WGU Applied-Probability-and-Statistics - Applied Probability and Statistics (FZO1 C955)

The random variable X counts the number of heads obtained in 4 coin flips. Since it is a count, it can take only specific whole-number values: 0, 1, 2, 3, or 4. It cannot take fractional values such as 2.5 heads. A random variable with countable possible outcomes is classified as discrete. This situation also fits a binomial framework because there is a fixed number of independent trials, each trial has two outcomes, and the probability of heads remains constant for a fair coin. A continuous random variable, by contrast, can take any value over an interval, such as time, weight, or height. The number of heads is not measured on a continuum; it is counted. Therefore, the correct classification is discrete. Study Guide references/topics: discrete random variables, binomial setting, coin-flip outcomes, probability distributions.

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This question uses the complement rule of probability. If the probability that it rains is 0.25, then the probability that it does not rain is the complement of that event. The complement rule states that P(not A) = 1 − P(A). Here, A represents the event “it rains,” so P(not rain) = 1 − 0.25 = 0.75. This means there is a 75% probability that rain does not occur. Option B repeats the probability of rain rather than its complement. Option C would imply rain and no rain are equally likely, which is not supported by the given probability. Option D would mean no rain is certain, but a 0.25 probability of rain means rain is still possible. The key technical point is that an event and its complement must sum to 1. Study Guide references/topics: probability, complement rule, event probability, sample-space total.

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In a linear regression equation, the slope represents the predicted change in the response variable Y for each one-unit increase in the explanatory variable X. In slope-intercept form, ŷ = b₀ + b₁x, the slope is b₁. For example, if a regression equation predicts cost as ŷ = 25 + 4x, the slope 4 means the predicted cost increases by 4 units for each additional unit of x. The intercept, b₀, is the predicted value of Y when X = 0, so option B describes a different component. R² measures the proportion of variation in Y explained by the regression model, not the rate of change. Correlation measures strength and direction of linear association, but it is not the same as the slope because it is unitless and standardized. The slope is the operational rate of change in the model. Study Guide references/topics: linear regression, slope interpretation, response variable, explanatory variable.

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In a linear regression equation of the form y = mx + b, the slope m gives the predicted change in the response variable y for a one-unit increase in the explanatory variable x. Here, y represents total ice cream sales and x represents average maximum temperature. The equation y = 13x − 598 has slope 13, so the model predicts that ice cream sales increase by about $13 for every 1°F increase in temperature. This is not the sales value at 0°F; that would be the intercept, −598, which is generally not meaningful in this practical context. It is also not the predicted sales at 100°F, because that would require substituting x = 100 into the regression equation. The positive slope matches the upward trend in the scatterplot: higher temperatures are associated with higher ice cream sales. References/topics from the Study Guide: correlation and regression, slope interpretation, explanatory and response variables.

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A boxplot summarizes the distribution of a quantitative variable using key positional statistics. It displays the median, first quartile, third quartile, and typically the minimum and maximum non-outlier values. It may also mark outliers separately. The box itself spans from Q1 to Q3, representing the interquartile range, or middle 50% of the data. The line inside the box marks the median. Whiskers extend to values within the non-outlier range, depending on the graphing convention. A boxplot does not primarily show frequency counts; histograms, dot plots, and bar charts are better for frequencies. A scatterplot displays the relationship between two quantitative variables. Probability is not directly displayed by a standard boxplot, although distributional interpretation can be supported by it. The correct answer is therefore median, quartiles, and outliers. Study Guide references/topics: boxplots, five-number summary, quartiles, outliers.

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For independent events, the probability that both events occur is the product of their individual probabilities. The rule is P(A and B) = P(A) × P(B). Independence means that the occurrence of one event does not change the probability of the other. For example, if a coin is tossed and a die is rolled, the probability of heads and rolling a 4 is 1/2 × 1/6 = 1/12. Option B is an addition expression used in “or” problems, not “and” problems. Option C applies to mutually exclusive events, not independent events. Option D would mean both events are certain to occur together, which is not generally true. The word “and” signals intersection, and independence allows the intersection to be calculated by multiplication. Study Guide references/topics: independent events, multiplication rule, joint probability, compound probability.

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A confidence interval has the structure estimate ± margin of error. The margin of error is calculated as critical value × standard error. For a 95% confidence interval using the normal approximation, the critical value is approximately 1.96, often rounded to 2 in introductory settings. The interval is given as 50 ± 2, so the margin of error is 2. Using the approximate 95% critical value of 2, we solve 2 = 2 × SE, giving SE = 1. More exactly, using 1.96 gives SE = 2/1.96 ≈ 1.02, which rounds to 1. Option B confuses the margin of error with the standard error. Option C is too large, and option D would produce a margin of error near 1 under a 95% critical value. The correct standard error is approximately 1. Study Guide references/topics: confidence intervals, standard error, critical value, margin of error.

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Standard deviation measures how spread out data values are around the mean. A smaller standard deviation indicates that values are clustered closely near the mean, while a larger standard deviation indicates greater variability. Standard deviation is based on deviations from the mean, squared deviations, variance, and then the square root of variance. It is expressed in the original units of measurement, making it easier to interpret than variance. Option B is incorrect because central tendency is measured by statistics such as the mean, median, and mode. Option C is incorrect because probability measures likelihood, not data dispersion. Option D refers to frequency or the number of observations, not their variability. For example, two classes could have the same mean test score but different standard deviations; the class with the larger standard deviation has more varied scores. Study Guide references/topics: standard deviation, variance, spread, descriptive statistics.

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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.

The phrase “given that they are enrolled in the science course” indicates conditional probability. The reference group is not all 30 students; it is only the 12 students enrolled in science. Among those 12 science students, 8 are also enrolled in math. Therefore, the conditional probability is P(math | science) = P(math and science) ÷ P(science), or more directly, 8 ÷ 12. This simplifies to 2/3, which is approximately 0.6667, rounded to .67. Option A, .27, would come from comparing the 8 students to the entire class of 30, but that ignores the given condition. Option B, .40, corresponds to 12/30, the proportion enrolled in science. Option C, .44, is approximately 8/18, which reverses the condition and would represent the probability of science given math. The correct answer uses the science group as the denominator. References/topics from the Study Guide: conditional probability, two-way events, joint enrollment, probability ratios.

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