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ST2217 Inferential Statistical Methods for Business Assignment Sample NUI Galway Ireland

ST2217 Inferential Statistical Methods for Business is a course offered by the National University in Ireland, Galway. The objective of the course is to provide students with an understanding of inferential statistical methods and to develop their skills in using these methods for business decision-making.

The course covers the following topics: sampling and polling, estimation, hypothesis testing, correlation and regression analysis, decision theory, and Bayesian inference. In addition to learning about the theoretical underpinnings of these techniques, students will also learn how to apply them using software such as Microsoft Excel or SPSS.

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

Assignment Activity 1: Define and identify in applications, basic terms: experimental unit, population, sample, variables, and their types, parameter, statistic, descriptive statistics, and inferential statistics.

An experimental unit is the smallest unit of observation in a study. In many applications, an experimental unit is a person, but it could be any other type of entity, such as a group of people, a company, or even an ecosystem. The population is the set of all experimental units that could be studied. A sample is a subset of the population that is studied.

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There are two types of variables in statistics: categorical and quantitative. Categorical variables take on values that can be placed into categories, such as gender (male/female) or political affiliation (Democrat/Republican). Quantitative variables take on numeric values that can be measured, such as height or weight. There are two types of quantitative variables: discrete and continuous. Discrete variables can only take on certain values (such as whole numbers), while continuous variables can take on any value within a certain range (such as all real numbers between 0 and 1).

A parameter is a population characteristic that is estimated by a statistic. For example, the population mean μ is a parameter, and the sample mean x is a statistic that estimates μ.

Descriptive statistics are measures that summarize data. Examples include the mean, median, and mode. Inferential statistics are methods that allow us to make conclusions about a population based on data from a sample. Examples include hypothesis testing and estimation.

Assignment Activity 2: Define the term standard error and define, discuss and identify common sampling distributions and the definition and apply the Central Limit Theorem in the context of the sampling distribution of the mean and the sampling distribution of the proportion of successes. Discuss the sampling distribution of the mean for large and small samples and discuss and check any assumptions that apply in those cases.

The standard error (SE) is a statistic that measures the variability of a sample estimate. SE is computed as the standard deviation of the sampling distribution of the estimate. 

The central limit theorem states that, given certain conditions, the sampling distribution of a statistic will be approximately normally distributed. This theorem is used in statistics to make inferences about population parameters based on samples. The conditions required for the central limit theorem to hold are that the sample size must be large and the population must be approximately normally distributed.

Assumptions: The sampling distribution of the mean is normal if the following conditions are met:

  • The population is normally distributed.
  • The sample size is large (n > 30).

If these conditions are not met, the sampling distribution of the mean will still be approximately normal if the population is large (N > 10,000).

The sampling distribution of the proportion of successes is normal if the following conditions are met:

  • The population is a Bernoulli trial (i.e., each individual has only two possible outcomes).
  • The sample size is large (n > 30).

If these conditions are not met, the sampling distribution of the proportion of successes will still be approximately normal if the population is large (N > 10,000) and the proportion of successes is not close to 0 or 1.

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Assignment Activity 3: Construct and interpret a confidence interval for a population mean for large and small samples. Discuss and check any assumptions that apply in doing so. Construct confidence intervals at varying levels of confidence and discuss the implications of changes in the confidence level and the sample size on the resulting interval.

A confidence interval for a population mean is a statistical measure that tells us how likely it is that the population mean falls within a certain range.

There are two types of confidence intervals: large and small. A large-sample confidence interval is based on data from a relatively large sample size, while a small-sample confidence interval is based on data from a relatively small sample size.

Both types of confidence intervals require us to make certain assumptions about the data to be accurate. For example, we must assume that the sampling distribution of the sample mean is normal (or at least close to normal). We must also assume that the population standard deviation is known.

If these assumptions are not met, then our confidence interval will not be accurate.

The level of confidence is the probability that the population means will fall within the confidence interval. The most common levels of confidence are 90%, 95%, and 99%.

The sample size affects the width of the confidence interval. In general, the larger the sample size, the narrower the confidence interval.

Changes in the level of confidence will also affect the width of the confidence interval. For example, a 95% confidence interval will be wider than a 99% confidence interval because there is a greater chance that the population means will fall outside of the 95% interval.

