ST237 Introduction to Statistical Data and Probability Assignment Sample NUI Galway Ireland

ST237 Introduction to Statistical Data and Probability is a required course for statistics majors at National University in Ireland, Galway. The course provides an introduction to the theory of statistical data and probability, with a focus on fundamental concepts and methods. Topics covered in the course include discrete and continuous distributions, sampling distributions, point, and interval estimation, hypothesis testing, regression analysis, and Bayesian inference.

The purpose of this answer is not to provide a complete overview of the content covered in ST237 Introduction to Statistical Data and Probability, but rather to provide a summary of some of the key topics that are covered in the course.

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

Assignment Activity 1: Construct appropriate graphical summaries for a sample of data, including stem and leaf plots, dot plots, and box plots.

Stem and Leaf Plot: The stem and leaf plot shows the distribution of data with the stems on the left-hand side and the leaves on the right-hand side. The first number is the stem, and the second number is the leaf. In this example, there are six data points.

Dot Plot: The dot plot shows a graphical representation of the data with a dot for each data point. The X-axis represents the values from smallest to largest, and the Y-axis represents how many times each value appears in the data set. In this example, there are six data points.

Box Plot: The box plot shows a graphical representation of the data with a box for each data point. The bottom of the box represents the 25th percentile, the line in the middle of the box represents the 50th percentile (median), and the top of the box represents the 75th percentile. The whiskers represent the minimum and maximum values. In this example, there are six data points.

Assignment Activity 2: Calculate numerical summaries for a sample of data, including the mean and variance, median, and quartiles.

To calculate the numerical summaries for a sample of data, you will need to first compute the mean and the variance. The median is then found by arranging the data in ascending order and finding the middle value. The quartiles are then found by dividing the data into four equal parts, with the lower quartile being the value that corresponds to the first quarter of the data, and so on.

The following table shows how to compute these values for a sample of six data points:

Data Point Mean Variance Median Quartiles 

1 2 2 1.5 1 

2 4 8 2.5 2 

3 6 12 3.5 3 

4 8 16 4.5 4 

5 10 20 5.5 5 

6 12 24 6.5 6

As you can see, the mean is simply the sum of the data points divided by the number of data points (6). The variance is computed by taking the sum of the squared deviations from the mean and dividing it by the number of data points (6). The median is the value that is in the middle of the data when it is sorted in ascending order. The quartiles are the values that divide the data into four equal parts.

In this example, the mean is 7, the variance is 5, the median is 4.5, and the quartiles are 2, 4, and 6.

Assignment Activity 3: Use simple counting and combinatorial arguments to calculate probabilities.

There are several different ways to calculate probabilities, but one of the simplest is to use counting and combinatorial arguments. For example, let’s say you have a deck of cards with 52 cards in it. If you draw one card at random from the deck, what is the probability that it will be a king?

There are 4 kings in the deck, so the probability of drawing a king is 4/52 = 1/13.

Similarly, if you have a bag with 10 balls in it, and you draw 2 balls at random, what is the probability that both balls will be blue?

There are 6 blue balls in the bag, so the probability of drawing two blue balls is 6/10 × 6/9 = 36/90 = 2/15.

Assignment Activity 4: Calculate probabilities for combinations of events, including unions, intersections, and complements, using the laws of probability.

To calculate the probability of a combination of events, we need to first understand the three fundamental concepts of probability:

1) The Probability of an Event: This is simply the likelihood that an event will occur. We typically express this as a percentage or in terms of odds.

2) The Intersection of Two Events: This is the likelihood that both events will occur simultaneously. We calculate it by multiplying the individual probabilities together.

3) The Complement of an Event: This is the likelihood that the event will not occur. We calculate it by subtracting the individual probability from 1.

Assignment Activity 5: Calculate conditional probabilities and use the Bayes theorem to reverse conditioning.

The Bayes theorem is a mathematical formula that allows you to calculate the probability of an event, based on the prior probability of that event and the likelihood of observing evidence for that event. In other words, it allows you to calculate the probability of something happening, after taking into account what you already know about it.

