Statistics | Questions and Answers

1)Use the dataset-black chp2 to calculate the following: range, inter quartile range (IQR), mean absolute deviation (MAD), Mean, Mode, Median, Variance, Coefficient of Variation and Coefficient of Skewness. 2) Interpret the difference between the Mode, Median and Mean, as well as the Coefficient of Skewness: what do it's value say? 3) If 3 out of 10 Wittenborg students is a female, what is the probability of finding 14 females in a class of 20 students? 4) Calculate or determine the following probabilities, given that the average IQ is 100, and the standard deviation is 15: - p (someones IQ is the population average or less) - p (someones IQ is less than 115) - p (someones IQ is less than 85 or bigger than 115) - p( someone IQ is bigger than 124 -or fill in your own IQ if known) 4) List three examples of sampling and three of non-sampling errors in research. Discuss how to change them into correct actions for investigation. 5) Take the dataset wage-example, taken from ch 10

STATISTICS
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Question 1
Numerical summary of the given dataset
Question 2
The mean value is 40.43 while the median is 37 and mode is 30. The difference in these
measures of central tendency implies that the given data is not normally distributed. Also,
higher value of mean than median is also on account of some abnormally higher values which
may be present in the dataset. The skew value is 0.85 which indicates that dataset is non-
normal with longer tail on the right (Flick. 2015).
Question 3
Number of female students = 3
Total students = 10
Probability that female student is selected = 3/10 = 0.30
Probability to find 14 females in class of 20 students =?
This is binomial distribution and the requisite probability is computed below.
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Question 4
Average IQ = 100
Standard deviation = 15
Normal distribution
 Probability (X <= 100)
 Probability (X <= 115)
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 Probability (85<X<115)
 Probability (X > 124)
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Question 4
Three examples of sampling errors along with corrective action are given below (Hillier,
2016).
 Simple random sampling technique is used where the underlying population is
heterogeneous owing to which the attributes representation is essential. In order to rectify,
check whether sample is representative of population of interest.
 Selection of smaller sample size than required. This may result in loss of reliability since
the underlying sample would not be representative of the underlying population. In order
to rectify, it is imperative to use atleast minimum sample size.
 Use of non-probabilistic sampling techniques – If the non-probability based techniques
such as convenience sampling is used, then there is inherent bias in the sample owing to
which result reliability is low. The researcher should therefore try to use probability based
sampling techniques.
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Three examples of non-sampling errors along with corrective action are given below
(Eriksson and Kovalainen, 2015).
 Selection error – This may take place since in a survey only those who are interested in
taking the survey are represented. In order to reduce this error, it is imperative to enhance
the overall participation levels.
 Non-response error – It is likely that the actual sample of people whose data is collected
may be different from those originally chosen for the data. This may happen since a
sizable chunk does not participate in the survey. Follow up surveys may be done in order
to reduce this.
 Sample frame error – This happens when a wrong sub-population is used for extracting
sample owing to which the results obtained lack validity. In order to avoid this, it is
imperative to cross check is appropriate sample frame has been selected considering the
underlying study.
Question 5
The requisite hypotheses are as stated below.
Null Hypothesis: μ1 = μ2 i.e. there is no significant difference between the average wages of
the two managers with 1 being advertising.
Alternative Hypothesis: μ1 ≠ μ2 i.e. there is significant difference between the average wages
of the two managers with 1 being advertising.
Level of significance = 0.05
The appropriate test to be conducted is two sample independent t test. Based on the variance
for the two groups, the unequal variance form is preferred. The result from Excel is shown
below.
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Based on the above result, relevant two tail p value is 0.02 which is lower than 0.05 level of
significance. Hence, reject null hypothesis and alternative hypothesis is accepted (Hair et. al.,
2015). This implies that there is significant difference in wages of the two groups of
managers.
Even if the sample size is reduced to 33 managers each, then also the same test would be used
since the two samples would continue to be independent only and thus two sample
independent t test would be the correct choice.
Question 6
Sample size = 25
Mean IQ = 80
Standard deviation of IQ = 10
95% Confidence interval =?
Degree of freedom = 25-1 = 24
The t value for 95% Confidence interval and 24dof = 2.0639
Standard error = Standard deviation / SQRT(Sample size) = 10/ SQRT(25) = 2
Margin of error = z value * Standard error = 2.0639*2 = 4.1278
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Upper limit of 95% CI= Mean + Margin of error=80+4.1278=84.13
Lower limit of 95% CI= Mean - Margin of error=80-4.1278=75.87
Hence,
95% Confidence interval = [75.87 84.13]
The conclusion can be made with 95% confidence that the mean IQ will be between 75.87
and 84.13.
 When sample size = 49
Degree of freedom = 49-1 = 48
The t value for 95% Confidence interval and 24dof = 2.0106
Standard error = Standard deviation / SQRT(Sample size) = 10/ SQRT(49) = 1.4286
Margin of error = z value * Standard error = 2.0106*1.4286= 2.8723
Upper limit of 95% CI= Mean + Margin of error=80+2.8723=82.87
Lower limit of 95% CI= Mean - Margin of error=80-2.8723=77.13
Hence,
95% Confidence interval = [77.13 82.87]
The conclusion can be made with 95% confidence that the mean IQ will be between 77.13
and 82.87 when the sample size is 49. Clearly, in comparison with the confidence interval
before, the current confidence interval is narrower.
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References
Eriksson, P. and Kovalainen, A. (2015) Quantitative methods in business research. 3rd ed.
London: Sage Publications.
Flick, U. (2015) Introducing research methodology: A beginner's guide to doing a research
project. 4th ed. New York: Sage Publications.
Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., and Page, M. J. (2015) Essentials
of business research methods. 2nd ed. New York: Routledge.
Hillier, F. (2016) Introduction to Operations Research.6th ed. New York: McGraw Hill
Publications.
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