Response to the following questions:
Question 1.1:
GDP data for some countries across 3 consecutive years are recorded in the following table:
|
Country
|
GDP (Sb)
|
|
2014
|
2015
|
2016
|
|
China
|
834
|
935
|
1035
|
|
U.S.
|
582
|
732
|
895
|
|
Russia
|
495
|
631
|
788
|
|
Japan
|
486
|
621
|
799
|
|
Germany
|
339
|
456
|
552
|
|
Canada
|
201
|
335
|
453
|
|
Australia
|
304
|
444
|
565
|
|
Turkey
|
703
|
819
|
919
|
(i) Conduct appropriate calculations to show in a tabular format:
a) Proportion of change from 2014 to 2015 and 2015 to 2016
b) Share of GDP for the countries across three years
(ii) Use appropriate graphs to depict:
a) Volume of GDP for the countries across three years;
b) Share of GDP for the countries across three years;
c) Proportion of change (from 2014 to 2015) and comment on this.
Question 1.2
In a computer assembly line, workers take different time to assemble the parts. The amount of time needed (in seconds) for the assembly was recorded for 100 workers.
|
360
|
378
|
78
|
104
|
125
|
|
216
|
299
|
119
|
154
|
174
|
|
317
|
244
|
313
|
452
|
210
|
|
196
|
210
|
388
|
225
|
422
|
|
468
|
456
|
160
|
171
|
175
|
|
320
|
235
|
310
|
106
|
222
|
|
200
|
109
|
281
|
195
|
160
|
|
213
|
260
|
280
|
242
|
222
|
|
140
|
101
|
272
|
121
|
223
|
|
290
|
246
|
213
|
236
|
234
|
|
195
|
262
|
110
|
211
|
255
|
|
220
|
310
|
119
|
155
|
95
|
|
255
|
360
|
222
|
435
|
145
|
|
289
|
194
|
110
|
124
|
433
|
|
194
|
237
|
210
|
123
|
210
|
|
374
|
203
|
380
|
123
|
349
|
|
382
|
240
|
105
|
135
|
220
|
|
185
|
275
|
103
|
315
|
111
|
|
219
|
455
|
280
|
167
|
455
|
|
215
|
291
|
188
|
223
|
224
|
a) Determine the approximate number of classes and the class width (round-up the width if necessary) you would use to represent the above data? Explain your reasons.
b) Create a frequency distribution of time for 100 workers
c) Draw a frequency histogram
d) Calculate the relative frequency and draw relevant histogram
e) Draw an Ogive
f) Find the proportion of time taken for assembly between 100-150 seconds; 200-250 seconds. Explain your answer fully.
Question 2.1.
(i) Giovanni is thinking of opening a pizza stall in an upcoming local carnival. Based on past experience he estimates the probability distribution of the number of pizzas he will sell each day. The probability distribution is given in the table below:
|
Number of Pizza
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
|
Probability
|
0.03
|
0.18
|
0.21
|
0.26
|
0.14
|
0.11
|
0.07
|
Giovanni sells a pizza at a price of $12. His costs include a fixed cost of $50 for renting a pizza oven and the cost of raw materials for preparing each pizza is $5.
(a) Calculate the probability of selling less than 20 pizzas in a day.
(b) What is the probability that Giovanni will make a profit of more than $100 in a day?
(ii) In the state of Wyoming, the speed of motorists travelling on the state highway is uniformly distributed between 55 and 115 miles per hour.
(a) Derive the density function for speed of motorists. Draw a graph to explain your answer.
(b) If the speed of a motorist was checked at random, what is the probability that it will be travelling between 65 and 85 miles per hour?
(c) A study by highway safety professionals find that motorists travelling at speeds above the third quartile (75th percentile) are highly prone to accidents. They ask the Governor of Wyoming to ban travelling at speeds above the third quartile. What should be the speed limit set by the Governor?
Question 2.2.
An investor is looking at shares of two companies - Coal Energy Ltd and Green Energy Ltd for possible investment. Based on past performance the investor knows that the share returns of both companies follow a Normal distribution. In addition, their returns are negatively correlated with a correlation coefficient of -0.90. The mean and standard deviations of the returns are provided in the table below:
|
Share
|
Coal Energy
|
Green Energy
|
|
Mean
|
5
|
12
|
|
Standard Deviation
|
2
|
8
|
(a) Consider the shares of Coal Energy. Construct a symmetric interval around the mean return such that the probability of getting a return within that interval is 0.90.
(b) Suppose a fixed deposit would provide a guaranteed return of 3% to the investor. What is the probability of getting a higher return if the investor decides to invest in Coal Energy shares?
(c) Suppose the investor buys Green Energy shares. Calculate the probability of getting a higher return compared to the fixed deposit? Compare your answer to part (b) and provide a brief explanation.
(d) Suppose we construct a portfolio with 70% of Coal Energy and 30% of Green Energy shares. Compute the mean and variance of the portfolio.
(e) Does the portfolio provide a superior investment option compared to investing in Coal Energy shares? Provide a brief explanation.
Question 3.1
(i) The recent average starting salary for new college graduates in computer information systems is $47 500. Assume that salaries are normally distributed, with a standard deviation of $4500.
a. What is the probability of a new graduate receiving a salary between $45 000 and $50 000?
b. What is the probability of a new graduate getting a starting salary in excess of $55 000?
c. What percentage of starting salaries are no more than $42 250?
d. What is the cut-off for the bottom 5% of the salaries?
(ii) The probability distribution for X, daily demand of a particular newspaper at a local news agency, (in hundreds) is as follows:
|
x
|
1
|
2
|
3
|
4
|
|
p(x)
|
0.05
|
0.42
|
0.44
|
0.09
|
a. Find and interpret the expected value of X.
b. Find V(X).
c. Find and interpret σ.
Question 3.2
(i) A sample of 30 observations is drawn from a normal population with mean of 750 and a standard deviation of 300. Suppose the population size is 600.
a. Find the expected value of the sample mean.
b. Find the standard error of the sample mean.
c. Find P (sample mean > 790).
d. Find P (sample mean < 650).
e. Find P (760 < sample mean < 810).
(ii) Find and interpret a 98% confidence interval for the mean number of animals visited by a veterinarian per day. A random sample of 35 veterinarians, found that they had a sample mean of 25.3 animals and a sample variance of 2.8 animals.