Exam: Management Research Methods 1 (MRM1)
Friday 13 January 2017, 09:00–12:00 (3 hours)
Solutions version 2 – corrected some typos
Question 1
1a. (3) What does 92% stand for in “92% confidence interval of a population mean”?
When samples are taken repeatedly, and each time a 92% confidence interval is calculated using the
sample, then 92% of all the calculated confidence intervals will be contain the true value of μ.
1b. (3) In a hypothesis test, a p-value of 0.02 means that there is only probability of 0.02 that the
null hypothesis is correct. True or false? Explain.
False. The p-value is the one or two-sided tail probability of the sample outcome. Or: The p-value is
the probability of finding the observed results or more extreme results, when the null hypothesis H0
is true – the definition of 'extreme' depends on how the hypothesis is being tested.
1c. (3) Plot three scatter diagrams, each with a correlation coefficient of approximately zero:
𝑟𝑟𝑋𝑋𝑋𝑋 ≈ 0.0. Each scatter diagram should contain at least twelve data points (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ) and the
three diagrams should be clearly different formations of points.
Many possible scatter diagrams, such as:
1
, Question 2
2a. (4) Use Tukey’s method to draw, on your answer sheet, a very precise boxplot of the following
bi-modal distribution of 28 data points. Is this distribution called positively skewed or
negatively skewed or not skewed?
• •
• • •
• • • •
• • • • •
• • • • • • • • • • • • • •
5- • •
4- • • •
3- • • • •
2- • • • • •
1- • • • • • • • • • • • • • •
* *
Skewness: POSITIVE
There is also another variable, called Volumes, from which three new variables are generated:
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = √𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑙𝑙𝑙𝑙(𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉2 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 2
The following output is made:
Statistics
Volumes VolumesSR VolumesLN VolumesX2
Valid 25 25 25 25
N
Missing 0 0 0 0
Mean 8.4844 2.6740 1.7959 133.8238
Std. Deviation 8.02594 1.17885 0.84025 268.86295
Skewness 2.048 1.199 0.110 3.123
Std. Error of Skewness 0.464 0.464 0.464 0.464
Kurtosis 4.218 1.367 0.060 10.149
Std. Error of Kurtosis 0.902 0.902 0.902 0.902
2b. (2) How can we conclude from this table that Volumes consist of only positive values?
Negative or zero values cannot be transformed using LN. There would be values “missing” under
VolumesLN. Since there are no values missing, there are no negative or zero Volumes.
“Is the population median of Volumes not equal to 8.5?” We will test this research question with a
significance level of 5%, by breaking the whole process down in three questions 2c), 2d) and 2e).
(NB. You might want to read all three questions before answering them)
2c. (6) First, motivate in detail why it is better to test the median of Volumes, instead of the mean.
Motivation why to test the median and not the mean of Volumes:
2
Friday 13 January 2017, 09:00–12:00 (3 hours)
Solutions version 2 – corrected some typos
Question 1
1a. (3) What does 92% stand for in “92% confidence interval of a population mean”?
When samples are taken repeatedly, and each time a 92% confidence interval is calculated using the
sample, then 92% of all the calculated confidence intervals will be contain the true value of μ.
1b. (3) In a hypothesis test, a p-value of 0.02 means that there is only probability of 0.02 that the
null hypothesis is correct. True or false? Explain.
False. The p-value is the one or two-sided tail probability of the sample outcome. Or: The p-value is
the probability of finding the observed results or more extreme results, when the null hypothesis H0
is true – the definition of 'extreme' depends on how the hypothesis is being tested.
1c. (3) Plot three scatter diagrams, each with a correlation coefficient of approximately zero:
𝑟𝑟𝑋𝑋𝑋𝑋 ≈ 0.0. Each scatter diagram should contain at least twelve data points (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ) and the
three diagrams should be clearly different formations of points.
Many possible scatter diagrams, such as:
1
, Question 2
2a. (4) Use Tukey’s method to draw, on your answer sheet, a very precise boxplot of the following
bi-modal distribution of 28 data points. Is this distribution called positively skewed or
negatively skewed or not skewed?
• •
• • •
• • • •
• • • • •
• • • • • • • • • • • • • •
5- • •
4- • • •
3- • • • •
2- • • • • •
1- • • • • • • • • • • • • • •
* *
Skewness: POSITIVE
There is also another variable, called Volumes, from which three new variables are generated:
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = √𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑙𝑙𝑙𝑙(𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉2 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 2
The following output is made:
Statistics
Volumes VolumesSR VolumesLN VolumesX2
Valid 25 25 25 25
N
Missing 0 0 0 0
Mean 8.4844 2.6740 1.7959 133.8238
Std. Deviation 8.02594 1.17885 0.84025 268.86295
Skewness 2.048 1.199 0.110 3.123
Std. Error of Skewness 0.464 0.464 0.464 0.464
Kurtosis 4.218 1.367 0.060 10.149
Std. Error of Kurtosis 0.902 0.902 0.902 0.902
2b. (2) How can we conclude from this table that Volumes consist of only positive values?
Negative or zero values cannot be transformed using LN. There would be values “missing” under
VolumesLN. Since there are no values missing, there are no negative or zero Volumes.
“Is the population median of Volumes not equal to 8.5?” We will test this research question with a
significance level of 5%, by breaking the whole process down in three questions 2c), 2d) and 2e).
(NB. You might want to read all three questions before answering them)
2c. (6) First, motivate in detail why it is better to test the median of Volumes, instead of the mean.
Motivation why to test the median and not the mean of Volumes:
2
