**ST1109 Introductory Statistics for Business Assignment Sample NUIG Ireland**

ST1109 Introductory Statistics for Business is offered by the National University in Ireland, Galway and is a 5-credit unit course. This course aims to provide students with an introduction to statistics and its application in business. The topics covered in this course include exploratory data analysis, numerical measures, probability, statistical distributions, hypothesis testing, goodness-of-fit tests, regression, and correlation.

This course will give you a broad understanding of how businesses use statistics daily to make strategic decisions. You will learn how to summarise data sets using numerical measures such as means and standard deviations. You will also be introduced to basic concepts of probability theory which will help you understand risk and uncertainty. In addition, you will learn how to use statistical techniques to test hypotheses and make predictions.

This course is designed for students who have little or no prior knowledge of statistics. No mathematical background beyond basic algebra is required. However, some exposure to computers would be beneficial as the course makes use of Excel for data analysis.

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

**Assignment Task 1: Describe data using numerical and graphical summaries.**

There are two main ways to summarise data: numerical and graphical.

**Numerical summaries** use statistical methods to condense data into a few key numbers, such as the mean, median, mode, and standard deviation. These are useful for getting a quick overview of a dataset, but can sometimes be misleading (for example, the mean can be heavily influenced by outliers).

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**Graphical summaries**, on the other hand, use visual methods to communicate information about the data. Common types of graphical summaries include histograms, box plots, and scatterplots. These are often more effective than numerical summaries at detecting patterns and identifying outliers.

**Assignment Task 2: Define basic probability terms, such as sample spaces and events, state properties of probabilities, and calculate probabilities of events, using counting theory, and basic probability rules.**

A sample space is a set of all potential outcomes of an event. For example, when flipping a coin, the sample space would be {heads, tails}. An event is any subset of the sample space. So, in our coin flipping example, the event “heads” would be the subset {heads}, and the event “tails” would be the subset {tails}.

The probability of an event is simply defined as the number of favourable outcomes divided by the total number of possible outcomes. In our coin flipping example, there are two favourable outcomes (heads and tails), so the probability of each would be 1/2 (or 50%). Note that if we wanted to calculate the probability of an event that doesn’t have equal chances of occurring, we would use a different method called counting theory.

There are three basic properties of probability that always hold:

- The probability of any event is always between 0 and 1 (inclusive).
- The probability of the sample space is always 1.
- The probability of the complement of an event is always 1 minus the probability of the event.

An example of a counting theory problem would be if we wanted to calculate the probability of getting two heads in a row when flipping a coin three times. In this case, we would use a method called the multiplication rule, which says that the probability of two events occurring is equal to the probability of the first event occurring multiplied by the probability of the second event occurring. So, in our example, the probability of getting two heads in a row would be 1/2 x 1/2, or 1/4.

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**Assignment Task 3: Describe and use discrete probability models to calculate expectation, variance, and probabilities.**

Discrete probability models can be used to calculate expectation, variance, and probabilities. These models are based on the assumption that there is a finite number of outcomes for a given situation. For example, if you were to flip a coin three times, there would be eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT.

The expectation of a discrete random variable is defined as the sum of the weighted outcomes of the random variable. In other words, it is the mean of the possible values that the random variable can take on. The variance of a discrete random variable is defined as the sum of the squared differences between each possible value and the mean.

To calculate the probability of an event using a discrete probability model, we simply count the number of favourable outcomes and divide it by the total number of possible outcomes. For example, if we wanted to calculate the probability of getting two heads in a row when flipping a coin three times, we would have to enumerate all of the possible outcomes, as we did above. There are eight possible outcomes in total, and two of them are favourable (HHT and THH). So, the probability of getting two heads in a row would be 2/8, or 1/4.

**Assignment Task 4: Describe and use continuous probability models to calculate expectation, variance, and probabilities.**

A continuous probability model is a mathematical modelling tool used to predict the likelihood of an event occurring over a particular period. It can be used to calculate expectation, variance, and probabilities.

The most basic form of a continuous probability model is the normal distribution curve, which is used to model many real-world phenomena, from Daily Life magazine subscription rates to Shape magazine cover sales. The normal curve is “bell-shaped”, with the bulk of observations falling in the middle and tapering off at either end.

Three main parameters define a normal distribution: mean ( μ ), standard deviation ( σ ), and variance ( σ2 ). Mean is the average value of all observations, while standard deviation is a measure of how spread out the values are. Variance is simply the square of the standard deviation.

To calculate the probability of an event occurring using a continuous probability model, we integrate the curve over the interval that represents the event. For example, if we wanted to calculate the probability of a Daily Life magazine subscription rate falling between 2% and 4%, we would integrate the normal curve over the interval from 2% to 4%. The result would be the probability of the event occurring.

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**Assignment Task 5: Discuss and implement methods in data collection and sampling.**

Data collection and sampling are important methods in data analysis. To get an accurate understanding of the data, it is necessary to collect data from a representative sample. This means that the data collected should be representative of the population under study.

There are several ways to** collect data**. The most common method is to ask people directly about their opinions or behaviour. However, this can be expensive and time-consuming, and may not always provide accurate information. A more efficient method is to use surveys or questionnaires.

Another common method is observation. This involves watching people’s behaviour and taking note of what they do. It can be useful for obtaining information on how people interact with each other or their environment. However, it can be difficult to observe people without them knowing they are being observed, which may affect their behaviour.

A third method is to use existing data sources. This could include government records, company reports, or data from research studies that have already been conducted. This is often the most efficient way to collect data, as it does not require any new data to be collected. However, it is important to make sure that the data are reliable and representative of the population under study.

When collecting data, it is important to consider how the data will be sampled. Sampling is the process of selecting a representative subset of the population under study. There are several different sampling methods, and the most appropriate method will depend on the type of data being collected and the population under study.

The most common **sampling method **is simple random sampling, which involves selecting a unit from the population at random and then collecting data from that unit. This is the most straightforward way to collect data, but it can be difficult to achieve if the population is large or if the units are not evenly distributed.

Another common sampling method is stratified sampling, which involves dividing the population into subgroups (strata) and then selecting a unit from each stratum at random. This ensures that the units in the sample are representative of the population as a whole.

A third common sampling method is cluster sampling, which involves dividing the population into groups (clusters) and then selecting a unit from each cluster at random. This is often used when it is difficult or expensive to collect data from individual units.

When choosing a sampling method, it is important to consider the type of data being collected and the population under study. The most appropriate method will depend on these factors.

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