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| Section | Objectives |
|---|---|
| Probability Theory | - Fundamental Probability Concepts
|
| Regression and Modeling | - Linear Relationships
|
| Statistical Inference | - Estimation and Confidence Intervals
|
| Descriptive Statistics | - Single Variable Data Analysis
|
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NEW QUESTION # 90
Standard error decreases when:
Answer: B
Explanation:
The standard error of the sample mean is calculated as SE = s/#n, where s is the sample standard deviation and n is the sample size. Since n appears in the denominator, increasing the sample size reduces the standard error, assuming the standard deviation stays constant. This reflects the fact that larger samples produce more stable estimates of the population mean. Option B is incorrect because decreasing sample size increases standard error. Option C is incorrect because a larger standard deviation increases standard error by increasing the numerator. Option D is incorrect because the mean affects the center of the distribution but does not directly determine the standard error. Standard error is about precision: smaller standard error means less sampling variability and a more precise estimate. That is why larger samples are preferred in inference. Study Guide references/topics: standard error, sample size, standard deviation, sampling variability.
NEW QUESTION # 91
A dataset: 2, 4, 6, 8, 10. Variance = ?
Answer: A
Explanation:
Variance measures the average squared distance of data values from the mean. For the dataset 2, 4, 6, 8, 10, the mean is (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6. The deviations from the mean are #4, #2, 0, 2, and 4.
Squaring these deviations gives 16, 4, 0, 4, and 16. The sum of squared deviations is 40. Using the sample variance formula, divide by n # 1, where n = 5. Thus, sample variance = 40 ÷ 4 = 10. Option A is correct.
Option C, 6.25, does not result from the standard sample variance computation for these values. The important distinction is whether a problem is using sample variance or population variance; the provided answer set and calculation use the sample variance formula. Study Guide references/topics: variance, mean, squared deviations, sample variance.
NEW QUESTION # 92
Independent events: P(A|B) = ?
Answer: A
Explanation:
For independent events, the occurrence of one event does not change the probability of the other. Conditional probability P(A|B) means the probability of A occurring given that B has occurred. If A and B are independent, knowing B occurred gives no new information about A. Therefore, P(A|B) = P(A). This is one of the defining properties of independence. For example, if a coin toss and a die roll are independent, knowing the die landed on 4 does not change the probability that the coin landed heads; it remains 1/2. Option B reverses the event being measured. Option C would imply that A becomes impossible after B, which describes a different situation. Option D is an addition expression and does not represent conditional probability. The correct relationship is that the conditional probability equals the original probability when events are independent. Study Guide references/topics: independent events, conditional probability, probability rules, event relationships.
NEW QUESTION # 93
In a class of 30 students, 18 are enrolled in a math course, and 12 are enrolled in a science course. Among the students enrolled in the science course, 8 are also enrolled in the math course.
What is the probability that a student is enrolled in the math course given that they are enrolled in the science course?
Answer: B
Explanation:
The phrase "given that they are enrolled in the science course" indicates conditional probability. The reference group is not all 30 students; it is only the 12 students enrolled in science. Among those 12 science students, 8 are also enrolled in math. Therefore, the conditional probability is P(math | science) = P(math and science) ÷ P(science), or more directly, 8 ÷ 12. This simplifies to 2/3, which is approximately 0.6667, rounded to .67. Option A, .27, would come from comparing the 8 students to the entire class of 30, but that ignores the given condition. Option B, .40, corresponds to 12/30, the proportion enrolled in science. Option C, .44, is approximately 8/18, which reverses the condition and would represent the probability of science given math.
The correct answer uses the science group as the denominator. References/topics from the Study Guide:
conditional probability, two-way events, joint enrollment, probability ratios.
NEW QUESTION # 94
Sample mean = best point estimate of population mean?
Answer: A
Explanation:
The sample mean is the standard point estimate for the population mean. A point estimate is a single statistic calculated from sample data and used to estimate an unknown population parameter. If the parameter of interest is the population mean #, the corresponding sample statistic is x#. Therefore, the statement is true.
This does not mean the sample mean is guaranteed to equal the population mean exactly; sampling variability can cause sample means to differ from the true population mean. However, the sample mean is unbiased under random sampling and becomes more stable as sample size increases. Confidence intervals expand this idea by placing a margin of error around the sample mean. Option B is incorrect because the sample mean is precisely the conventional point estimator for #. Options C and D impose invalid restrictions. Study Guide references/topics: sample mean, population mean, point estimate, sampling variability.
NEW QUESTION # 95
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