To use the Bayes theorem, you need to know three things: the prior probability of an event happening (usually represented by P), the likelihood of observing evidence for that event (represented by L), and the conditional probability of observing evidence for an event, given that it has already happened (represented by P(E|H)). Once you have these three values, you can use them to calculate the posterior probability of the event happening (represented by P(H|E)).

The posterior probability is simply the product of the prior probability and the likelihood, divided by the conditional probability. In other words, it is:

Posterior Probability = (Prior Probability × Likelihood) / Conditional Probability

For example, let’s say you’re trying to calculate the probability that it will rain tomorrow. The prior probability of rain is 0.3 (30%), the likelihood of rain is 0.9 (90%), and the conditional probability of rain is 0.8 (80%). Plugging these values into the formula, we get:

P(Rain|Tomorrow) = (0.3 × 0.9) / 0.8

P(Rain|Tomorrow) = 0.27 / 0.8

P(Rain|Tomorrow) = 3.375

Therefore, the posterior probability that it will rain tomorrow is 3.375 (33.75%).

This is just a simple example to illustrate how the Bayes theorem works. In reality, you would need to gather a lot more data to accurately calculate the probability of an event happening. But as you can see, the Bayes theorem is a powerful tool that can be used to calculate the probability of an event, given evidence for that event.

Assignment Activity 6: Construct probability distributions for random variables in simple settings.

There are a few different types of probability distributions, each with its characteristics. To construct a probability distribution for a random variable, you first need to know the type of distribution it follows. Let’s take a look at some common distributions and their properties.

The uniform distribution is the simplest type of distribution. It is characterized by having all values within a specified range have the same probability of occurring. For example, if you toss a coin, the probability of getting heads is 1/2 regardless of how often you’ve tossed heads before. This is because the coin has no memory and flipping it again doesn’t change its probability of coming up heads or tails.

Another simple distribution is the binomial distribution. This is the distribution of a discrete random variable that can only take on two values, such as heads or tails. The binomial distribution is characterized by two parameters: the probability of success (p) and the number of trials (n). For example, if you flip a coin 10 times, the binomial distribution would be used to calculate the probability of getting heads a certain number of times.

The normal distribution is the most common type of distribution. It is characterized by having a bell-shaped curve, with the majority of values clustered around the mean (average) value. The normal distribution is also characterized by two parameters: the mean (μ) and the standard deviation (σ).

Assignment Activity 7: Calculate means and variances of random variables.

Two main types of averages are often used when dealing with random variables – the mean and the variance. The mean is simply the average of all the values of the random variable, while the variance is a measure of how to spread out those values. In other words, it tells you how much variation there is in the dataset.

To calculate the mean, you simply add up all the values and divide by the number of data points. For example, let’s say you have a dataset of five numbers: 1, 2, 3, 4, and 5. The mean would simply be (1 + 2 + 3 + 4 + 5) / 5 = 3.

To calculate the variance, you take each data point and subtract the mean. You then square this value and add up all the squares. Finally, you divide by the number of data points – 1 (n – 1). This may seem like a lot of steps, but it’s quite simple. Let’s take a look at an example:

Let’s say you have a dataset of five numbers: 1, 2, 3, 4, and 5. The mean would be (1 + 2 + 3 + 4 + 5) / 5 = 3. To calculate the variance, you first subtract the mean from each data point:

1 – 3 = -2

2 – 3 = -1

3 – 3 = 0

4 – 3 = 1

5 – 3 = 2

Then, you square each of these values:

(-2)^2 = 4

(-1)^2 = 1

(0)^2 = 0

(1)^2 = 1

(2)^2 = 4

Finally, you add up all the squares and divide by n – 1:

(4 + 1 + 0 + 1 + 4) / (5 – 1) = 10 / 4 = 2.5

And that’s it! You’ve just calculated the variance of a dataset.

As you can see, the variance is a measure of how much the values in a dataset differ from the mean. The larger the variance, the more spread out the data is. Conversely, a small variance means that the data is clustered closely around the mean.

The standard deviation is simply the square root of the variance. In our example above, the standard deviation would be √2.5 = 1.58.

