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| Section | Objectives |
|---|---|
| Statistical Inference | - Estimation and Confidence Intervals
|
| Descriptive Statistics | - Single Variable Data Analysis
|
| Probability Theory | - Probability Distributions
|
| Regression and Modeling | - Linear Relationships
|
>> Clearer Applied-Probability-and-Statistics Explanation <<
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NEW QUESTION # 65
95% CI = 50 ± 2. SE = ?
Answer: A
Explanation:
A confidence interval has the structure estimate ± margin of error. The margin of error is calculated as critical value × standard error. For a 95% confidence interval using the normal approximation, the critical value is approximately 1.96, often rounded to 2 in introductory settings. The interval is given as 50 ± 2, so the margin of error is 2. Using the approximate 95% critical value of 2, we solve 2 = 2 × SE, giving SE = 1. More exactly, using 1.96 gives SE = 2/1.96 # 1.02, which rounds to 1. Option B confuses the margin of error with the standard error. Option C is too large, and option D would produce a margin of error near 1 under a 95% critical value. The correct standard error is approximately 1. Study Guide references/topics: confidence intervals, standard error, critical value, margin of error.
NEW QUESTION # 66
Best chart for categorical data?
Answer: B
Explanation:
A bar chart is the best display for categorical data because it shows the frequency or relative frequency of each category using separate bars. Categorical variables classify observations into groups, such as favorite color, type of transportation, political affiliation, or product category. A bar chart allows the viewer to compare category counts directly by bar height. A histogram is not appropriate because it is used for quantitative data grouped into numerical intervals. A scatterplot is used to examine the relationship between two quantitative variables, such as height and weight. A boxplot summarizes a quantitative distribution using quartiles, median, spread, and outliers. Since the question asks specifically about categorical data, the display must preserve category labels rather than numerical intervals or continuous measurement. Therefore, the bar chart is the correct choice. Study Guide references/topics: categorical data, bar charts, frequency displays, descriptive statistics.
NEW QUESTION # 67
Dataset: 5, 7, 7, 10. Standard deviation?
Answer: B
Explanation:
The standard deviation measures the typical distance of data values from the mean. For the dataset 5, 7, 7, 10, the mean is (5 + 7 + 7 + 10) ÷ 4 = 29 ÷ 4 = 7.25. The deviations from the mean are #2.25, #0.25, #0.25, and
2.75. Squaring these deviations gives 5.0625, 0.0625, 0.0625, and 7.5625. The sum of squared deviations is
12.75. If treated as a sample, the variance is 12.75 ÷ 3 = 4.25, and the sample standard deviation is #4.25 #
2.06. If rounded using the answer set provided, the closest available value is 2.16. The key conceptual point is that standard deviation cannot be 0 unless all values are identical, and it is not as large as 3 because the observations are relatively close to the mean. Study Guide references/topics: mean, variance, standard deviation, spread of data.
NEW QUESTION # 68
Sum of probabilities in sample space = ?
Answer: B
Explanation:
The probabilities of all outcomes in a complete sample space must sum to 1. A sample space contains every possible outcome of a probability experiment, and one of those outcomes must occur. For example, when rolling a fair six-sided die, the outcomes are 1, 2, 3, 4, 5, and 6. Each has probability 1/6, and the sum is 1/6 +
1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1. A total probability of 0 would mean no outcome can occur, which is impossible for a valid experiment. A total greater than 1 violates probability rules because probabilities cannot exceed certainty. "Cannot exceed 2" is too broad and mathematically invalid, since the exact total must equal
1. This principle is foundational for checking probability distributions and validating whether assigned probabilities are coherent. Study Guide references/topics: sample space, probability axioms, total probability, theoretical probability.
NEW QUESTION # 69
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 # 70
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