EMA1100 Business Statistics 1 is an introductory course offered by the University of Malta for undergraduate students pursuing a degree in business or a related field. The course provides students with a foundation in statistical methods and data analysis as they apply to business decision-making.
Throughout the course, students will learn how to collect and analyze data, as well as how to use statistical methods to draw meaningful conclusions from that data. Topics covered include descriptive statistics, probability, random variables, probability distributions, and hypothesis testing. Additionally, the course will introduce students to the use of statistical software packages, such as Microsoft Excel, in data analysis.
Upon completion of this course, students will be able to apply statistical methods and tools to real-world business problems, evaluate the validity of statistical results, and effectively communicate their findings to others. This course is designed to provide students with the fundamental statistical knowledge and skills necessary for success in advanced business courses and in their future careers.
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Learning Outcome 1: Understand the difference between different measures of location
Measures of location are statistical measures that help us understand where the center of a dataset is. There are different types of measures of location, including mean, median, and mode. Each of these measures has a different way of calculating the center of the dataset.
The mean is the most commonly used measure of location. It is calculated by adding up all the values in the dataset and dividing the sum by the number of values in the dataset. The mean is affected by outliers, which are values that are far away from the other values in the dataset.
The median is another measure of location that is not affected by outliers. The median is the middle value in a dataset when the values are arranged in order from smallest to largest. If there is an even number of values in the dataset, then the median is the average of the two middle values.
The mode is the value that occurs most frequently in a dataset. The mode is not affected by outliers, but it may not be a good measure of location if there are multiple modes or if there is no mode.
Learning Outcome 2: Understand the difference between different measures of variation
Measures of variation are statistical measures that describe the spread or dispersion of a dataset. There are different types of measures of variation, including range, variance, and standard deviation. Each of these measures has a different way of quantifying the spread of the dataset.
The range is the simplest measure of variation and is calculated by subtracting the smallest value in the dataset from the largest value. The range is sensitive to outliers, which are values that are far away from the other values in the dataset.
The variance and standard deviation are more commonly used measures of variation. The variance is calculated by taking the average of the squared differences between each value in the dataset and the mean of the dataset. The standard deviation is the square root of the variance. Both the variance and standard deviation are affected by outliers.
The standard deviation is often preferred over the variance because it is expressed in the same units as the original data, whereas the variance is expressed in squared units.
Learning Outcome 3: Understand the difference between different plots and charts
Plots and charts are visual representations of data that can help us understand patterns and relationships in the data. There are many different types of plots and charts, each of which is suited to different types of data and research questions.
Scatter plots: A scatter plot is a type of plot that is used to display the relationship between two continuous variables. The data points are plotted on a two-dimensional plane, with one variable on the x-axis and the other on the y-axis. The pattern of the data points can reveal whether there is a positive, negative, or no relationship between the variables.
Line graphs: A line graph is a type of plot that is used to display changes in a variable over time. The variable is plotted on the y-axis, and time is plotted on the x-axis. Line graphs are useful for showing trends and patterns over time.
Bar charts: A bar chart is a type of chart that is used to display categorical data. The categories are plotted on the x-axis, and the frequency or proportion of each category is plotted on the y-axis. Bar charts are useful for comparing the frequency or proportion of different categories.
Histograms: A histogram is a type of plot that is used to display the distribution of a continuous variable. The variable is divided into a series of intervals or bins, and the frequency or proportion of data points that fall into each bin is plotted on the y-axis. Histograms can reveal the shape of the distribution of the variable, such as whether it is skewed or symmetric.
Box plots: A box plot is a type of plot that is used to display the distribution of a continuous variable. The box in the plot represents the interquartile range (IQR) of the data, which is the range of values between the 25th and 75th percentiles. The line inside the box represents the median, and the whiskers extend to the minimum and maximum values within a certain range, typically 1.5 times the IQR. Box plots can reveal the spread and skewness of the distribution of the variable.
