**ST1112 Statistical Methods Assignment Sample NUIG Ireland**

ST1112 is an introductory-level statistics course that covers methods for describing and analyzing data. In this course, students learn how to use statistical methods to make sense of data and to answer questions about relationships among variables. Topics include descriptive statistics, basic probability, sampling distributions, statistical inference (including hypothesis testing and confidence intervals), linear regression, and ANOVA.

This course is ideal for students who are interested in learning how to use statistics to understand patterns in data. It is also a prerequisite for many other courses in the social sciences and business.

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

**Assignment brief 1: Exploratory Data Analysis, summarise data numerically and graphically, including measures of central tendency, spread, and shape of the observed data. Produce output of descriptive statistics and plots with the use of statistical software, R programming, and Minitab.**

Exploratory data analysis is a process of summarising data numerically and graphically to gain an understanding of the underlying distribution of the data. Measures of central tendency, spread, and shape are typically used to characterize the data.

There are various measures of central tendency that can be used, including the mean, median, and mode. The mean is simply the average of all the values in the data set. The median is the middle value when all the values are sorted from smallest to largest. The mode is the most frequently occurring value.

The spread of the data can be measured using measures such as the range, interquartile range, variance, and standard deviation. The range is simply the difference between the largest and smallest values in the data set. The interquartile range is the difference between the first and third quartiles (the 25th and 75th percentiles). The variance is a measure of how far the values in the data set are spread out from the mean. The standard deviation is the square root of the variance.

The shape of the data can be summarized using measures such as skewness and kurtosis. Skewness is a measure of how symmetric the data are about the mean. Kurtosis is a measure of how peaked the data are (how much they resemble a normal distribution).

There are many ways to summarize data numerically and graphically. In this assignment, you will be asked to produce an output of descriptive statistics and plots with the use of statistical software, R programming, and Minitab.

**Assignment brief 2: Explain what is meant by a sample, a population, a statistic, a parameter, sampling distribution, standard error, and statistical inference.**

When we talk about statistics, we are usually referring to one of two things: a population or a sample.

A population is the complete set of all data that we’re interested in. A sample is a subset of data from the population. So, when researchers talk about “sample size”, they’re referring to how many people are in the sample.

A statistic is simply a numerical measure that describes some characteristic of the data in a sample. For example, the mean or average is a statistic that describes the typical or average value in a sample.

A parameter is a specific numerical measure that describes some characteristic of the population. For example, the mean or average is a parameter that describes the typical or average value in the population.

The sampling distribution is simply the distribution of a statistic (such as the mean) calculated from a sample. It tells us how that statistic is likely to vary from sample to sample.

The standard error is a measure of how accurately the statistic estimates the population parameter. It is calculated by taking the standard deviation of the sampling distribution and dividing it by the square root of the sample size.

Statistical inference is the process of using the sampling distribution to make conclusions about the population parameter. For example, if we want to know what the average height of all adults is, we could take a sample of adults and calculate the mean. But we would not expect that mean to be the same as the population parameter (the true average height of all adults). We would expect it to be close, but not the same. This is because samples are usually only a small subset of the population, so they will usually contain some variability.

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**Assignment Brief 3: Derive a confidence interval for an unknown parameter using a given sampling distribution and interpret this interval estimate.**

A confidence interval for an unknown parameter can be derived using a given sampling distribution and interpreting this interval estimate. The amount of precision desired will determine how close the confident bounds need to be to the estimated value. If we desire a high degree of precision, then we will want the confidant bounds to be close to the estimated value. On the other hand, if we are less concerned with precision, then we may be willing to have confidence bounds that are further from the estimated value. The trade-off is that as we require more precision, the width of our confidence interval will increase.

There are many ways to construct a confidence interval, but one way is to use Chebyshev’s inequality. This tells us that for any given confidence level, the width of our interval will be at least as wide as:

(1/2)*((1-confidence level)/(z-score))^2

where z is the z-score that corresponds to the desired confidence level. For example, if we want a 95% confidence interval, then z = 1.96.

So, using Chebyshev’s inequality, we can see that for a 95% confidence interval, the width of our interval will be at least:

(1/2)*((1-0.95)/(1.96))^2 = 0.02

This means that our interval will be at least 2% wide.

We can also use the Central Limit Theorem to construct a confidence interval. The Central Limit Theorem tells us that the distribution of the sample mean will be normal, even if the population is not. So, we can use the properties of a normal distribution to construct a confidence interval for the population mean.

**Assignment brief 4: Demonstrate how probability is used in hypothesis testing, explain what is meant by the terms null and alternative hypotheses, simple and composite hypotheses, type I and type II errors, level of significance, power of a test, test statistic, critical region, p-value, and likelihood ratio.**

Probability is the cornerstone of hypothesis testing. In very simple terms, probability can be thought of as a measure of how likely something is to happen. In hypothesis testing, we use probability to calculate the likelihood that our null hypothesis is true.

The null and alternative hypotheses are two competing hypotheses about a population parameter. The null hypothesis is typically the “default” or “standard” outcome, while the alternative hypothesis represents some kind of deviation from that default.

For example, let’s say we’re interested in whether or not people prefer brand A over brand B. Our null hypothesis might be that there is no preference (i.e., p=0.5), while our alternative hypothesis might be that people prefer brand A (i.e., p>0.5).

A simple hypothesis makes a single claim about a population parameter, while a composite hypothesis makes multiple claims about different population parameters.

For example, let’s say we’re interested in the average height of all adults. A simple hypothesis might be that the average height is 72 inches, while a composite hypothesis might be that the average height is 72 inches and the average weight is 180 pounds.

