MAT 2058 Statistics COMPLETED EXAM 2026 (12pages)

1. What is the difference between a population and a sample? How are they related to the concepts of parameters and statistics? - A population is the entire group of individuals or objects that we want to study, while a sample is a subset of the population that we actually collect data from. Parameters are numerical measures that describe some aspect of the population, such as the mean or the standard deviation, while statistics are numerical measures that describe some aspect of the sample, such as the sample mean or the sample standard deviation. We use statistics to estimate parameters and make inferences about the population. 2. What is a sampling distribution? What is its relationship to the central limit theorem? - A sampling distribution is the distribution of a statistic based on repeated sampling from the population. It shows how the statistic varies from sample to sample. The central limit theorem states that if the sample size is large enough, the sampling distribution of the sample mean (or any other statistic that is a linear combination of independent random variables) will be approximately normal, regardless of the shape of the population distribution. 3. What is a confidence interval? How do you interpret a 95% confidence interval for a population mean? - A confidence interval is an interval estimate that gives a range of plausible values for a population parameter, based on a sample statistic and a level of confidence. A 95% confidence interval for a population mean means that if we repeated the sampling process many times and calculated a 95% confidence interval for each sample, about 95% of those intervals would contain the true population mean. 4. What is a hypothesis test? What are the null and alternative hypotheses? How do you make a decision based on a p-value? - A hypothesis test is a procedure that allows us to evaluate a claim about a population parameter, based on sample data. The null hypothesis is the statement that represents the status quo or no effect, while the alternative hypothesis is the statement that represents the research question or some effect. The p-value is the probability of obtaining the observed or more extreme results, assuming that the null hypothesis is true. We compare the p-value to a significance level, which is a predetermined