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

The null hypothesis represents the default claim or baseline assumption in hypothesis testing. For a new drug, a typical null hypothesis states that the drug has no effect, no difference from placebo, or no improvement over the existing standard. The testing process begins by assuming the null hypothesis is true, then evaluates whether the sample evidence is strong enough to reject it. Therefore, the correct statement is that researchers assume no effect until evidence suggests otherwise. This does not mean the drug is proven ineffective; it means the burden of evidence lies with demonstrating an effect. Option B is the alternative hypothesis, not the null. Option C overstates the conclusion because failing to reject the null does not prove the drug has no effect. Option D is false because this type of claim is testable using experimental data and statistical inference. Study Guide references/topics: null hypothesis, alternative hypothesis, hypothesis testing, statistical evidence.

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A Type I error occurs when a statistical test rejects the null hypothesis even though the null hypothesis is actually true. This is often described as a false positive. In practical terms, the test concludes that there is an effect, difference, or relationship when none truly exists under the null condition. The probability of committing a Type I error is denoted by α, the significance level, often set at 0.05. Option B describes a Type II error, where the test fails to reject a false null hypothesis. Option C is not an error at all. Option D is too general; Type I error is not a data-entry issue but a formal decision error in hypothesis testing. The defining phrase is “reject H₀ when true.” Study Guide references/topics: hypothesis testing, Type I error, null hypothesis, significance level.

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For a binomial random variable, the expected value is E(X) = np, where n is the number of trials and p is the probability of success on each trial. The binomial model applies when there is a fixed number of independent trials, each trial has two outcomes, and the probability of success remains constant. The expected value represents the long-run average number of successes across repeated binomial experiments. For example, if a test has 20 independent questions and the probability of success on each is 0.75, the expected number correct is 20 × 0.75 = 15. Option B, n/p, and option C, p/n, do not represent the binomial mean. Option D, n + p, incorrectly adds the number of trials and success probability, which are not comparable units. The formula np captures both how many opportunities there are and how likely success is on each opportunity. Study Guide references/topics: binomial distribution, expected value, success probability, discrete random variables.

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For a normal distribution, the empirical rule, also called the 68-95-99.7 rule, describes how data are distributed around the mean. Approximately 68% of observations fall within 1 standard deviation of the mean, approximately 95% fall within 2 standard deviations, and approximately 99.7% fall within 3 standard deviations. The question asks specifically for the percentage within 2 standard deviations, so the correct value is 95%. This interval is written as μ − 2σ to μ + 2σ, where μ is the mean and σ is the standard deviation. Option B corresponds to one standard deviation, not two. Option C is an imprecise version of the three-standard-deviation rule, and option D has no standard empirical-rule interpretation in this context. This rule is frequently used to estimate ranges in bell-shaped distributions. Study Guide references/topics: normal distribution, empirical rule, standard deviation, distribution spread.

For a continuous uniform distribution from 0 to 10, probability is proportional to interval length because every subinterval of equal length has equal probability density. The total interval length is 10 − 0 = 10. The event X < 4 corresponds to the interval from 0 to 4, which has length 4. Therefore, P(X < 4) = 4/10 = 0.4. This means 40% of the distribution lies below 4. Option B, 0.6, would correspond to P(X > 4), since the interval from 4 to 10 has length 6. Option C would represent half the interval, such as P(X < 5). Option D does not match the interval proportion. The essential uniform-distribution principle is area equals probability, and with constant density, area reduces to interval length divided by total length. Study Guide references/topics: uniform distribution, continuous probability, interval length, probability density.

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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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