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

The correlation coefficient r measures the direction and strength of a linear association between two quantitative variables. Since r = 0.72 is positive, the relationship is positive: as daylight hours increase, gym attendance tends to increase. The value 0.72 also indicates a moderately strong to strong linear association because it is closer to 1 than to 0. However, correlation alone does not establish causation. Even though daylight hours and gym attendance move together, other variables could influence attendance, such as weather, seasonal routines, work schedules, or fitness promotions. Therefore, the valid conclusion is that there is a positive association, not a proven causal relationship. Options A and B are incorrect because they describe a negative relationship, which conflicts with the positive correlation coefficient. Option D overstates the evidence by claiming causation. References/topics from the Study Guide: correlation coefficient, positive association, linear relationship, correlation versus causation.

============

Probability values must lie between 0 and 1, inclusive. A probability of 0 means an event is impossible, while a probability of 1 means an event is certain. Values between 0 and 1 represent events with varying degrees of likelihood. A negative probability has no valid interpretation in standard probability theory because probability measures relative likelihood or long-run frequency, and neither can be less than zero. Option B is allowed because impossible events have probability 0. Option C is allowed because certain events have probability 1. Option D is allowed because ordinary uncertain events have probabilities strictly between 0 and 1. For example, the probability of rolling a 3 on a fair die is 1/6, which lies between 0 and 1. The only invalid option listed is negative probability. Study Guide references/topics: probability axioms, probability range, impossible events, certain events.

============

A Type II error occurs when the null hypothesis is false but the test fails to reject it. Many introductory answer choices phrase this as “accept H₀ when false,” though the more technically careful wording is “fail to reject H₀ when false.” This is a false negative: the real effect or difference exists, but the test does not detect sufficient evidence. The probability of a Type II error is denoted β. Option B describes a Type I error, which is rejecting a true null hypothesis. Option C is incorrect because 1 − α is not β; α is the Type I error probability, while 1 − β is statistical power. Option D is too vague and does not identify a formal hypothesis-testing error. The defining condition for Type II error is missing a real effect by failing to reject a false null hypothesis. Study Guide references/topics: Type II error, beta, hypothesis testing, statistical power.

============