EC275 Statistics for Economics assignment sample NUIG Ireland

EC275 Statistics for Economics is a great unit for students interested in economics and data analysis. The unit covers a wide range of topics, including probability theory, statistical inference, econometrics, and game theory. The material is presented in a clear and concise manner, making it easy to follow along.

The unit starts with an introduction to statistics and probability theory. This is essential background information for understanding the rest of the material covered in the unit. You’ll learn about different types of data, how to collect and analyze data, and how to make valid inferences from your data. This section also covers important concepts such as sample space, events, Random Variables (RVs), etc. 

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We will go with some assignment briefs. These are

Assignment Brief 1: Descriptive Statistics: 

1.1 Distinguish between qualitative variables and quantitative variables.

Qualitative variables are usually categorical, and they can be either measured or unmeasured. For example, hair color (red, blond, brown, black), political affiliation (Republican, Democrat, Independent), or religious affiliation (Catholicism, Protestantism, Judaism) are all qualitative variables.

Quantitative variables are usually numerical values that can be measured. For example, weight (in pounds), height (in inches), or age (in years).

1.2 Describe how a discrete variable is different from a continuous variable.

A continuous variable is a variable that can be measured in any unit of measurement, while a discrete variable is a variable that can only be measured in specific units.

Continuous variables can take on any value between two extremes, while discrete variables can only take on certain specific values. For example, weight is a continuous variable because it can be measured in pounds, grams, ounces, etc., while age is a discrete variable because it can only be measured in years.

1.3 Distinguish among the nominal, ordinal, interval, and ratio levels of measurement.

There are four levels of measurement: nominal, ordinal, interval, and ratio. Each level has its own unique characteristics and uses.

Nominal level data is used to label or name something. It is the lowest level of measurement and does not involve any numerical values. Examples of Nominal data include gender (male/female), social class (upper, middle, lower), or eye color (brown, green, blue). 

Ordinal level data involves rank or order. It is one step up from Nominal data as it now includes numerical values, but these numbers only indicate the relative position of each value and do not show how much difference there is between them. An example of Ordinal data would be a customer satisfaction survey where respondents are asked to rate their level of satisfaction on a scale from 1 to 5 (1 being the lowest and 5 being the highest). 

Interval level data is similar to Ordinal data in that it involves rank or order, but Interval data also includes information about the amount of difference between each value. This level of measurement is often used in temperature measurements, as there can be an equal amount of difference between each degree (e.g., the difference between 60 degrees and 70 degrees is the same as the difference between 80 degrees and 90 degrees). 

Ratio level data is the highest level of measurement as it includes all of the information from the previous levels, but also has a true zero point. This means that there is an absolute value for each data point and we can compare the ratios of two or more values. An example of Ratio level data would be measurements of length, weight, or time. 

1.4 Organize and present qualitative data in a frequency table, bar chart, and pie chart.

Organizing and presenting qualitative data can be tricky, but frequency tables, bar charts, and pie charts are all great ways to do it. Here’s a quick rundown of each:

Frequency Tables

A frequency table is simply a list of all the different values that your data takes on, along with the number of times each value occurs. So, if you’re looking at data on people’s favorite colors, you might have a table that looks something like this:

Interpreting a frequency table is pretty straightforward – the most common value (in this case, “red”) is the mode, and the value that occurs least often (in this case, “purple”) is the median.

Bar Charts

A bar chart is a graphical way of representing data from a frequency table. Each value from the table is represented by a bar, and the height of the bar corresponds to how many times that value occurs. So, the bar chart for the favorite colors data would look something like this:

Pie Charts

A pie chart is a graphical way of representing data from a frequency table, but unlike a bar chart, a pie chart shows the data as a “slice” of a circle. So, if you were looking at data on people’s favorite pizza toppings.

1.5 Organize and present quantitative data in a frequency distribution, histogram, frequency polygon, and cumulative frequency polygon.