Assignment Activity 8: Calculate marginal and conditional distributions of bivariate discrete distributions, calculate the correlation, and assess independence.

To calculate the marginal distributions, we need to first calculate the conditional distributions. We can use the formula for conditional probability, P(B|A), which is

P(B|A) = P(AB)/P(A)

We can also use the law of total probability to find P(B):

P(B) = P(A|B)P(B) + P(A|~B)P(~B)

Now that we have both formulas, we can calculate them using our data. The marginal distribution of A is found by summing over all values of B, while the marginal distribution of B is found by summing over all values of A. The correlation is simply the covariance divided by the product of the standard deviations.

First, let’s calculate the conditional distribution of A, given B. We’ll use the formula P(A|B) = P(AB)/P(B):

P(A|B) = P(AB)/P(B)

= (0.2 * 0.6)/(0.3)

= 0.4

Now, let’s calculate the conditional distribution of B, given A. We’ll use the formula P(B|A) = P(AB)/P(A):

P(B|A) = P(AB)/P(A)

= (0.2 * 0.6)/(0.4)

= 0.3

Now, let’s calculate the marginal distribution of A. We’ll use the law of total probability:

P(A) = P(A|B)P(B) + P(A|~B)P(~B)

= (0.4 * 0.3) + (0.6 * 0.7)

= 0.5

Finally, let’s calculate the marginal distribution of B. We’ll use the law of total probability:

P(B) = P(A|B)P(B) + P(A|~B)P(~B)

= (0.3 * 0.5) + (0.7 * 0.5)

= 0.5

Now, let’s calculate the correlation between A and B. The correlation is simply the covariance divided by the product of the standard deviations. We can calculate the covariance using the formula:

Cov(A,B) = E((A – μA)(B – μB))

where μA is the mean of A and μB is the mean of B.

Cov(A,B) = E((A – μA)(B – μB))

= (0.4 * 0.2) + (0.6 * 0.8) – [(0.5 * (0.4 + 0.6)]

= -0.04

Now, we can calculate the correlation:

Corr(A,B) = Cov(A,B)/(σA * σB)

where σA is the standard deviation of A and σB is the standard deviation of B. We can calculate the standard deviations using the formula:

σ = √(Var(X))

where Var(X) is the variance of X.

Corr(A,B) = Cov(A,B)/(σA * σB)

= (-0.04)/((0.5 * 0.5) * (0.5 * 0.5))

= -0.16

Finally, let’s assess the independence of A and B. We’ll use the formula:

P(A|B) = P(A)

We can see that P(A|B) ≠ P(A), so A and B are not independent.

Assignment Activity 9: Calculate probabilities from standard distributions (Binomial, Poisson, Normal) using tables.

There are a few different ways to calculate probabilities from standard distributions. One way is to use tables, like the one below. This table can be used to find the probability of getting a specific number of successes out of a given number of trials when the probability of success is p (column 2) and the number of trials is n (column 1).

The other way to calculate probabilities from standard distributions is through the use of formulas. The formulas for the Binomial, Poisson, and Normal distributions can be found online or in any good statistics textbook. With a little practice, you’ll be able to quickly calculate probabilities using these formulas.

Assignment Activity 10: Use Minitab to explore data both numerically and graphically and to calculate probabilities from standard probability models.

Minitab can be used to explore data numerically and graphically, as well as to calculate probabilities from standard probability models. Probability is the branch of mathematics that deals with chance or uncertainty. Probability theory forms the basis for statistics, which is concerned with concluding data that are subject to random variation (chance).

Minitab provides three main ways to analyze data: descriptive statistics, graphical techniques, and inferential statistics. Descriptive statistics summarize data using measures of central tendency (mean, median, and mode) and dispersion (range, interquartile range, variance, and standard deviation). Graphical techniques help us visualize relationships among variables by constructing scatterplots, histograms, and boxplots. Inferential statistics allow us to make predictions or inferences about a population based on a sample.

To calculate probabilities from standard distributions, we need to use the Probability Distributions menu in Minitab. This menu can be found by clicking on the Stat tab and then selecting Probability Distributions from the drop-down menu.

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