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

A correlation coefficient of r = 0.8 indicates a strong positive linear relationship between two quantitative variables. The positive sign means that as one variable increases, the other tends to increase. The magnitude, 0.8, is close to 1, which indicates a strong linear pattern. It is not perfect, because perfect positive correlation would be r = 1, but it is clearly stronger than a weak or moderate association. Option B is incorrect because the interpretation matches both the sign and magnitude of r. Option C is incorrect because correlation is used for quantitative variables, not categorical variables. Option D is incorrect because a negative r would indicate a negative relationship. Importantly, correlation does not prove causation; it describes the direction and strength of linear association. Study Guide references/topics: correlation coefficient, positive association, linear relationship, scatterplot interpretation.

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The correlation coefficient r measures the direction and strength of a linear relationship between two quantitative variables. It ranges from −1 to 1. A value close to −1 indicates a strong negative linear relationship, meaning that as one variable increases, the other tends to decrease in a highly consistent linear pattern. Since r = −0.9 is very close to −1, the relationship is strong and negative. Option B is incorrect because the sign is negative, not positive. Option C identifies the direction correctly but understates the strength; −0.9 is not weak. Option D would correspond to r near 0, where there is little or no linear association. A scatterplot associated with r = −0.9 would show points clustered closely around a downward-sloping line. This does not prove causation; it describes linear association only. Study Guide references/topics: correlation coefficient, negative association, strength of relationship, scatterplots.

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A histogram and a bar chart both use bars, but they display different types of data. A histogram is used for quantitative numerical data grouped into intervals, such as ages, weights, test scores, or commute times. The bars represent continuous or ordered numeric ranges, and the bars usually touch to show that the scale is continuous. A bar chart is used for categorical data, such as favorite sport, political party, product type, or survey response category. The bars are separated because the categories are distinct labels rather than continuous intervals. Option B reverses the correct uses. Option C is incorrect because the graphs are not the same even though both contain bars. Option D is invalid because option A states the standard distinction. Correct graph selection depends on identifying whether the variable is categorical or quantitative. Study Guide references/topics: histograms, bar charts, categorical data, quantitative data displays.

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The equation 4 + x = 6 − x is a linear equation in one variable. To solve it, collect variable terms on one side and constant terms on the other. Add x to both sides to eliminate the negative x on the right: 4 + 2x = 6. Then subtract 4 from both sides: 2x = 2. Finally, divide both sides by 2, giving x = 1. This value can be verified directly in the original equation. Substituting x = 1 gives 4 + 1 = 5 on the left and 6 − 1 = 5 on the right, so both sides are equal. Option A gives 4 = 6, which is false. Option C gives 6 = 4, also false. Option D gives 9 = 1, which is not valid. The correct solution is the value that balances both sides of the equation. References/topics from the Study Guide: basic algebra, solving linear equations, inverse operations, substitution checks.

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A standard deck has 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red, while clubs and spades are black. Each suit has 13 cards, so the number of red cards is 13 + 13 = 26. The probability of drawing a red card is therefore the number of favorable outcomes divided by the total number of outcomes: 26/52. This simplifies to 1/2. Option B, 1/4, would describe the probability of drawing one specific suit, such as hearts only. Option C, 1/13, would describe one specific rank in the deck, such as a particular value across suits. Option D does not correspond to the card structure of a standard deck. Since exactly half the deck is red, the correct probability is 26/52 = 1/2. Study Guide references/topics: card probability, favorable outcomes, sample space, theoretical probability.

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For two events, the general addition rule is P(A or B) = P(A) + P(B) − P(A and B). Because A and B are independent, their joint probability is P(A and B) = P(A)P(B). Here, P(A) = 0.3 and P(B) = 0.2, so P(A and B) = 0.3 × 0.2 = 0.06. Then P(A or B) = 0.3 + 0.2 − 0.06 = 0.44. The subtraction is necessary because the overlap, where both A and B occur, is counted twice when adding P(A) and P(B). Option A, 0.5, incorrectly adds the two probabilities without correcting for overlap. Option B gives the probability of both events, not either event. Option D is unrelated to the addition rule. Study Guide references/topics: independent events, addition rule, multiplication rule, union of events.

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This question requires comparing conditional percentages from a two-way table. For residents, 150 out of 200 are in favor of the proposed transportation system. The resident support rate is therefore 150/200 = 0.75, or 75%. For business owners, 30 out of 100 are in favor. The business owner support rate is 30/100 = 0.30, or 30%. The difference is 75% − 30% = 45 percentage points, which is a large gap. Therefore, a resident is much more likely than a business owner to favor the new transportation system. Option A understates the magnitude of the difference by calling it slight. Options B and C reverse the direction of the comparison because residents have the higher support rate, not the lower one. The appropriate statistical tool here is conditional percentage because support is being compared within each respondent group. References/topics from the Study Guide: contingency tables, conditional percentages, two-variable categorical data, comparative proportions.

The median is the measure of central tendency least affected by outliers because it depends only on the ordered position of values, not on the numerical magnitude of extreme observations. To find the median, the data are arranged from least to greatest and the middle value is identified, or the average of the two middle values is taken when the sample size is even. If an extremely large or extremely small value is added to a data set, the mean can shift substantially because every value contributes to the arithmetic average. The median, however, usually remains stable unless the extreme value changes the middle position. The mode is also not directly affected by magnitude, but it may be absent or non-unique and is often less informative for numerical data. Range is not a measure of central tendency and is highly sensitive to outliers because it uses the minimum and maximum. Study Guide references/topics: descriptive statistics, measures of center, median, resistance to outliers.

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The correlation coefficient r describes the direction and strength of a linear relationship between two quantitative variables. Its values range from −1 to +1. A negative value indicates that as one variable increases, the other tends to decrease. Since r = −0.5, the direction is negative. Its magnitude, 0.5, is not close enough to zero to indicate no relationship, but it is also not close enough to −1 to represent an extremely strong relationship. Therefore, the best description is a moderate negative linear relationship. Option B is incorrect because the sign is negative, not positive. Option C is incorrect for both direction and strength. Option D would be appropriate only when r is near 0, indicating little or no linear association. A scatterplot with r = −0.5 would generally show a downward trend, but with noticeable scatter around the trend. Study Guide references/topics: correlation coefficient, negative association, scatterplots, linear relationship strength.

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This study involves two categorical variables: customer age group and preferred shopping method. Age group is categorical because customers are classified into groups such as teens, adults, and seniors. Shopping method is also categorical because customers are classified as preferring either online or in-store shopping. When two categorical variables are analyzed together, conditional percentages are the most appropriate numerical summary because they describe how the distribution of one categorical variable changes within levels of another categorical variable. For example, the analyst could compute the percentage of teens who prefer online shopping, the percentage of adults who prefer online shopping, and the percentage of seniors who prefer online shopping. A five-number summary is used for quantitative data, not categorical data. Mean difference requires numerical measurements. A correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables, so it is not appropriate here. References/topics from the Study Guide: two-variable categorical data, conditional percentages, categorical association, numerical summaries.

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