A. A researcher, Dr. Smith, is conducting a study on the academic performance of students at a local high school. The results of this study will provide valuable insights into the academic performance of students at the high school and inform potential interventions to support student success.

To investigate the average test scores of students, Dr. Smith randomly selects a sample of 8 students from the population of all students at the school. After collecting the data, Dr. Smith calculates the sample mean test score to be 68, with a standard deviation of 50.

Required:

Using the 6-step process of hypothesis testing learned in HM6007, assess whether he would be able to conclude with 90% confidence that the population mean is less than 85.

B. Innovate Enterprises' sales team is launching a new product and has initiated a survey of potential customers as a preliminary step.

Required:

Determine the necessary sample size for the company to estimate, with 99% confidence, the proportion of individuals likely to purchase the product within a margin of 3%.