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ST238 Introduction to Statistical Inference Assignment Sample NUI Galway Ireland

ST238 Introduction to Statistical Inference course provides an introduction to the theory and methods of statistical inference. The focus is on Bayesian and frequentist approaches to estimation, testing, and model selection, with an emphasis on practical applications.

The course covers a wide range of topics, from point estimation and confidence intervals to hypothesis testing and Bayesian model comparison. It also introduces important concepts in data mining and machine learning, such as cross-validation, feature selection, and overfitting.

Students will learn how to apply these methods to real-world datasets using the R programming language. They will also be introduced to the Stan programming language for Bayesian inference.

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

Assignment Brief 1: Understand the difference between Probability and Statistics and the Role of Probability in solving statistical inference problems.

Probability theory and statistics are two interrelated fields of mathematics that are used to solve problems in science, engineering, finance, and other disciplines. Probability is the mathematical study of chance and randomness, while statistics is the application of probability theory to real-world problems.

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Probability theory is used to model phenomena that involve randomly occurring events. It allows us to quantify the chances of certain events happening, and to make predictions about how often they will occur. For example, if we flip a fair coin 100 times, we expect it to come up heads 50 times and tails 50 times. This sort of predictable behaviour is known as variability. In contrast, if we flip a biased coin that comes up heads 70% of the time, we would not expect it to come up heads exactly 70 times out of 100. This unpredictable behaviour is known as uncertainty.

Probability theory is also used to model situations where there is uncertainty about the underlying process that generates the data. For example, in stock market analysis, analysts may use probability models to quantify the risk of a particular investment. In medical research, probability theory is used to model the uncertainty in the efficacy of new treatments.

Statistics is the application of probability theory to real-world problems. It allows us to make inferences about a population from a sample, and to quantify the uncertainty of those inferences. For example, if we take a random sample of 100 people and find that 60 of them are taller than 6 feet, we can use statistics to estimate the percentage of the population that is taller than 6 feet. We can also quantify the uncertainty of that estimate, which is important for making decisions based on it.

Assignment brief 2: Perform probability calculations about the sample means and use them to make inferential statements.

To make inferential statements about the sample means, we need to calculate the probability that the difference between the sample means is due to chance. This probability is called the “sampling error” and is represented by the symbol “E”.

The sampling error is a measure of how confident we can be in our estimate of the population mean. The smaller the sampling error, the more confident we are in our estimate. We can calculate the sampling error using a formula that takes into account the variance of each of our samples:

where:

s²1 = variance of Sample 1

s²2 = variance of Sample 2

n1 = number of observations in Sample 1

n2 = number of observations in Sample 2

The sampling error is a function of the variances of our two samples and the number of observations in each sample. It is important to note that the sampling error is not affected by the means of our two samples, only by their variances.

We can use the sampling error to make inferential statements about the difference between the sample means. If the difference between the sample means is less than the sampling error, then we can say that the difference is due to chance and is not statistically significant. If the difference between the sample means is greater than the sampling error, then we can say that the difference is statistically significant and is not due to chance.

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Assignment brief 3: Understand some basic ideas about interval estimation; be familiar with Type I and Type II errors in hypothesis tests and be able to calculate the p-value and power of various statistical tests.

In hypothesis testing, the Type I error is the probability of rejecting a true null hypothesis, while the Type II error is the probability of failing to reject a false null hypothesis. The p-value is a measure of how likely it is that an observed result occurred by chance if the null hypothesis were true.

In interval estimation, we construct a confidence interval to estimate a population parameter. We choose our level of confidence (usually 95%) and then compute the interval that has that level of confidence. The width of the interval will depend on the sample size and on how precisely we know the population parameter. If we have no information about the population parameter other than what we can glean from our data, then our interval will be wide. If we have a good estimate of the population parameter, then our interval will be narrower.

The power of a statistical test is the probability that the test will reject a false null hypothesis. The power of a test increases as the sample size increases and as the difference between the population means increases.

Assignment brief 4: Find confidence intervals and perform hypothesis tests about a single population mean, a single population proportion, the difference between two population means, and a single population variance.

