Business resource and Statistics practical exam
Question one-
1. What is the Central Limit Theorem? Why and when is it used? Please elaborate on how the Central Limit Theorem and the Empirical Rule are connected. Please use a graph to illustrate your answer.
2. In probability theory, if population mean is μ and standard deviation is σ, we know that the standard deviation of a sampling distribution is σ/√n If a random sample of size n = 49 is obtained from a population with µ = 75 and (population standard deviation) = 14
What can you say about the distribution of the sampling dataset in terms of?
i. The shape of the distribution curve?
ii. The value of the mean?
iii. The value of the standard deviation
Quewstion two -
i. What is meant by the term interquartile range
ii. Describe the steps you need to take to calculate the interquartile range
iii. Find the median, lower quartile, upper quartile and interquartile range of the following data set:
18 20 23 20 23 27 24 23 29
iv. Draw a box and whisker plot to visualise the five number summary of the data set in iii. above
iv. What is a binomial distribution and when is it used? Please list the five key aspects of the binomial distribution
vi. For a binomial distribution we know that the standard deviation, σ = √np(1-P)
Where n is the number of trials and p is the probability of success.
Calculate the standard deviation of the binomial distribution if n=4 and P=0.25
Question Three-
1. Explain the difference between the following terms. It would be useful to explain your answer with the diagram.
A. Data (or Raw Data)
B. Information (or records)
Question four-
Consider the following three data sets; A, B and C
A. = {9, 10, 11, 7, 13}
B. = {10, 10, 10, 10, 10}
C. = {{1, 1, 10, 19, 19}
I. Calculate the mean of each data set
II. Calculate the standard deviation of each data set, given the formula:
III. Which set has the largest standard deviation?
IV. Is it possible to answer question iii) without calculations of the standard deviation? Explain your answer (show the rough or working)
C. Statistics
D. Knowledge
2. List four basic ways of visualising data.
3. Briefly explain the difference between descriptive statistics and inferential statistics.
4. Univariate Analysis involves the examination across cases of one variable at a time. In analysing a variable, there are three major characteristics that we tend to look at:
Distribution
Central Tendency
Dispersion/Variability
Briefly explain each of these terms and give examples of each. You can use drawings to illustrate your answer
5. With regard to the standard deviation of data distributions, what is the Empirical Rule?
Question Five-
Fill in the missing words on the answer sheet provided.
1. A study is under way in Lough Key Forest Park to determine the adult height of oak Trees. Specifically, the study is attempting to determine what factors aid a tree in reaching heights greater than 15m tall. It is estimated that the park contains 25,000 adult oak trees. The study involves collecting heights from 250 randomly selected adult oak trees and analysing the results. Identify the population from which the study was sampled.
A. The 250 randomly selected adult oak trees.
B. The 25,000 adult oak trees in the park.
C. All the adult oak trees taller than 15m.
D. All chestnut trees, of any age, in the park.
2. From the same study identify the variable of interest:
A. The age of an oak tree in Lough Key Forest Park.
B. The height of an oak tree in Lough Key Forest Park.
C. The number of oak trees in Lough Key Forest Park.
D. The species of trees in Lough Key Forest Park.
3. Again from the same study above what is the sample in the study?
A. The 250 randomly selected adult oak trees.
B. The 25,000 adult oak trees in the park.
C. All the adult oak trees taller than 15m.
D. All oak trees, of any age, in the park.
4. To monitor campus security, the security superintendent is taking a survey of the number of students in a concourse each 30 minutes of a 24-hour period with the goal of determining when patrols of the concourse would best serve the most students. If X is the number of students in the concourse each period of time, then X is an example of
A. A categorical random variable.
B. A discrete random variable.
C. A continuous random variable.
D. A statistic.
Question Six-
1. What is the Central Limit Theorem? Why and when is it used? Please elaborate on how the Central Limit Theorem and the Empirical Rule are connected. Please use a graph to illustrate your answer.
2. In probability theory, if population mean is μ and standard deviation is σ, we know that the standard deviation of a sampling distribution is σ/√n If a random sample of size n = 49 is obtained from a population with µ = 75 and (population standard deviation) = 14
What can you say about the distribution of the sampling dataset in terms of?
i. The shape of the distribution curve?
ii. The value of the mean?
iii. The value of the standard deviation?