A frequency distribution is a table or graph that shows how often a particular value (or range of values) appears in a data set. The most common type of frequency distribution is the histogram, which is used to display the distribution of numerical data.

A histogram is created by dividing the range of data into intervals, and then counting how many data points fall within each interval. The resulting graph has one bar for each interval, with the height of the bar representing the frequency (number of data points) in that interval.

A frequency polygon is similar to a histogram, but it’s created by connecting the points on a graph that represent the frequencies in each interval. This gives a more visual representation of the shape of the data.

A cumulative frequency polygon is created by connecting the points on a graph that represents the cumulative frequencies in each interval. This gives a more visual representation of how the data is distributed.

1.6 Calculate the mean, median, and mode. 

There are a few different ways to calculate the mean, median, and mode.

The mean is simply the average of all the numbers in a set. To calculate it, add up all the numbers in the set and then divide by the number of items in the set.

The median is the middle value in a set. To calculate it, first, sort the set from smallest to largest (or vice versa), and then find the value that is in the middle of the set. If there are an even number of values in the set, you can take the average of the two middle values.

The mode is simply the most common value in a set. To calculate it, just count how many times each value occurs and then find the value that occurs the most.

1.7 Explain the characteristics, uses advantages, and disadvantages of each measure of location.

There are a variety of measures of location, each with its own advantages and disadvantages. The most common measures are the mean, median, and mode.

The mean is the most commonly used measure of location and is simply the average of all the data points. The advantage of using the mean is that it is easy to calculate and understand. However, the main disadvantage is that it is very sensitive to outliers, or data points that are far from the rest of the data. For example, if we have a data set with values 1, 2, 3, 4, 1000, our mean would be 204 (4+3+2+1+1000)/5=204), which clearly does not accurately represent our data set.

The median is the middle value in a set. To calculate it, first, sort the set from smallest to largest (or vice versa), and then find the value that is in the middle of the set. If there are an even number of values in the set, you can take the average of the two middle values. The advantage of using the median is that it is not as sensitive to outliers as the mean. The main disadvantage is that it can be more difficult to calculate, especially if you are working with a large data set.

The mode is simply the most common value in a set. To calculate it, just count how many times each value occurs and then find the value that occurs the most. The advantage of using the mode is that it is not as sensitive to outliers as the mean or median. The main disadvantage is that it can be more difficult to calculate, especially if you are working with a large data set.

1.8 Calculate the range, mean deviation, variance, and standard deviation.

The range, mean deviation, variance, and standard deviation are important concepts in statistics. They are used to summarize data sets and to calculate various measures of central tendency and dispersion.

Range: The range is the difference between the highest and lowest values in a data set. It is a measure of dispersion.

Mean Deviation: The mean deviation is the average distance of each value from the mean. It is a measure of dispersion. 

Variance: The variance is a measure of how far each value in a data set is from the mean. It is a measure of dispersion. 

Standard Deviation: The standard deviation is the square root of the variance. It is a measure of dispersion. 

Assignment Brief 2: Inferential Statistics: 

2.1 Describe the classical, empirical, and subjective approaches to probability.

The classical approach to probability is based on the idea of counting cases. In this approach, we consider all the possible outcomes of an event and then count up the number of times each outcome occurs. This gives us a mathematical way of calculating the probability of an event.

The empirical approach to probability is based on actual experience or data. In this approach, we observe how often a particular event happens and calculate the probability from that data.

The subjective approach to probability is based on personal beliefs or opinions. In this approach, we assign a probability to an event based on our feelings or intuition about it.

2.2 Explain the terms experiment, event, outcome, permutations, and combinations.

An experiment is a process or set of steps conducted in order to test a hypothesis, observe an outcome, or explore a new phenomenon. In statistics, an experiment is often seen as the process of randomly assigning treatments to subjects and observing the resulting outcomes. 

An event is an occurrence that happens during the course of an experiment.

An outcome is the result of an event occurring.

Permutations are all possible ways that a set of items can be arranged.

Combinations are all possible ways that a set can be selected from a larger set.

