

1. State appropriate hypotheses for the test, defining any symbols that you use. [2]

The sample mean wingspan is denoted by \(\bar{x}\) cm and the sample variance is \(1.2\) cm\(^{2}\).

2. Find the set of values of \(\bar{x}\) for which the null hypothesis would be rejected at \(5\%\) level of significance[4]

The lepidopterist later discovered some old records and noted that the wingspan of the butterfly has a population variance of \(1.3\) cm\(^{2}\). Using this information, he carried out an appropriate test at \(5\%\) significance level to determine if the size of the butterflies has changed.

3. Find the set of values of \(\bar{x}\) for which the null hypothesis would not be rejected. [4]