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| Section | Objectives |
|---|---|
| Probability | - Probability rules
|
| Probability Distributions | - Discrete distributions
|
| Regression and Correlation | - Relationship analysis
|
| Statistical Inference | - Hypothesis testing
|
| Descriptive Statistics | - Data visualization
|
>> Applied-Probability-and-Statistics問題数 <<
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質問 # 71
Given a normal distribution with a mean of 40 and a standard deviation of 6, what is the range that contains approximately 95% of the data?
正解：C
解説：
For a normal distribution, the empirical rule states that approximately 95% of the data fall within two standard deviations of the mean. The mean is 40, and the standard deviation is 6. Two standard deviations equal 2 × 6
= 12. To construct the 95% interval, subtract and add 12 from the mean: 40 # 12 = 28 and 40 + 12 = 52.
Therefore, the range containing approximately 95% of the data is 28 to 52. Option D, 34 to 46, represents one standard deviation from the mean and corresponds to about 68% of the data. Option B, 22 to 58, represents three standard deviations from the mean and corresponds to approximately 99.7% of the data. Option C is not based on the stated standard deviation. The correct interval must be symmetric around the mean and extend exactly two standard deviations in each direction. References/topics from the Study Guide: normal distribution, empirical rule, mean, standard deviation.
質問 # 72
Standard deviation squared = ?
正解：D
解説：
Variance and standard deviation are directly connected measures of spread. Variance is calculated by averaging squared deviations from the mean, while standard deviation is the square root of variance.
Therefore, reversing that relationship, standard deviation squared equals variance. Symbolically, if the standard deviation is s, then the sample variance is s². This relationship matters because variance is expressed in squared units, while standard deviation is expressed in the original units of measurement. For example, if test-score standard deviation is 4 points, the variance is 4² = 16 square points. The mean and median are measures of center, not squared spread. The standard error measures the variability of a sample statistic, usually the sample mean, across repeated samples. The correct answer is variance because it is the squared form of standard deviation. Study Guide references/topics: standard deviation, variance, measures of spread, descriptive statistics.
質問 # 73
Standard deviation measures:
正解：B
解説：
Standard deviation measures the spread, variability, or dispersion of data values around the mean. A small standard deviation indicates that the data values are clustered closely around the mean, while a large standard deviation indicates that the values are more widely dispersed. It is calculated from deviations from the mean, squared deviations, variance, and then the square root of variance. Standard deviation is not a measure of central tendency; measures of central tendency include the mean, median, and mode. It is also not a probability, although it is used in probability distributions such as the normal distribution. Frequency refers to how often a value or category occurs, which is summarized with tables, bar charts, histograms, or dot plots.
The standard deviation is essential because it quantifies how consistent or variable a dataset is. Study Guide references/topics: standard deviation, variance, spread, descriptive statistics.
質問 # 74
A t-distribution is used when:
正解：C
解説：
A t-distribution is used for inference about a population mean when the population standard deviation # is unknown, especially with a small sample. In that situation, the sample standard deviation s must estimate #, creating extra uncertainty. The t-distribution accounts for this added uncertainty with heavier tails than the standard normal distribution. As sample size increases, the t-distribution becomes closer to the normal distribution. Option B describes a z-procedure setting, where # is known and the normal distribution may be used. Option C is incorrect because categorical data are usually analyzed with proportions or chi-square procedures, not t-distributions for means. Option D is incomplete: t-procedures typically require quantitative continuous or approximately continuous data, but the decisive condition in the answer set is unknown # with a small sample. Study Guide references/topics: t-distribution, unknown standard deviation, small samples, inference for means.
質問 # 75
Type I error = ?
正解：B
解説：
A Type I error occurs when a statistical test rejects the null hypothesis even though the null hypothesis is actually true. This is often described as a false positive. In practical terms, the test concludes that there is an effect, difference, or relationship when none truly exists under the null condition. The probability of committing a Type I error is denoted by #, the significance level, often set at 0.05. Option B describes a Type II error, where the test fails to reject a false null hypothesis. Option C is not an error at all. Option D is too general; Type I error is not a data-entry issue but a formal decision error in hypothesis testing. The defining phrase is "reject H# when true." Study Guide references/topics: hypothesis testing, Type I error, null hypothesis, significance level.
質問 # 76
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