s A to G ADM2303 Fall 2015 Assignment #4 (19 Marks) Question 1. (4 points) With the upcoming holiday season, the national postal corporation uses an automated sorter which scans postal codes in order to separate letters sent by mail. However in some cases, the scanner misclassifies letters and a manual inspection is also undertaken in order to ensure quality control standards are met. The management team is concerned that with the hiring of new employees for the holiday period, the speed of the conveyor belt may not provide enough time for the inspectors to determine which letters are misclassified. In the following table, data from an experiment in which the same batch of letters (with known number of misclassified letters) was inspected using different conveyor belt speeds. Conveyor speed (ft/min) 10 12 15 17 20 22 25 28 30 32 35 No. of misclassified letters found 27 21 19 14 13 15 12 11 9 7 6 a) (1 point) Using MINITAB, plot a scatterplot with the conveyor speed on the x-axis. b) (1 point) What does the scatterplot found in (a) indicate about the relationship between the two variables? c) (2 points) Use MINITAB to calculate the correlation coefficient between the two variables. Interpret the result. Does the management team's concern seem justified? Question 2.(4 points) A manager of a restaurant in a commercial building would like to offer a new tea drink to customers. She randomly polled 100 customers and asked how many of them drink tea on a regular basis. Of the 100 customers, 41 reported to be tea drinkers. a) (1 point) Calculate , the estimate of the true population proportion of customers who drink tea. b) (3 points) Before the poll was conducted, the manager believed that 52% of customers were tea drinkers. Assuming this assertion is true, find (.)14.0 1 Sections A to G ADM2303 Fall 2015 Question 3. (6 Points) The inorganic mercury content in a single cigarette of a particular brand is a random variable with mean 25 ng and a population standard deviation of 12 ng. A random sample of n = 100 cigarettes is taken for analysis. a) (2 points) What is the approximate distribution of the sample mean ? b) (2 points) The inorganic mercury content for a cigarette is considered high when the content is greater than 29 ng. What is the probability that the resulting sample mean content will be greater than 29 ng? c) (2 points) Now suppose that the market analyst would like to test another brand of cigarettes and selects a random sample of = 25 cigarettes from a population with mean inorganic mercury content 25 ng, but the standard deviation is now estimated from the sample as 10\tng. What is the probability that the sample mean content will be greater than 29 mg for this brand? Question 4. (5 Points) Using MINITAB, generate observations from an Exponential (=1) distribution. Generate 200 samples of 30 observations each by generating 200 rows of data and storing the results in columns C1-C30. Refer to the MINITAB Instructions provided at the end of the assignment. i) Now, to simulate a random sample of size n = 3, select data from columns C1 - C3. Find the mean rowwise and store the result in column C31. ii) Similarly, simulate a random sample of size n = 10 by selecting data from columns C1-C10. Find the mean row-wise and store the result in column C32. iii) Finally, simulate a random sample of size n = 30 by selecting data from columns C1-C30. Find the mean row-wise and store the result in column C33. Submit responses to the following two questions with your assignment: a) (4 points) Plot a histogram of the sample means obtained in each of i), ii) and iii), and describe the shape of the distribution for each case. What do you notice as the sample size increases? b) (1 point) What theoretical result is illustrated by this procedure? 2 Sections A to G ADM2303 Fall 2015 MINITAB INSTRUCTIONS 1. Sampling Distributions i) To generate data from an Exponential (=1) distribution, we use the commands Calc > Random Data > Exponential Using the dialog box, we wish to generate 200 rows of data and store the results in column(s) C1-C30. With Exponential (=1), we select Scale = 1.0 and Threshold = 0.0. Select OK. We note that in the Worksheet, there are 30 columns of data. ii) Now, to take random samples of size n = 3 and find the mean row-wise, we use the following function: Calc > Row Statistics 3 Sections A to G ADM2303 Fall 2015 Select Mean from the Row Statistics box and for Input variables type C1-C3 (i.e. n=3). Store result in C31. Select OK. Since we selected a sample of size 3 many times and recorded the mean of each sample, a histogram of C31 will show us, approximately, the sampling distribution of the sample mean. To find the graph, select Graph > Histogram > Simple. Under GraphVariables type C31 to produce a histogram of the sampling distribution. iii) In a similar manner, to take samples of size n = 10 and find the mean, we repeat Step (ii) using the function Calc > Row Statistics again. As before, we select the Mean from the Row Statistics box, but this time we input variables C1-C10 (i.e. n = 10) and store result in C32. We may then select Graph > Histogram > Simple and type C32 under Graph Variables to graph the sampling distribution of the mean for n=10. Similarly, repeat Step (ii) for a sample size of n = 30, using input variables C1-C30 and store the result in C33. 4