这部分需要会什么？
- 区分总体、样本和样本统计量
- 理解随机抽样与随机分组各自支持的结论
- 解释误差范围并识别样本偏差
核心概念与公式
总体是希望研究的全体对象，样本是实际观测的一部分。随机抽样有助于把结果推广到所抽取的总体；随机分配处理的设计良好的对照实验，有助于判断处理造成的因果影响。这两种随机化用途不同。
估计值加减误差范围构成报告区间，例如 52% ± 3 个百分点给出 49% 至 55%。相同抽样方式和置信水平下，样本量增大通常减小抽样误差；常见情况下误差范围约与样本量平方根成反比。
做题时怎样使用
- 先确定研究希望推广到谁，以及样本实际上来自谁
- 检查随机抽样、非应答和自愿参与等因素
- 只作研究设计支持的结论，读清误差范围的单位与置信条件
容易混淆的地方
样本很大不能自动消除抽样偏差。观察到相关关系不能直接断言因果；误差区间也不是对总体参数位置的绝对保证。
用一个例题串起方法
知识点教学例题从某校全体学生中随机抽取 400 人，估计支持某方案的比例为 58%，误差范围为 4 个百分点。报告区间是什么，能直接推广到全国学生吗？
展开解题思路 ＋
报告区间为 \(58\%-4\%=54\%\) 至 \(58\%+4\%=62\%\)。样本来自这一所学校，推断范围应对应这所学校的学生，不能直接推广到全国学生。
本知识点题库
先独立作答，再查看答案与解析。
抽样・推断与误差范围挑战
To estimate the proportion of a population that has a certain characteristic, a random sample was selected from the population. Based on the sample, it is estimated that the proportion of the population that has the characteristic is 0.2, with an associated margin of error of 0.08. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of the population that has the characteristic?
原库参考解析
抽样・推断与误差范围进阶
A company studied a random sample of 70 lawns in a small town. The company found that 10 of these lawns contain acidic soil. If there are 910 lawns in the town, which of the following is the best estimate of the number of lawns that contain acidic soil?
原库参考解析
抽样・推断与误差范围挑战
To investigate the effect of magnesium supplementation on the sleep quality of adults, a researcher selected 190 adults at random from a community center to participate in a study. The researcher first measured the sleep efficiency of each participant. Each participant was then randomly assigned to take a magnesium supplement or a placebo each day for 6 weeks. At the end of the 6 weeks, the researcher measured the sleep efficiency of all participants again and found that the magnesium supplement caused statistically significant improved sleep quality for the adults at the community center. What feature of this study allowed the researcher to conclude that the magnesium supplement caused the improved sleep quality?
原库参考解析
抽样・推断与误差范围进阶
A survey was conducted in Madison to estimate the percent of adult residents who support the addition of streetlights at every corner in their town. Based on the survey, it is estimated that 40% of all adult residents of Madison support the addition of streetlights at every corner in their town, with an associated margin of error of 6%. Which of the following is the most appropriate conclusion about all adult residents of Madison?
原库参考解析
抽样・推断与误差范围进阶
A group of 40 employees selected at random from all the employees at a company were surveyed about the number of books they read last year. From the data collected, it was estimated that the mean number of books employees at the company read last year is 6.8, with an associated margin of error of 1.1. Which of the following is the most appropriate conclusion?
原库参考解析
抽样・推断与误差范围进阶
For a survey, a sample of 332 people was selected at random from all registered voters in a county. Based on the survey, it is estimated that 84% of all registered voters in the county are in favor of a new law, with an associated margin of error of 3.94%. Which of the following best describes the plausible values for the percent of all registered voters in the county who are in favor of the new law?
原库参考解析
抽样・推断与误差范围挑战
A science club from a certain high school in Ohio conducted an experiment to study the effects of smell on taste. A sample of 80 students was selected at random from the high school to participate in the experiment. First, the students tasted a pureed food sample while wearing covered goggles and nose plugs. Then, the students tasted the same pureed food sample without nose plugs. 32 students were able to correctly identify the pureed food sample while wearing nose plugs. Which of the following is the largest population to which the results of the experiment can be generalized?
原库参考解析
抽样・推断与误差范围挑战
A researcher is designing a study to investigate the average number of hours students at a high school spend reading per day. The researcher will report an estimated average number of hours students at the high school spend reading per day with an associated margin of error. The researcher is considering using a random sample of either 90 or 180 students from the high school. Which of the following would be the most likely effect of using the larger random sample compared to the smaller random sample?
原库参考解析
抽样・推断与误差范围挑战
A researcher surveyed a random sample of 1,151 people from a certain state to estimate the percentage of people who support the governor’s recent approval of a bill. From the survey, the researcher estimated that 67% of people from the state support the governor’s approval of the bill, with an associated margin of error of 2.37%. The researcher repeated the survey with a random sample of people from the state that was double the original sample size. Assuming the margins of error were calculated in the same way, which of the following is true about the margin of error associated with the estimate from the larger sample?
原库参考解析
抽样・推断与误差范围挑战
A research manager selected 2 random samples of ovens of a certain type to estimate the average amount of time this type of oven takes to preheat to 350 degrees Fahrenheit \( (^{∘}F) \) . The research manager recorded the amount of time, in minutes, each oven takes to preheat to \( 350^{∘}F \). Based on the first sample, the research manager estimated that this type of oven takes an average of 14.2 minutes to preheat to \( 350^{∘}F \), with an associated margin of error of 1 minute. Based on the second sample, the research manager estimated that this type of oven takes an average of 14.4 minutes to preheat to , with an associated margin of error of 2.2 minutes. Assuming the margins of error were calculated the same way, which of the following best explains why the first sample obtained a smaller margin of error than the second sample?
原库参考解析
抽样・推断与误差范围挑战
A researcher selected 25 guinea pigs from one habitat and 19 wild cavies from another habitat at random in Venezuela. A field of a specific size within each habitat was divided into equally-sized virtual squares, and the animals were allowed to wander in the field for a fixed amount of time. To observe their behavior, the researcher counted the number of times the guinea pigs and wild cavies crossed the virtual squares in their field during early adolescence and again during late adolescence. The researcher found that in early adolescence, the number of virtual squares the wild cavies crossed was significantly more than the number of virtual squares the guinea pigs crossed. The researcher also found that the number of virtual squares crossed by both the wild cavies and the guinea pigs decreased from early adolescence to late adolescence.
Based on the researcher's findings, which of the following statements is an appropriate conclusion that can be drawn from this study?
原库参考解析
抽样・推断与误差范围挑战
An inspector checked a sample of 370 scales selected at random from a population of 8,000 scales to estimate what percentage of the 8,000 scales were inaccurate. From this sample, the inspector estimated that 9% of the 8,000 scales were inaccurate, with an associated margin of error of 2.9%. Which of the following is the most appropriate conclusion?
原库参考解析
