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| Section | Objectives |
|---|---|
| Probability Distributions | - Discrete distributions
|
| Probability | - Fundamental probability concepts
|
| Statistical Inference | - Estimation
|
| Descriptive Statistics | - Data summarization
|
| Regression and Correlation | - Relationship analysis
|
>> Standard Applied-Probability-and-Statistics Answers <<
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NEW QUESTION # 14
Regression intercept represents:
Answer: A
Explanation:
In a linear regression equation, usually written as # = b# + b#x, the intercept b# represents the predicted value of Y when X equals 0. It is the point where the regression line crosses the vertical axis. For example, if a model is # = 12 + 3x, the intercept 12 means the predicted response is 12 when x = 0. The slope, b#, is different; it represents the predicted change in Y for each one-unit increase in X. Correlation measures the strength and direction of a linear association, not the intercept. The mean is a measure of center for a variable and is not the same as a regression intercept. In applied settings, the intercept is meaningful only when X = 0 is within a realistic or relevant range of the data. Study Guide references/topics: linear regression, intercept interpretation, slope-intercept form, response prediction.
NEW QUESTION # 15
In a normal distribution, 95% of data lies within:
Answer: B
Explanation:
For a normal distribution, the empirical rule states that approximately 68% of data fall within one standard deviation of the mean, approximately 95% fall within two standard deviations, and approximately 99.7% fall within three standard deviations. The notation ±2 SD means two standard deviations below the mean to two standard deviations above the mean, or # # 2# to # + 2#. Therefore, the interval containing about 95% of normally distributed observations is ±2 standard deviations. Option B corresponds to approximately 68%, not
95%. Option C corresponds to approximately 99.7%, and option D extends beyond the standard empirical- rule benchmarks. This concept is central when estimating the typical spread of bell-shaped data, such as test scores, biological measurements, or repeated measurement errors. The correct answer is ±2 SD because it matches the 95% portion of the 68-95-99.7 rule. Study Guide references/topics: normal distribution, empirical rule, standard deviation, distribution spread.
NEW QUESTION # 16
In a city's public transportation system, the distribution of commuters by mode of transport is as follows: 40% use buses, 25% use trains, 20% use bicycles, 10% walk, and 5% use other means of transportation.
What is the probability that a randomly selected commuter uses either a bus or a train?
Answer: A
Explanation:
The event is that a randomly selected commuter uses either a bus or a train. These categories are mutually exclusive because one commuter is classified under one primary transportation mode in the given distribution.
For mutually exclusive events, the probability of "A or B" is found by adding the individual probabilities. The probability of using a bus is 40%, and the probability of using a train is 25%. Therefore, P(bus or train) = 40%
+ 25% = 65%. The bicycle, walking, and other transportation categories are not included because the question restricts the event to bus or train only. Option B, 45%, would incorrectly combine bus with the 5% "other" category or otherwise omit part of the train group. Option C is too low, and option A is unrelated to the stated categories. References/topics from the Study Guide: probability, mutually exclusive events, addition rule, percentage distributions.
NEW QUESTION # 17
Z-test used when:
Answer: C
Explanation:
A z-test for a population mean is used when the population standard deviation # is known and the sampling distribution of the test statistic can be treated as normal. The test statistic has the form z = (sample statistic # hypothesized parameter) divided by the standard error. Knowing # allows the standard error to be computed using #/#n rather than estimating it with the sample standard deviation. Option B describes the common setting for a t-test, not a z-test. Option C is not sufficient by itself; small samples generally require stronger normality assumptions and often favor t-procedures when # is unknown. Option D is unrelated to the classic z- test for a mean, though z-tests can also be used for proportions under appropriate large-sample conditions. In the provided answer set, the defining condition is that the population standard deviation is known. Study Guide references/topics: z-test, population standard deviation, standard error, hypothesis testing.
NEW QUESTION # 18
A commuter has a .4 probability of taking the bus to work, a .4 probability of driving a car, and a .2 probability of cycling. The probability of being late is .15 when taking the bus, .05 when driving a car, and .1 when cycling.
What is the probability of taking the bus and being on time, or driving a car and being on time?
Answer: A
Explanation:
This item requires multiplying conditional probabilities and then adding mutually exclusive outcomes.
"Taking the bus and being on time" means the commuter takes the bus and is not late. Since the probability of being late by bus is .15, the probability of being on time by bus is 1 # .15 = .85. Therefore, P(bus and on time)
= .4 × .85 = .34. For driving, the probability of being late is .05, so the probability of being on time is 1 # .05
= .95. Therefore, P(car and on time) = .4 × .95 = .38. Because a commuter cannot both take the bus and drive a car on the same trip, the events are mutually exclusive. Add the two joint probabilities: .34 + .38 = .72. The cycling information is not used because the question asks only about bus-on-time or car-on-time outcomes.
References/topics from the Study Guide: probability rules, complements, conditional probability, mutually exclusive events.
NEW QUESTION # 19
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