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# ST2001 Statistics for Data Science 1 Assignment Sample NUI Galway Ireland

ST2001 Statistics for Data Science 1 is an introductory statistics course that covers basic concepts and methods in statistics. The topics covered include exploratory data analysis, probability, distributions, inference, and regression. This course is designed for students who are interested in data science and want to learn more about how to use statistical methods to analyze data. The course is also useful for students who are considering a career in data science or who want to learn more about statistical methods for data analysis.

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In this section, we are describing some assigned tasks. These are:

### Assignment Task 1: Calculate conditional probabilities and probabilities for random variables from standard distributions (Binomial, Poisson, Normal).

In probability theory, conditional probability is the probability of an event given that another event has occurred. For example, the conditional probability of rolling a 6 on a die, given that the first roll was a 4, can be calculated as follows:

P(6|4) = P(4 and 6)/P(4)

= (1/6 x 1/6)/(1/6)

= 1/36

There are three main types of standard distributions – binomial, Poisson, and normal. The calculation of conditional probabilities for these random variables involves using their respective probability mass functions or density functions.

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The binomial distribution is used to model outcomes that can only have two possible values, such as success or failure. The probability mass function for the binomial distribution is given by:

P(x|n,p) = P(n,x)p^x(1-p)^(n-x)

where n is the number of trials and p is the probability of success.

The Poisson distribution is used to model the number of events that occur in a given period. The probability mass function for the Poisson distribution is given by:

P(x|λ) = λ^xe^-λ/x!

where λ is the mean number of events per unit of time.

The normal distribution is used to model data that is continuous. The probability density function for the normal distribution is given by:

P(x|μ,σ) = 1/(σ√2π)e^-((x-μ)^2/2σ^2)

where μ is the mean and σ is the standard deviation.

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### Assignment Task 2: Summarise data numerically (centre and spread) and graphically (e.g. bar charts, line, area, boxplots, histograms, density plots, scatterplots) with an emphasis on best practices for communication.

There are many ways to summarize data, both numerically and graphically. Numerical summary methods include computing the mean, median, mode, and range. Graphical summary methods include using bar charts, line graphs, area graphs, boxplots, histograms, density plots, and scatterplots.

The mean is the average of all the values in the data set. To calculate it, add up all the values and then divide by the number of values. The median is the middle value in a data set. To find it, arrange all the values from smallest to largest (or vice versa) and then pick the one in the middle. The mode is the most common value in a data set. To find it, count how often each value occurs and then pick the one that occurs the most. The range is the difference between the largest and smallest values in a data set.

Bar charts are used to visualize categorical data, while line graphs, area graphs, and boxplots are used to visualize numerical data. Histograms and density plots are used to visualize the distribution of data, and scatterplots are used to visualize the relationships between variables.

When summarizing data, it is important to choose the best method for the type of data being summarized and to communicate the results clearly and concisely. For example, when summarizing categorical data, a bar chart would be more appropriate than a line graph. When summarizing the distribution of data, a histogram or density plot would be more appropriate than a boxplot. And when summarizing the relationships between variables, a scatterplot would be more appropriate than an area graph.

### Assignment Task 3: Summarise the importance of probabilistic-based sampling schemes (e.g. simple random sampling, stratified sampling, cluster sampling).

Probabilistic-based sampling schemes are important because they help researchers ensure that the samples they collect are representative of the population as a whole. With simple random sampling, every member of the population has an equal chance of being selected for the sample; stratified sampling ensures that each subgroup within the population (e.g. males and females) is represented in proportion to its size in the population, and cluster sampling makes it possible to study smaller groups within the population in more detail. All of these methods help reduce bias and increase the accuracy of results.

In general, probabilistic-based sampling schemes are considered to be more reliable than non-probabilistic methods (such as convenience or snowball sampling). This is because probabilistic methods help to ensure that the sample is representative of the population, which means that results are more likely to be accurate.

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### Assignment Task 4: Summarise the difference between observational and experimental studies and the principles of experimental design.

There are two main types of scientific studies: observational and experimental. Observational studies watch and describe what happens, while experimental studies test a specific hypothesis by manipulating variables and observing the results.

The principles of experimental design help researchers control for confounding variables so that they can accurately isolate the cause and effect relationship they’re interested in. By randomly assigning subjects to different treatment conditions, controlling for known differences between groups, and using blind or double-blind procedures, scientists can be confident that their results are due to the manipulated variable and not other extraneous factors.

Experimental studies are important because they allow scientists to test specific hypotheses and isolate cause-and-effect relationships. observational studies can be useful for generating hypotheses, but experimental studies are necessary for testing them. The principles of experimental design help to ensure that results are accurate and that conclusions can be drawn about cause and effect relationships.

### Assignment Task 5: Perform probability calculations about the sample means and use them to make inferential statements using the Central Limit Theorem.

