1. [16 marks You are a member of a team of quality assurance specialists. Your team's immediate objective is to determine whether a product meets engineering design specifications provided by the product design team. Your team gathers a sample of 49 units of product, and you measure the height of each unit, in millimetres. + a) [1 mark] Find the sample mean of these 49 units. 2.1 3. Do not round your answer. Enter your command and your final answer in the space below. For example, if you were instead calculating the sample standard deviation, and the data were in cells A1:A49, your command and your final answer would be ustdev.s(A1:A49)= number. I b) [4 marks] Suppose the population standard deviation is 5 millimetres, and the engineering design specifications state that the population mean height must be at least 90 mm. What is the probability of obtaining a sample mean height of at least 2 (calculated from part a) of this question), if the population mean height is at least 90 mm? Declare the random variable of interest, show the probability you are asked to calculate and any tricks you might choose to use), how you standardize, your Z-score, and your final answer rounded to 4 decimal places. Hint: use u = 90 in your calculation. + c) [6 marks] Calculate 68%, 95%, and 99.7% tolerance intervals for the sample mean height, and interpret each of your intervals. Since you know its value, use the population standard deviation () instead of the sample standard deviation (S). Do not round your answers. Show your work. 1 d) [2 marks] Suppose the product design team changes their design specification: now, they say that at least 95% of all units of product must have a height of at least 95mm. Based on your tolerance intervals from part c), do you believe that the new design specification is being met? Why or why not? Please answer in at most 3 sentences. e) [3 marks] Suppose your team collects a new sample with 150 units. Notice the population standard deviation does not change, so your tolerance intervals from part c) still apply. How many units from your new sample of 150 do you expect to lie in your 68% tolerance interval? Your 95% tolerance interval? Your 99.7% tolerance interval? Do not round any of your answers. Show your work: show cach distribution you use, how you calculate your answers, and your final answers.