WGU Updated Applied-Probability-and-Statistics Exam Questions and Answers by nayla

The z-score formula is z = (x − μ)/σ, where x is the observed value, μ is the population mean, and σ is the population standard deviation. A z-score standardizes a raw value by measuring how many standard deviations it lies above or below the mean. The numerator, x − μ, gives the deviation from the mean. Dividing by σ converts that deviation into standard deviation units. If z is positive, the value is above the mean. If z is negative, the value is below the mean. If z = 0, the value equals the mean. Option B incorrectly adds the value and mean, which does not measure deviation. Option C ignores the mean, and option D ignores the observed value entirely. The z-score is central for normal distribution calculations because it allows values from different scales to be compared using the standard normal distribution. Study Guide references/topics: z-scores, standard normal distribution, mean, standard deviation.

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The mean is the arithmetic average of a quantitative data set. To calculate it, add all observed values and divide by the number of observations. The jelly bean counts are 50, 60, 80, 80, 100, 100, 100, 120, 120, and 140. Their sum is 950. There are 10 jars, so the mean is 950 ÷ 10 = 95. This means the average number of jelly beans per jar is 95. Option A, 100, may seem plausible because 100 appears frequently and is near the center of the data, but frequency alone determines the mode, not the mean. Option C, 80, is one of the repeated values but not the average. Option B, 120, is too high because several observations fall well below 120. The mean uses every value in the data set, including the low values 50 and 60 and the high value 140. References/topics from the Study Guide: mean, measures of center, quantitative data, arithmetic average.

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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Variance and standard deviation are directly connected measures of spread. Variance is calculated by averaging squared deviations from the mean, while standard deviation is the square root of variance. Therefore, reversing that relationship, standard deviation squared equals variance. Symbolically, if the standard deviation is s, then the sample variance is s². This relationship matters because variance is expressed in squared units, while standard deviation is expressed in the original units of measurement. For example, if test-score standard deviation is 4 points, the variance is 4² = 16 square points. The mean and median are measures of center, not squared spread. The standard error measures the variability of a sample statistic, usually the sample mean, across repeated samples. The correct answer is variance because it is the squared form of standard deviation. Study Guide references/topics: standard deviation, variance, measures of spread, descriptive statistics.

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