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Statistics and Probability
QUESTION:
Weekly sales of copy paper at Cubicle Suppliers are provided in the table below. Compute a three-period moving average, a three-period weighted moving average (most recent period 05, last two period = 0.3, last three period = 0.2) and an exponential smoothing forecast (initial forecast = 26, α = 0.2). Compute the MAD, MSE, MAPE for each forecasting method. Which model is more accurate for week 8 forecast? (please use 3 decimal points)
Week Sales (cases) Forecast
1 27 26
2 28
3 28
4 29
5 29
6 27
7 28
8
Answer:
3MA:
Week Sales (cases) 3MA |MAD| |MSE| |MAPE|
1 27
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
3WMA:
Week Sales (cases) 3WMA
(0.5,0.3, 0.2) |MAD| |MSE| |MAPE|
1 27
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
Exponential Smoothing:
Week Sales (cases) F
(α = 0.2) |MAD| |MSE| |MAPE|
1 27 26
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
Weekly sales of copy paper at Cubicle Suppliers are provided in the table below. Compute a three-period moving average, a three-period weighted moving average (most recent period 05, last two period = 0.3, last three period = 0.2) and an exponential smoothing forecast (initial forecast = 26, α = 0.2). Compute the MAD, MSE, MAPE for each forecasting method. Which model is more accurate for week 8 forecast? (please use 3 decimal points)
Week Sales (cases) Forecast
1 27 26
2 28
3 28
4 29
5 29
6 27
7 28
8
Answer:
3MA:
Week Sales (cases) 3MA |MAD| |MSE| |MAPE|
1 27
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
3WMA:
Week Sales (cases) 3WMA
(0.5,0.3, 0.2) |MAD| |MSE| |MAPE|
1 27
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
Exponential Smoothing:
Week Sales (cases) F
(α = 0.2) |MAD| |MSE| |MAPE|
1 27 26
2 28
3 28
4 29
5 29
6 27
7 28
8
Ʃ
Statistics and Probability
1. An operator trainee is attempting to monitor a filling process that has an overall average of 650 cc. The average range is 22 cc. If you use a sample size of 8, what are the upper and lower control limits for the x-bar chart and R-chart?
A2 = 0.373, D4 = 1.864, D3 = 0.136
2. A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=100 each day. Data from the previous 10 days indicate the following number of out-of-state license plates:
Day Out-of-state Plates
1 6
2 4
3 5
4 7
5 8
6 3
7 4
8 15
9 3
10 11
(a) Calculate the overall proportion of "tourists" (cars with out-of-state plates) and the standard deviation of proportions.
(b) Using ±3σ limits, calculate the LCL and UCL for these data.
(c) Is the process under control? Explain.
A2 = 0.373, D4 = 1.864, D3 = 0.136
2. A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=100 each day. Data from the previous 10 days indicate the following number of out-of-state license plates:
Day Out-of-state Plates
1 6
2 4
3 5
4 7
5 8
6 3
7 4
8 15
9 3
10 11
(a) Calculate the overall proportion of "tourists" (cars with out-of-state plates) and the standard deviation of proportions.
(b) Using ±3σ limits, calculate the LCL and UCL for these data.
(c) Is the process under control? Explain.
Statistics and Probability
Three cards are drawn randomly from a deck of cards one after the other with replacement each time. Find the probability that the three cards are a 5, a heart and a king.
a. 0.0156
b. 0.125
c. 0.0015
a. 0.0192
a. 0.0156
b. 0.125
c. 0.0015
a. 0.0192
Statistics and Probability
If you know the mean and standard deviation of a normally distributed population, and somebody asks you questions about the highest 1% and lowest 1% of numbers in that population, what could you tell them? Pick your own values for the mean and standard deviation and then answer the question for those values. Choose one of these R functions to answer the question: dnorm(), pnorm(), qnorm(), rnorm().
Statistics and Probability
A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3.
The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be,
a. More than 2 daisies
b. Either 5 or 6 daisies.
Statistics and Probability
Find the area under the normal curve between a. z = -1.22 and z = 2.32 b. z =1.27 and z= 1.77, c. z =2.0 and z = -0.50.
Statistics and Probability
Following is a sequence of heads (H) and tails (T)
in tossing a coin 14 times.
HTTHHHTHTTHHTH
Test whether the heads and tails occur in random
order.
Statistics and Probability
A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is 1.5 ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below.
Weight
Day Package 1 Package 2 Package 3 Package 4
Monday 23 22 23 20
Tuesday 23 21 19 21
Wednesday 20 19 20 21
Thursday 18 19 20 17
Friday 18 20 22 22
(a) Calculate all sample means and the mean of all sample means.
(b) Calculate upper and lower 2-sigma x-bar chart control limits that allow for natural variations.
(c) Based on the x-bar chart, is this process in control? Explain either natural variation causes or assignable variation causes.
Weight
Day Package 1 Package 2 Package 3 Package 4
Monday 23 22 23 20
Tuesday 23 21 19 21
Wednesday 20 19 20 21
Thursday 18 19 20 17
Friday 18 20 22 22
(a) Calculate all sample means and the mean of all sample means.
(b) Calculate upper and lower 2-sigma x-bar chart control limits that allow for natural variations.
(c) Based on the x-bar chart, is this process in control? Explain either natural variation causes or assignable variation causes.
Statistics and Probability
A traffic survey on a busy highway reveals that one of every four vehicles is a truck. This survey also established that one-eighth of all automobiles are unsafe to drive and one-twentieth of all trucks are unsafe to drive.
a. What i s the probability that the next vehicle to pass a given point i s an unsafe truck?
b. What is the probability that the next vehicle will be a truck, given that it is unsafe?
c. What is the probability that the next vehicle that passes a given point will be a truck, given that the previous vehicle was an automobile?
a. What i s the probability that the next vehicle to pass a given point i s an unsafe truck?
b. What is the probability that the next vehicle will be a truck, given that it is unsafe?
c. What is the probability that the next vehicle that passes a given point will be a truck, given that the previous vehicle was an automobile?
Statistics and Probability
The school board is trying to evaluate a new math program introduced to second- graders in five elementary schools across the county this year. A sample of the student
scores on standardized math tests in each elementary school yielded the following data:
SCHOOL NO OF TEST ERRORS
A 52
B 27
C 35
D 44
E 55
Construct a c-chart for test errors, and set the control limits to contain 99.73% of the
random variation in test scores. What does the chart tell you? Has the new math program
been effective
scores on standardized math tests in each elementary school yielded the following data:
SCHOOL NO OF TEST ERRORS
A 52
B 27
C 35
D 44
E 55
Construct a c-chart for test errors, and set the control limits to contain 99.73% of the
random variation in test scores. What does the chart tell you? Has the new math program
been effective