Question seven
i. What is meant by the term interquartile range
ii. Describe the steps you need to take to calculate the interquartile range
iii. Find the median, lower quartile, upper quartile and interquartile range of the following data set:
18 20 23 20 23 27 24 23 29
iv. Draw a box and whisker plot to visualise the five number summary of the data set in iii. above
i. What is a binomial distribution and when is it used? Please list the five key aspects of the binomial distribution
ii. For a binomial distribution we know that the standard deviation, σ = √np(1-P) Where n is the number of trials and p is the probability of success.
Calculate the standard deviation of the binomial distribution if n=4 and P=0.25
Question Eight
1. What is a "Normal Distribution" of data, and how would you recognise it if it was graphed on a histogram?
2. What are two branches of statistics in brief and explain statistical terms in detail.
2. Explain population parameters and sample statistics. Also define measures of central tendency.
3. Calculate mean, median, mode, variance and standard deviation of the following dataset:
45, 50, 55, 55, 55, 60, 40
. Explain percentiles, quartiles and Inter Quartiles. Describe population, sample proportions and various types of charts used in statistics
4. Explain the concept of probability, and detail calculations of population, variance and standard showing formulas.
5.What is data analytics? Explain data analytics at workplace and provide various methods of turning data to knowledge.
6. Explain data mining and business reporting. Give appropriate examples of business reporting tools.
7.Explain central limit theorem in detail by giving examples.
8. A quiz consists of four multiple-choice questions, each with four possible answer choices (A, B, C, or D). One of which is correct. Suppose that an unprepared student does not read the question, but simply makes a random guess for each question. Let the random variable X equal the number of correct guesses the student makes for the five questions. Is this binomial? And if so what is n and what is p? Calculate variance and standard deviation.
The level of economic activity in a region has been recorded over a period of four years and the data is presented below:
Year Quarter Activity Level
1 1 105
2 99
3 90
4 110
2 1 111
2 104
3 93
4 119
3 1 118
2 109
3 96
4 127
4 1 126
2 115
3 100
4 135
a. Construct a graph of this data
b. Find a centred four-point moving average and place it on your graph
c. Calculate the corresponding seasonal components(*show the rough or working of it)
Question nine
Sales on article B (‘000 units)
Q1 Q2 Q3 Q4
2010 24.8 36.3 38.1 47.5
2011 31.2 42.0 43.4 55.9
2012 40.0 48.8 54.0 69.1
2013 54.7 57.8 60.3 68.9
a. Plot the time series of the sales figures
b. Find a centred four-point moving average and place it on your graph
c. Calculate the seasonal component for each quarter
Comment on the following:
‘The process of polling is often mysterious, particularly to those who don't see how the views of 1,000 people can represent those of hundreds of millions'.
Question 11
Discuss the relative advantages and disadvantages of the postal questionnaire and the personal interview as a means of collecting data.
b. Compare simple random sampling and quota sampling as methods of selecting a representative sample from a population.
Question 12
Sampling methods are frequently used for the collection of data. State which type of sampling method is being described in the following situations.
i. One school in an area is selected at random and then all pupils in that school are surveyed.
ii. The local authority has a list of all pupils in the area and the sample is selected in such a way that all pupils have an equal probability of selection
iii. An interviewer surveys pupils emerging from every school in the area, attempting to question them randomly but in line with specified numbers of boys and girls in the various age groups.
iv. The local authority has a list of all pupils in the selected area. The first pupil is selected randomly from the list and then after every 100th pupil thereafter is selected for the survey.
B. List the advantages and disadvantages of Quota Sampling.
Question 13
The following transactions have been recorded on an automatic cash dispenser:
Value of transactions
(€): Number:
10 46
20 57
30 68
50 56
100 47
150 39
200 34
Required:
a. Determine the mean
b. What value has the mode
c. Draw an appropriate graph for the above data
Question 14
A. Give two examples of variables which might be correlated
B. Distinguish between positive and negative correlation
C. What range of values can the product moment correlation coefficient use?
D. When should Spearman's rank correlation coefficient be used?
E. What are the advantages and disadvantages of the least squares method of linear regression?