Type I and type II errors are both ways that we can mistakenly reject or fail to reject the null hypothesis. A Type I error occurs when we reject the null hypothesis when it is true, while a Type I error occurs when we fail to reject the null hypothesis when it is false.

The level of significance is the probability of making a Type I error. In other words, it’s the probability of rejecting the null hypothesis when it is true. The level of significance is usually set at 0.05, which means that there is a 5% chance of making a Type I error.

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it’s the probability of not making a Type II error. The power of a test is usually set at 0.80, which means that there is an 80% chance of correctly rejecting the null hypothesis.

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**Assignment Brief 5: Identify and perform one-sided and two-sided tests for parametric models; infer the value of a population mean in cases of large and small samples, a population proportion, chi-square goodness of fit test, two-population inference comparing two population means in cases of large and small samples, two-population inference test for the value of the ratio of two variances, paired-samples analysis for comparison of population means, comparing two population proportions.**

A one-sided test is a hypothesis test in which the null and alternative hypotheses are not symmetric. In other words, one side of the alternative hypothesis is either greater than or less than the corresponding side of the null hypothesis.

For example, let’s say we’re interested in whether or not people prefer brand A over brand B. Our null hypothesis might be that there is no preference (i.e., p=0.5), while our alternative hypothesis might be that people prefer brand A (i.e., p>0.5). This would be a one-sided test because we are only testing for a difference in favour of brand A.

A two-sided test is a hypothesis test in which the null and alternative hypotheses are symmetric. In other words, both sides of the alternative hypothesis are either greater than or less than the corresponding side of the null hypothesis.

For example, let’s say we’re interested in whether or not people prefer brand A over brand B. Our null hypothesis might be that there is no preference (i.e., p=0.5), while our alternative hypothesis might be that people prefer brand A (i.e., p≠0.5). This would be a two-sided test because we are testing for a difference in either direction (i.e., favour of brand A or brand B).

The level of significance is the probability of making a Type I error. In other words, it’s the probability of rejecting the null hypothesis when it is true. The level of significance is usually set at 0.05, which means that there is a 5% chance of making a Type I error.

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it’s the probability of not making a Type II error. The power of a test is usually set at 0.80, which means that there is an 80% chance of correctly rejecting the null hypothesis.

**Assignment Brief 6: Relationships of two variables; conduct a chi-square association test from observed data for two qualitative variables, define and fit a simple linear regression model to sample data including interpretation of the coefficient of determination, statistical inference for population slope parameter, prediction intervals for individual response and confidence intervals for a meaningful response.**

A **chi-square association test** is a statistical test used to determine if there is a relationship between two qualitative variables. The chi-square test is used to calculate a statistic that measures the strength of the relationship between the two variables.

The chi-square statistic is based on the difference between the expected values and the observed values. The expected values are the values that would be expected if there was no relationship between the two variables.

The chi-square statistic is used to determine whether or not the difference between the expected and observed values is statistically significant. If the chi-square statistic is statistically significant, then we can conclude that there is a relationship between the two variables.

The **coefficient **of determination is a statistical measure that shows how well a linear regression model explains the variation in the data. It is also known as the R-squared value.

The coefficient of determination is used to determine how much of the variation in the data is explained by the linear regression model.

A high coefficient of determination means that the linear regression model explains a lot of the variation in the data. A low coefficient of determination means that the linear regression model does not explain much of the variation in the data.

**Statistical inference** is a process of using statistics to make predictions about a population based on a sample. Statistical inference is used to estimate population parameters, test hypotheses, and make predictions.

The population slope parameter is the average value of the independent variable in the population. The population slope parameter is used to estimate the population mean and to make predictions about the population.

Prediction intervals are used to predict the value of a future observation. Prediction intervals are calculated using the standard error of the estimate.

Confidence intervals are used to estimate population parameters. Confidence intervals are calculated using the standard error of the estimate.

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**Assignment Brief 7: Apply the permutation approach to non-parametric hypothesis tests.**

The permutation approach to non-parametric hypothesis tests is a method of statistical inference that allows for the examination of the null hypothesis in situations where the distribution of the test statistic is unknown.

The permutation approach works by generating all possible samples from the population under study and then calculating the test statistic for each sample. The p-value for the permutation test is then determined by comparing the observed value of the test statistic to the distribution of values generated by all possible samples.

If the p-value is less than or equal to the significance level, then the null hypothesis is rejected. If the p-value is greater than the significance level, then the null hypothesis cannot be rejected.

The permutation approach to non-parametric hypothesis tests is a powerful tool for examining the null hypothesis in situations where the distribution of the test statistic is unknown. This approach has the advantage of being able to account for all possible samples from the population, which makes it less likely that the results are due to chance.

**Assignment Brief 8: Produce and interpret the output of statistical inference procedures with the use of statistical software, R programming, and Minitab.**

Several different software programs can be used for statistical inference, including R programming and Minitab. Each program has its strengths and weaknesses, so it’s important to choose the one that will work best for your specific needs.

In general, however, all of these programs allow you to perform a wide range of statistical tests, as well as to graphically represent the results of those tests. This can help understand the underlying data and in making decisions about how to proceed with further analysis.

R programming is a popular choice for statistical inference because it’s free and open-source, meaning that anyone can use it and contribute to its development. R also has a large community of users, which can help find answers to questions or in troubleshooting problems.

Minitab is another popular choice for statistical inference, and it has the advantage of being very user-friendly. Minitab also has a wide range of features, making it a good choice for more complex analyses.

No matter which software program you choose, however, all of them can be used to produce and interpret the output of statistical inference procedures.

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