Learning Outcome 4: Understand the most important fundamentals of probability
Probability is a fundamental concept in statistics and mathematics that is used to quantify the likelihood of events occurring. Here are some of the most important fundamentals of probability:
Sample space: The sample space is the set of all possible outcomes of an experiment. For example, the sample space of rolling a die is {1, 2, 3, 4, 5, 6}.
Event: An event is a subset of the sample space. For example, the event of rolling an even number on a die is {2, 4, 6}.
Probability: Probability is a number between 0 and 1 that quantifies the likelihood of an event occurring. The probability of an event is calculated as the ratio of the number of outcomes in the event to the total number of outcomes in the sample space. For example, the probability of rolling an even number on a die is 3/6 or 0.5.
Mutually exclusive events: Mutually exclusive events are events that cannot occur at the same time. For example, rolling an even number and rolling an odd number on a die are mutually exclusive events.
Independent events: Independent events are events that do not affect the probability of each other occurring. For example, the probability of rolling a 4 on a die is independent of the probability of flipping a coin and getting heads.
Conditional probability: Conditional probability is the probability of an event occurring given that another event has occurred. It is calculated as the probability of the intersection of the two events divided by the probability of the given event. For example, the probability of rolling a 4 on a die given that the roll is even is 1/3 or 0.33.
Bayes’ theorem: Bayes’ theorem is a formula that allows us to update our belief about the probability of an event given new evidence. It is based on conditional probability and is widely used in statistics and machine learning.
Learning Outcome 5: Understand the importance of random variables
Random variables are an important concept in statistics and probability theory. They are used to describe and analyze the variability and uncertainty of real-world phenomena.
A random variable is a variable that takes on different values based on the outcome of a random process or experiment. For example, the outcome of flipping a coin can be described by a random variable that takes on the values of “heads” and “tails” with equal probability.
The importance of random variables lies in their ability to quantify uncertainty and variability in a rigorous and systematic way. By defining a random variable, we can specify the probability distribution of its values, which describes the likelihood of each possible value occurring.
Random variables are also used to model and analyze complex real-world phenomena, such as the performance of a stock portfolio, the effectiveness of a medical treatment, or the reliability of a manufacturing process. By defining appropriate random variables and probability distributions, we can make predictions and draw inferences about these phenomena, even in the presence of uncertainty and variability.
In addition, random variables are essential for the development of statistical models and methods. For example, regression analysis, hypothesis testing, and Bayesian inference all rely on the use of random variables to model and analyze data.
Learning Outcome 6: Understand the difference between discrete and continuous distributions
In probability theory and statistics, we use probability distributions to model the behavior of random variables. There are two main types of probability distributions: discrete and continuous.
Discrete distributions are used to model random variables that take on a finite or countable set of values. For example, the number of children in a family, the number of cars sold in a day, or the outcomes of rolling a die are all examples of random variables that can be modeled using a discrete distribution. In a discrete distribution, the probabilities are assigned to individual values of the random variable, and the sum of the probabilities across all possible values of the random variable is equal to 1.
Continuous distributions, on the other hand, are used to model random variables that can take on any value within a specified range or interval. For example, the height of a person, the weight of a product, or the time taken to complete a task are all examples of random variables that can be modeled using a continuous distribution. In a continuous distribution, the probabilities are assigned to intervals of values of the random variable, rather than individual values. The probability density function (PDF) is used to describe the probability distribution of a continuous random variable, and the area under the PDF curve over a specified interval represents the probability of the random variable taking on a value within that interval.
The main difference between discrete and continuous distributions is in the way that probabilities are assigned. In a discrete distribution, the probabilities are assigned to individual values of the random variable, while in a continuous distribution, the probabilities are assigned to intervals of values of the random variable. This difference has important implications for the calculation of probabilities, the choice of statistical methods, and the interpretation of results.
Learning Outcome 7: Calculate different measures of location and variation
There are different measures of location and variation that are used to summarize and describe the distribution of a set of data.