2.3 Define the terms conditional probability and joint probability. 

The terms conditional probability and joint probability are both related to the concept of probability. Probability is a measure of the likelihood that an event will occur. The higher the probability of an event, the more likely it is to occur.

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. For example, the conditional probability of flipping a coin and getting heads might be 50% if we don’t know anything about the coin (that is if the coin is fair). However, if we know that the coin has been flipped 10 times already and has landed on heads every time, then the conditional probability of flipping it again and getting heads would be 100%.

A joint probability is a measure of the likelihood of two events occurring at the same time. For example, the joint probability of flipping a coin and getting heads and tails might be 50% if we don’t know anything about the coin (that is if the coin is fair). However, if we know that the coin has been flipped 10 times already and has landed on heads every time, then the joint probability of flipping it again and getting heads and tails would be 0%.

2.4 Calculate probabilities using the rules of addition and rules of multiplication. 

The rules of addition state that the probability of two events occurring is the sum of the probabilities of each event occurring. For example, if the probability of event A is 0.3 and the probability of event B is 0.4, then the probability of both event A and event B occurring is 0.7.

The rules of multiplication state that the probability of two events occurring is the product of the probabilities of each event occurring. For example, if the probability of event A is 0.3 and the probability of event B is 0.4, then the probability of both event A and event B occurring is 0.12.

2.5 Distinguish between discrete and continuous probability distributions.

A discrete probability distribution is a function that assigns probabilities to certain outcomes. For example, if you were tossing a coin, the discrete probability distribution would tell you the chances of getting heads or tails.

A continuous probability distribution, on the other hand, is a function that assigns probabilities to all points within a certain range. So if you were rolling a die, the continuous probability distribution would tell you the chances of getting any number between 1 and 6.

2.6 Calculate the mean, variance, and standard deviation of discrete and continuous probability distributions.

The mean, variance, and standard deviation of a discrete random variable can be found using its probability mass function. The mean is simply the sum of all the values in the distribution divided by the total number of values. The variance is the square of the standard deviation, and the standard deviation is just the square root of the variance.

For a continuous random variable, its probability density function can be used to find its mean, variance, and standard deviation. The mean is just the integral of the probability density function over all space. The variance is determined by taking the second derivative of the probability density function (which gives you a measure of how to spread out your data) and multiplying it by two. And finally, the standard deviation is just the square root of the variance.

2.7 Describe the characteristics of and compute probabilities using the binomial, poison, uniform, and the normal probability distributions.

The binomial distribution is a discrete probability distribution that gives the probability of k successes in n trials, where each trial is either a success or a failure. The binomial formula is:

P(x) = (n choose k) p^k q^{n-k} 

The poison distribution is a continuous probability density function that gives the pdf of the number of failures x before the first success in an infinite sequence of independent and identically distributed Bernoulli trials with parameter p. The poison function is:

f(x) = (1-p)^x 

The uniform distribution is a continuous probability density function that gives the joint PDF of two random variables X and Y which are both uniformly distributed between 0 and 1. The uniform function is:

f(x,y) = 1 

The normal distribution is a continuous probability density function that gives the PDF of a random variable X which is normally distributed with mean u and variance sigma^2. The normal function is:

f(x) = (1/sqrt(2pi sigma^2)) e^{-(x-u)^2/2sigma^2} 

2.8 Describe methods to select a sample. 

There are a variety of methods to select a sample, and the most appropriate method will depend on the type of data you are working with and the population you are interested in. For example, if you want to study the behavior of dogs, you might want to use a convenience sample of dogs that live near you. But if you want to study the behavior of all dogs, you would need to use a random sample.

If you are working with a population that is difficult to access, you might want to use a snowball sample, where you start with a few people who fit your criteria and then ask them to refer you to other people who fit your criteria.

Another common method is stratified sampling, where the population is divided into subgroups (strata) and then a random sample is taken from each stratum. This is often used when the population is heterogeneous, and it ensures that the subgroups are represented in the sample.

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