There are a few probability calculations that we can do to make inferential statements about the sample means. The first is the calculation of the standard error of the mean. This measures how much variability there is in the sample means and gives us an indication of how likely it is that the sample means are all drawn from the same population distribution.

The second calculation is known as the z-score, which tells us how many standard errors away from the population mean a particular sample mean is. And finally, we can use something called the Central Limit Theorem to determine whether or not our sample means are likely to be drawn from a population with a normal distribution.

The Central Limit Theorem states that, regardless of the shape of the population distribution, the sample means will be normally distributed if the sample size is large enough. This theorem is important because it allows us to make inferences about a population based on a sample, even if we don’t know anything about the distribution of that population.

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### Assignment Task 6: Calculate interval estimates for parameter estimation in one sample problem using classical and computational (i.e. bootstrap) approaches.

Parameter estimation is the process of estimating the value of a parameter, typically denoted by θ, from data. Interval estimation is a sub-field of parameter estimation in which one not only estimates the value of θ but also provides a range within which θ is estimated to lie with some specified probability. This probability is usually taken to be 95%.

There are two main approaches to interval estimation: classical and computational. Classical interval estimation procedures usually involve Closed-Form Estimators (CFEs), while computational methods make use of resampling techniques such as the bootstrap.

For the one-sample problem, let’s say we have n observations, x1,…xn ~ iid N(θ,1). We would like to estimate θ and construct a 95% confidence interval for it.

The classical approach would involve using the CFE for θ, which is given by:

θ̂=1n∑ni=1xi

and then using the fact that

θ̂−1.96√1nσθ̂≤θ≤θ̂+1.96√1nσθ̂

with probability 0.95, where σ is the population standard deviation.

The computational approach would involve resampling our data n times with replacement and calculating the mean for each resample. This would give us a distribution of sample means, from which we could then calculate the 2.5th and 97.5th percentiles as our confidence interval.

Both approaches would give us a valid 95% confidence interval for θ, but the computational approach is more robust and can be used even when the population distribution is not known.

### Assignment Task 7: Perform hypothesis testing (null and alternative hypotheses, type I and II errors, and p-values) in a variety of scenarios.

A hypothesis test is a statistical procedure used to make decisions about a population based on a sample. The null hypothesis (H0) is the Hypothesis that nothing has changed or that there is no difference between the two groups. The alternative hypothesis (HA) seeks to disprove the null hypothesis by showing that there is a difference between the two groups. A Type I error occurs when the Null Hypothesis is true but is rejected, while a Type II error occurs when the Null Hypothesis is false but fails to be rejected. The p-value helps us decide whether or not to accept or reject the null hypothesis; it represents the probability of seeing our data given that H0 is true. If the p-value is less than some threshold (usually 0.05), we reject H0 in favour of HA.

There are a variety of scenarios in which hypothesis testing can be used, but one of the most common is comparing two means. For example, we might want to know if there is a difference in the average height of men and women. In this case, our H0 would be that there is no difference between the two means (i.e. μmen=μwomen) and our HA would be that there is a difference (i.e. μmen≠μwomen). Another common scenario is comparing the means of two groups that have been randomly assigned to treatment and control groups.

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### Assignment Task 8: Fit and interpret a simple linear regression model.

A linear regression model is a mathematical equation that describes the relationship between two variables. The equation is used to predict one variable, known as the dependent variable, from the other variable, known as the independent variable.

The linear regression model is based on the assumption that the relationship between the two variables is linear. This means that it can be described by a straight line. The linear regression model can be used to predict values for the dependent variable given values for the independent variable.

The coefficient of determination (R2) is a measure of how well the linear regression model describes the data. An R2 value of 1 indicates that the model fits perfectly and an R2 value of 0 indicates that there is no relationship between the two variables.

### Assignment Task 9: Compile a statistical report, i.e. prepare a typed document that introduces the statistical research question being explored, describes the data collection mechanism, provides subjective impressions on relevant numerical and graphical summaries, and outlines conclusions from all formal statistical analyses undertaken.

It is important to understand the research question you are exploring as well as the data collection mechanisms you will use to address that question. In this section, you will detail both of those components. Keep in mind that your statistical report should be reproducible, so it is important to be clear and concise in your explanations.

When it comes to research questions, it is helpful to think about the variables you are interested in and how they might relate to one another. For example, if you are looking at the relationship between income and health outcomes, your independent variable would be income and your dependent variable would be health outcomes. From there, you can start to form a hypothesis about what you think the relationship might be between these two variables (e.g. “I think that higher income will lead to better health outcomes”).

Once you have a research question in mind, you need to decide how you are going to collect data on the variables you are interested in. There are a variety of ways to do this, but some common methods include surveys, observational studies, and experiments.

Once you have collected your data, it is important to summarize it in a way that is easy to understand. This can be done using numerical summaries, such as means and standard deviations, or graphical summaries, such as histograms and scatterplots.

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