Sections A to G ADM2303 Fall 2015

Sections A to G ADM2303 Fall 2015

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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)

No. of misclassified

letters found

10 27

12 21

15 19

17 14

20 13

22 15

25 12

28 11

30 9

32 7

35 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 �(�̂ ≤ 0.41).

Sections A to G ADM2303 Fall 2015

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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 ng. 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 (λ=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 row-

wise 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?

Sections A to G ADM2303 Fall 2015

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MINITAB INSTRUCTIONS

1. Sampling Distributions

i) To generate data from an Exponential (λ=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 (λ=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

Sections A to G ADM2303 Fall 2015

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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.