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

A Type II error occurs when the null hypothesis is false but the statistical test fails to reject it. In many answer choices, this is expressed as “accepting the null when false,” though the more precise language is “failing to reject the null.” This error is a false negative: the test misses a real effect, difference, or relationship. For example, if a medication truly improves recovery but a study fails to detect sufficient evidence of improvement, that is a Type II error. Option B describes a Type I error, which occurs when a true null hypothesis is rejected. Option C is not an error. Option D is too vague; statistical decision errors refer specifically to incorrect conclusions about hypotheses, not ordinary data-entry mistakes. The probability of a Type II error is denoted β, and statistical power is 1 − β. Study Guide references/topics: hypothesis testing, Type II error, null hypothesis, statistical power.

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Two coin flips produce four equally likely ordered outcomes: HH, HT, TH, and TT. Exactly one head occurs in two of these outcomes: HT and TH. Therefore, the probability is 2 favorable outcomes out of 4 total outcomes, or 2/4 = 1/2. This can also be computed using the binomial model. There are n = 2 independent trials, success probability p = 1/2, and exactly one success is required. The binomial calculation is C(2,1)(1/2)^1(1/2)^1 = 2 × 1/4 = 1/2. Option B, 1/4, counts only one of the two favorable sequences. Option C, 3/4, includes outcomes with at least one head rather than exactly one head. Option D would mean certainty, which is impossible because HH and TT do not satisfy the condition. Study Guide references/topics: sample spaces, coin-flip probability, binomial probability, independent events.

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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The central limit theorem states that, for sufficiently large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the shape of the original population distribution, provided observations are independent and drawn appropriately. This is why option A is correct. The theorem does not require the population itself to be normal; if the population is normal, the sample mean is normally distributed for any sample size, but the central limit theorem is especially powerful because it applies broadly for large n. Option C is false because sample variance is an estimate of population variance, not automatically equal to it. Option D is false because standard error generally decreases as sample size increases but does not become exactly zero unless sample size is infinite or variability is absent. The central limit theorem supports confidence intervals and hypothesis tests for means. Study Guide references/topics: central limit theorem, sampling distribution, sample mean, normal approximation.

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