Assignment Activity 4: Carry out a hypothesis test for a population mean for large and small samples. Discuss and check any assumptions that apply in carrying out the analysis. Define type I and type II error, the significance level, the test statistic, the power of the test, and the p-value and interpret each of these terms in the application. Complete the hypothesis test by either determining a rejection region for the test statistic, a rejection region for the sample estimate of the parameter, or a p-value. Identify and complete one and two-tailed testing procedures.

A hypothesis test for a population means is a statistical test that is used to determine whether or not there is evidence to support the claim that the population means is different from a specified value.

The null hypothesis for this test is that the population mean is equal to the specified value. The alternative hypothesis is that the population mean is not equal to the specified value.

Two types of error can occur in this test: type I and type II. Type I error occurs when the null hypothesis is rejected when it is true. Type II error occurs when the null hypothesis is not rejected when it is false.

The significance level for this test is the probability of making a type I error. The most common levels of significance are 0.05 and 0.01.

The test statistic for this test is the z-score. This is a measure of how many standard deviations away from the mean the sample mean is.

The power of the test is the probability of correctly rejecting the null hypothesis when it is false.

The p-value is the probability of observing a test statistic that is at least as extreme as the one that was observed, given that the null hypothesis is true.

A rejection region for this test can be either a rejection region for the test statistic or a rejection region for the sample estimate of the parameter. A rejection region for the test statistic is a set of values that are so extreme that we would reject the null hypothesis if the test statistic fell within that region. A rejection region for the sample estimate of the parameter is a set of values that are so extreme that we would reject the null hypothesis if the sample estimate of the parameter fell within that region.

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Assignment Activity 5: Expand the application of basic skills learned in constructing confidence intervals and carrying out hypothesis tests for inferring the value of a single population mean to other problems such as:

Inference for comparing means of two populations (large and small samples), independent samples.

To compare the means of two populations, we need to make sure that the samples are independent. This means that the two samples were not drawn from the same population.

If the samples are independent, then we can use a t-test to compare the means. This test is used when the data is normally distributed.

If the samples are not independent, then we cannot use a t-test to compare the means. In this case, we would need to use a different type of test.

Inference for comparing means between two populations (large and small samples), paired samples.

When comparing two populations, it is important to use a paired samples t-test rather than a regular t-test. This is because when there are only small sample sizes, the variance in the population can be much greater than the variance in the sample. This will cause the test statistic (t) to be large and lead to falsely concluding that there is a statistically significant difference between the means of the two populations.

The paired samples t-test takes into account the increased variability in the larger population and gives a more accurate estimate of whether or not there is a difference between the means of the two populations.

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Inference for comparing means of more than two populations using ANOVA.

The ANOVA statistic is a measure of the extent to which the means of the populations are different. It is calculated as the ratio of the within-group variance to the total variance.

The within-group variance is simply the variance of the individual observations within each population. The total variance is the sum of the variances within and between groups. Thus, the ANOVA statistic provides a measure of how much of the total variability in the data can be accounted for by differences between populations, and how much can be attributed to differences within populations.

Inference for a single population proportion of successes(large samples only) in a binomial experiment.

In a binomial experiment, we are interested in inferring the population proportion of successes (p) from a sample proportion of successes (x). In general, we can only make inferences for a single population proportion when the sample size is large.

There are two main methods for making inferences about a population proportion: the confidence interval approach and the hypothesis testing approach.

The confidence interval approach consists of constructing a confidence interval for the population parameter p based on the sample statistic x. We can then interpret the confidence interval to determine whether or not p is significantly different from some hypothesized value, represented by p0. 

Inference for population proportions in a multinomial experiment, the χ 2 goodness of fit test. 

The χ2 goodness of fit test is a measure of how well the data in a sample distribution matches the expected values for that distribution. It’s used to assess the statistical significance of differences between observed and expected frequencies.

The χ2 test is most commonly used in tests of independence, such as when testing whether or not two categorical variables are related. A multinomial experiment can be used to assess the goodness of fit for each possible combination of outcomes. This allows you to compare the observed proportions against the expected proportions and determine which pairs are significantly different.

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