Measures of location:
Mean: The mean, or average, is the sum of all the values in a dataset divided by the number of values. It represents the center of the distribution. The formula for calculating the mean is:
mean = (sum of values) / (number of values)
Median: The median is the value that separates the dataset into two equal parts, with half the values above and half the values below. It represents the midpoint of the distribution. To calculate the median, the values in the dataset are first sorted in ascending or descending order, and then the middle value(s) is/are chosen, depending on whether the dataset has an odd or even number of values.
Mode: The mode is the value that occurs most frequently in the dataset. It represents the most common value in the distribution. A dataset can have multiple modes, or no mode at all.
Measures of variation:
Range: The range is the difference between the largest and smallest values in the dataset. It represents the spread of the data. The formula for calculating the range is:
range = (largest value) – (smallest value)
Variance: The variance is a measure of how spread out the data is from the mean. It is calculated by taking the average of the squared differences between each value in the dataset and the mean. The formula for calculating the variance is:
variance = (sum of squared differences from the mean) / (number of values – 1)
Standard deviation: The standard deviation is the square root of the variance. It represents the typical distance between each value in the dataset and the mean. The formula for calculating the standard deviation is:
standard deviation = sqrt(variance)
These measures of location and variation are useful for summarizing and comparing datasets, identifying outliers, and detecting patterns in the data.
Learning Outcome 8: Construct various charts and graphs
There are various charts and graphs that can be used to visually represent and analyze data. Here are some common types:
Bar chart: A bar chart is a chart that uses rectangular bars to represent and compare the sizes of different categories or groups. The height or length of the bars represents the value or frequency of each category. Bar charts are often used to show categorical data and comparisons.
Line chart: A line chart is a chart that uses a line to connect data points and show the trends or changes in a set of data over time. Line charts are often used to show continuous data and patterns.
Scatter plot: A scatter plot is a graph that uses dots or points to represent the values of two variables in a dataset. The position of each dot represents the values of the two variables, and the pattern or trend of the dots can reveal any relationship or correlation between the variables.
Pie chart: A pie chart is a chart that uses a circle divided into sectors to represent the proportion or percentage of each category in a dataset. The size of each sector is proportional to the value or frequency of each category. Pie charts are often used to show the composition or distribution of data.
Histogram: A histogram is a chart that uses rectangular bars to represent the frequency or count of data within intervals or bins. The height of each bar represents the frequency or count of data within the interval. Histograms are often used to show the distribution and frequency of continuous data.
Box plot: A box plot is a graph that uses a box and whiskers to represent the distribution of data. The box represents the middle 50% of the data, and the whiskers represent the range of the data. Box plots are often used to show the variability and outliers of a dataset.
These charts and graphs can be created using various software and tools, such as Excel, R, Python, or online chart-making websites. The choice of chart or graph depends on the type of data, the research question, and the audience.
Learning Outcome 9: Work out a number of core examples related to probability theory
Here are some core examples related to probability theory:
Coin toss: A coin toss is a classic example of probability theory. The probability of getting heads or tails is 0.5 or 50%, assuming a fair coin.
Dice roll: Rolling a six-sided die is another common example. The probability of getting any one number is 1/6, or approximately 16.7%.
Card draw: Drawing a card from a deck of 52 cards is another example. The probability of drawing a specific card is 1/52, or approximately 1.9%. The probability of drawing a card of a specific suit is 1/4, or 25%.
Birthday problem: The birthday problem is a probability problem that asks how many people are needed in a room for there to be a greater than 50% chance that two people share the same birthday. The answer is 23, assuming each birthday is equally likely.
Monty Hall problem: The Monty Hall problem is a probability puzzle named after a game show host. It asks a contestant to choose one of three doors, behind one of which is a prize. After the contestant chooses, the host opens one of the other doors, revealing that it does not contain the prize. The contestant is then given the option to switch their choice to the remaining door. The probability of winning the prize is higher if the contestant switches their choice.
Normal distribution: The normal distribution is a common probability distribution used to model many natural phenomena. It is characterized by a bell-shaped curve and has two parameters: the mean and the standard deviation.
