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| Section | Objectives |
|---|---|
| Regression and Correlation | - Relationship analysis
|
| Statistical Inference | - Estimation
|
| Probability | - Fundamental probability concepts
|
| Descriptive Statistics | - Data summarization
|
| Probability Distributions | - Discrete distributions
|
>> Applied-Probability-and-Statistics Exam Quiz <<
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NEW QUESTION # 46
Probability of rolling 1, 2, or 3 on die = ?
Answer: C
Explanation:
A standard six-sided die has six equally likely outcomes: 1, 2, 3, 4, 5, and 6. The event "rolling 1, 2, or 3" has three favorable outcomes: 1, 2, and 3. The probability is therefore favorable outcomes divided by total outcomes: 3/6. This fraction simplifies to 1/2. Option B, 1/3, would correspond to two favorable outcomes out of six. Option C, 1/6, is the probability of rolling one specific number only. Option D, 2/3, would require four favorable outcomes out of six. Since exactly half of the die faces are 1, 2, or 3, the correct probability is one- half. This is a direct application of theoretical probability with equally likely outcomes. Study Guide references/topics: die probability, favorable outcomes, sample space, theoretical probability.
NEW QUESTION # 47
Z-test used when:
Answer: A
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 # 48
Probability sum for all outcomes = ?
Answer: D
Explanation:
In any valid probability model, the probabilities of all outcomes in the sample space must sum to 1. The sample space is the complete set of possible outcomes for an experiment, and exactly one outcome from that set must occur. A total probability of 1 represents certainty. For example, when rolling a fair six-sided die, the six outcomes each have probability 1/6, and their sum is 6 × 1/6 = 1. A total of 0 would imply no possible outcome can occur, which is not a valid probability model. A total greater than 1 violates the maximum boundary of probability, and a total less than 1 would mean the sample space is incomplete or some probability has been omitted. This rule is one of the axioms of probability and is used to verify distributions.
Study Guide references/topics: sample space, probability axioms, total probability, probability distributions.
NEW QUESTION # 49
Central limit theorem applies when:
Answer: B
Explanation:
The central limit theorem applies when the sample size is sufficiently large and observations are independent.
It states that the sampling distribution of the sample mean becomes approximately normal as sample size increases, even if the original population distribution is not normal. This makes option A correct. Option B is incorrect because the population does not have to be normal for the theorem to operate; large sample size is what permits the normal approximation. Option C is not the condition that defines the theorem. A population parameter may be unknown in inference, but that is not the trigger for the central limit theorem. Option D is incorrect because the classic central limit theorem for means concerns quantitative data, not purely categorical labels. The theorem is foundational because it justifies using normal-based confidence intervals and hypothesis tests for means when sample sizes are large. Study Guide references/topics: central limit theorem, sample size, sampling distribution, normal approximation.
NEW QUESTION # 50
Two dice rolled, probability of both even = ?
Answer: D
Explanation:
A standard six-sided die has three even outcomes: 2, 4, and 6. Thus, the probability that one die lands on an even number is 3/6 = 1/2. When two dice are rolled, the outcomes are independent because the result of one die does not affect the other. Therefore, the probability that both dice are even is 1/2 × 1/2 = 1/4.
Equivalently, there are 36 ordered outcomes for two dice. The even-even outcomes are formed from 3 even choices on the first die and 3 even choices on the second die, giving 3 × 3 = 9 favorable outcomes. Then 9/36 simplifies to 1/4. Option B gives the probability of one die being even, not both. Options C and D do not match the count or multiplication rule. Study Guide references/topics: independent events, dice probability, multiplication rule, sample space.
NEW QUESTION # 51
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