To look up probability values/critical cutoffs from a distribution table, please round the relevant z-scores or t-scores to two decimal places before using their respective table.

Round the numerical values in your answers to three decimal places.

A policymaker is conducting an investigation into the quality of life of the residents of a large district. Previously, validated questionnaires measuring subjective quality of life were given out to every single household in the district, and all of its districts returned completed versions of the questionnaires. The questionnaire is scored on an interval scale, with higher scores reflecting better subjective quality of life. The policymaker finds that the district's overall mean subjective quality of life score is 75, with a standard deviation of 16.

a. If the policymaker randomly selects a single resident from the district, what is the probability that this resident will have a quality of life score between 66 and 88? [5 marks]

b. If the policymaker randomly samples a group of 15 residents from the district, what is the probability that this sample will have a mean quality of life score either below 72 or above 80? [5 marks]

c. The policymaker wishes to compare whether residents living in the south of the district have a different subjective quality of life compared to the rest of the district. She randomly samples a group of 30 residents from the south of the district, and finds that their mean quality of life rating is 82. Using a significance criterion of 5%, run an appropriate statistical test (by hand-calculation) to address this research question, and state your conclusions. [6 marks]

d. Based on the information about the district, and in part c. above, calculate the 99% confidence interval of the mean subjective quality of life rating of residents who are living in the south of the district. [4 marks]