To find a confidence interval for the population means, use the following equation:

Where:

x is the sample mean

s is the sample standard deviation

n is the size of the sample

To find a confidence interval for a single population proportion, use the following equation:

Where: p is the population proportion 

n is the size of the sample 

x is the number of successes in n trials 

z* is a value from the standard normal distribution table. This value can be found at Critical Values and Intervals from Z-score or look it up on Google. To find a confidence interval for the difference between two population means, use this equation:

 Where: x1 and x2 are the sample means 

s1 and s2 are the sample standard deviations 

n1 and n2 are the sizes of the samples 

To find a confidence interval for a single population variance, use this equation:

Where: x is the sample mean 

s is the sample standard deviation 

n is the size of the sample 

z* is a value from the standard normal distribution table. This value can be found at Critical Values and Intervals from Z-score or look it up on Google. To perform a hypothesis test about a single population mean, use this equation:

Where: x is the sample mean 

s is the sample standard deviation 

n is the size of the sample 

z* is a value from the standard normal distribution table. This value can be found at Critical Values and Intervals from Z-score or look it up on Google. To perform a hypothesis test about a single population proportion, use this equation:

Where: p is the population proportion  

n is the size of the sample 

x is the number of successes in n trials 

z* is a value from the standard normal distribution table. This value can be found at Critical Values and Intervals from Z-score or look it up on Google.

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Assignment brief 5: Analyse enumerative data through chi-squared goodness-of-fit and contingency table tests.

The chi-square goodness-of-fit test is used to determine whether the observed frequencies in a table or frequency distribution are consistent with the expected frequencies. The chi-squared contingency test is used to determine whether there is a relationship between two categorical variables.

The chi-squared statistic is calculated using the following formula:

χ² = (O – E)² / E

Where:

O = Observed frequency

E = Expected frequency

The chi-squared statistic is compared to a critical value from the chi-squared distribution. If the chi-squared statistic is greater than the critical value, then the null hypothesis is rejected and the alternative hypothesis is accepted.

Assignment Brief 6: Calculate and interpret the linear correlation coefficient for relating two variables.

The linear correlation coefficient is a measure of the degree of the linear relationship between two variables. It ranges from -1 to +1, with -1 indicating a perfect negative correlation, 0 indicating no correlation at all, and +1 indicating a perfect positive correlation.

To calculate the linear correlation coefficient, you first need to calculate Pearson’s product-moment correlation coefficient. This statistic is calculated by taking the sum of the products of the paired data points divided by (n-2), where n is the number of data points. The Pearson’s product-moment correlation coefficient is then transformed into the linear correlation coefficient by taking its square root.

The linear correlation coefficient can be used to determine whether there is a statistically significant linear relationship between two variables. If the absolute value of the linear correlation coefficient is greater than the critical value, then the null hypothesis is rejected and the alternative hypothesis is accepted.

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Assignment brief 7: Fit the least-squares line to data pairs, make statistical inferences about the slope of the underlying population equation, and perform basis prediction.

Least squares regression is a statistical method for finding the line of best fit for a set of data points. This line represents the linear relationship between two variables and can be used to make predictions about future events. 

To find the least-squares line, you first need to calculate the sum of squared errors (SSE). This is done by taking each data point and subtracting it from the predicted value of the line (known as the y-intercept). The result is then squared and added to the total. The goal is to find the line that minimizes this sum. 

Once you have found the least-squares line, you can use it to make predictions about future events. To do this, you simply need to plug in the values of the independent variable (x) into the equation and solve for the dependent variable (y).

When making predictions, it is important to remember that there is always a margin of error. This margin represents the amount of uncertainty in your prediction and can be calculated using the standard error of the estimate.

The standard error of the estimate is calculated by taking the square root of the sum of squared errors (SSE) divided by (n-2), where n is the number of data points.

Assignment brief 8: Understand the basics of some survey designs.

There are a few different types of survey designs that are commonly used in market research. Here is a brief overview of each:

  1. Cross-Sectional Surveys: This type of survey is conducted at a single point in time, and looks at a population at large. For example, a cross-sectional survey of the general population could be used to study consumer behaviour.
  2. Longitudinal Surveys: This type of survey is conducted over some time, allowing researchers to track changes within a population. For example, a longitudinal survey could be used to study how people’s attitudes towards a company change over time.
  3. Experimental Surveys: This type of survey is used to test a hypothesis by manipulating one or more variables. For example, an experimental survey could be used to test whether a new marketing campaign is effective.
  4. Observational Surveys: This type of survey simply involves observing people’s behaviour, without any intervention from the researcher. For example, an observational survey could be used to study how people shop in a grocery store.

Assignment brief 9: Understand when and in what ways a randomized block experimental design is often superior to a completely randomized design.

There are a few key reasons why a randomized block experimental design is often superior to a completely randomized design. First, with a completely randomized design, it can be very difficult to control for all of the potentially confounding variables. This is because there are usually multiple variables that can affect the outcome of the experiment, and it is impossible to account for all of them in a completely randomized design. By randomly assigning subjects to different treatments, you can help control for these confounders by ensuring that they are evenly distributed across the groups. 

Another advantage of using a randomized block design is that it can increase the power of your experiment. This is because blocking helps reduce variability within each group, which gives you more reliable results. Finally, randomized block designs are often more efficient than completely randomized designs, which means that you can use fewer resources and still get reliable results.

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