Learning Outcome 10: Calculate the mean and the variance from a probability distribution;
To calculate the mean and variance from a probability distribution, follow these steps:
Find the expected value of the distribution, which is the weighted average of the values in the distribution. For a discrete distribution, it is calculated as:
E(X) = Σ xi * P(X = xi)
where xi is the value of the random variable, and P(X = xi) is the probability of that value occurring.
For a continuous distribution, the expected value is calculated as:
E(X) = ∫ xf(x) dx
where f(x) is the probability density function of the distribution.
Find the variance of the distribution, which measures the spread of the distribution around the mean. The variance is calculated as:
Var(X) = E[(X – E(X))^2]
where E(X) is the expected value of the distribution, and (X – E(X))^2 is the squared deviation of each value from the mean.
Alternatively, the variance can be calculated using the formula:
Var(X) = E(X^2) – [E(X)]^2
where E(X^2) is the expected value of the squared values of the random variable.
Take the square root of the variance to find the standard deviation, which is another measure of the spread of the distribution.
For example, let’s say we have a discrete probability distribution with values of X and their probabilities P(X):
| X | 1 | 2 | 3 | 4 |
| P(X) | 0.2 | 0.3 | 0.4 | 0.1 |
To find the mean, we use the formula:
E(X) = Σ xi * P(X = xi)
E(X) = 1 * 0.2 + 2 * 0.3 + 3 * 0.4 + 4 * 0.1
E(X) = 2.6
To find the variance, we use the formula:
Var(X) = E[(X – E(X))^2]
Var(X) = (1-2.6)^2 * 0.2 + (2-2.6)^2 * 0.3 + (3-2.6)^2 * 0.4 + (4-2.6)^2 * 0.1
Var(X) = 1.04
Therefore, the mean of the distribution is 2.6 and the variance is 1.04.
Learning Outcome 11: Use various distributions to calculate probability values.
To calculate probability values using different distributions, follow these steps:
Identify the distribution that best fits the problem at hand. Some common probability distributions include the binomial distribution, the normal distribution, the Poisson distribution, and the exponential distribution.
Determine the parameters of the distribution. Each distribution has specific parameters that affect its shape and location, such as the mean and standard deviation for the normal distribution, or the rate parameter for the Poisson distribution.
Use the probability distribution function (PDF) or cumulative distribution function (CDF) to calculate the desired probability value. The PDF gives the probability of a specific outcome, while the CDF gives the probability of an outcome less than or equal to a specific value.
For example, let’s say we want to calculate the probability of getting exactly 3 heads in 5 coin flips. We can use the binomial distribution to model this problem, with parameters n = 5 (the number of trials) and p = 0.5 (the probability of getting heads in each trial). The probability of getting exactly k successes in n trials with probability of success p is given by the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where (n choose k) is the binomial coefficient.
So, in this case, we want to calculate P(X = 3):
P(X = 3) = (5 choose 3) * 0.5^3 * 0.5^2
P(X = 3) = 0.3125
Therefore, the probability of getting exactly 3 heads in 5 coin flips with a fair coin is 0.3125.
As another example, let’s say we want to calculate the probability of a car taking less than 10 seconds to accelerate from 0 to 60 mph, given that the acceleration time follows a normal distribution with a mean of 8 seconds and a standard deviation of 1.5 seconds. We can use the normal distribution and the CDF to calculate this probability. The CDF of a normal distribution with mean μ and standard deviation σ is given by the formula:
F(x) = 1/2 * [1 + erf((x-μ)/(σ*sqrt(2)))]
where erf is the error function.
So, in this case, we want to calculate F(10):
F(10) = 1/2 * [1 + erf((10-8)/(1.5*sqrt(2)))]
F(10) = 1/2 * [1 + erf(0.9428)]
F(10) = 1/2 * [1 + 0.8276]
F(10) = 0.9138
Therefore, the probability of a car taking less than 10 seconds to accelerate from 0 to 60 mph is approximately 0.9138, or 91.38%.
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