Courses and Certificates Applied-Probability-and-Statistics WGU Study Notes

The Poisson distribution is used to model counts of events occurring within a fixed interval of time, space, area, or volume, especially when the events are relatively rare and occur at a constant average rate. Examples include the number of calls received per hour, accidents per month, defects per batch, or arrivals at a service counter per minute. Its parameter λ represents the average number of events per interval. Option B describes the binomial distribution, which models the number of successes in a fixed number of independent success/failure trials. Option C is incorrect because Poisson outcomes are discrete counts, not continuous measurements. Option D is also incorrect because categorical data classify observations into groups, while Poisson data count event occurrences. The phrase “rare events per interval” is the defining clue for the Poisson model. Study Guide references/topics: Poisson distribution, rare-event counts, λ parameter, discrete probability models.

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The correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables. Values close to 1 indicate strong positive linear association, values close to −1 indicate strong negative linear association, and values near 0 indicate no linear relationship. Therefore, r = 0 means there is no linear association detected by the correlation coefficient. It is important to interpret this precisely: r = 0 does not guarantee there is no relationship of any kind. A nonlinear relationship may still exist, but correlation measures only linear pattern. Option B and option C are incorrect because strong relationships require r to be close to 1 or −1. Option D is incorrect because perfect correlation occurs at r = 1 or r = −1, not at 0. The correct interpretation is no linear relationship. Study Guide references/topics: correlation coefficient, linear association, scatterplots, correlation interpretation.

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Ethnic group is a categorical variable because it classifies students into named groups rather than measuring a numerical quantity. Categorical data are commonly displayed with bar graphs or pie charts. A bar graph shows the frequency or relative frequency of each category using separate bars, making it easy to compare group sizes. A pie chart shows how the whole student population is divided among categories, making it useful for displaying proportions or percentages. Therefore, the correct pair is bar graph and pie chart. A box plot is used for quantitative data and summarizes values using quartiles and a median, so it is not appropriate for ethnic categories. A line graph is typically used to show change over time or ordered numerical trends. A histogram is used for distributions of quantitative data grouped into intervals, not named categories. Since the goal is to display category membership, both selected graphs must support categorical frequency or percentage displays. References/topics from the Study Guide: categorical data displays, bar graphs, pie charts, relative frequency.

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There are 10 green, 20 yellow, and 15 purple marbles, so the total number of marbles is 10 + 20 + 15 = 45. Because each marble is replaced after selection, the probabilities remain the same on every draw, making the events independent. The probability of selecting a green marble on one draw is 10/45 = 2/9. Selecting two green marbles in a row gives (2/9)(2/9). The probability of then selecting a yellow marble is 20/45 = 4/9. Multiply the probabilities because the question asks for green and green and yellow in that exact sequence: (2/9)(2/9)(4/9) = 16/729. Replacement is essential; without replacement, the denominators and numerators would change after each draw. Since replacement keeps the sample space constant at 45 marbles each time, the independent multiplication rule applies directly. References/topics from the Study Guide: independent events, replacement, multiplication rule, compound probability.