统计代写|AP统计辅导AP统计答疑|One-Sample t-Test for the Mean

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 AP统计学与大学的统计学课程在核心内容上是一致的,只是涉及的深度稍浅,AP统计学主要包含以下四部分内容。 第一部分 如何获取数据,获取数据的方式有哪些呢? 获取数据的方式主要包括普查、抽样调查、观测研究和实验设计等。

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
Predicting predator-prey populations
统计代写|AP统计辅导AP统计答疑|One-Sample t-Test for the Mean

统计代写|AP统计辅导AP统计答疑|One-Sample t-Test for the Mean

  • The following example (Example 2) will be used to outline the essentials of a one-sample t-test. This is an example of a hypothesis test, or test of significance. We use this form of statistical inference when we wish to test a claim that has been made concerning a population. As with confidence intervals, we use sample data to help us make decisions about the population of interest. In other words, we use the sample data to see if there is enough “evidence” to support the claim or to reject it.
  • We will use the same basic three-step method for hypothesis testing that we used for confidence intervals with some minor modifications. Remember, it’s not the numbering of the steps that’s important; it’s what’s in the three steps. Make sure that no matter how you solve inference problems, you include all the essentials.
  • Use the following three-step method when performing a hypothesis test for the mean of a population:
  1. Identify the parameter of interest, choose the appropriate inference procedure, and verify that the assumptions and conditions for that procedure are met. Define any variables of interest. State the appropriate null and alternative hypotheses.
  2. Carry out the inference procedure. Do the math! Calculate the test statistic and find the p-value.
  3. Interpret the results in context of the problem. This is by far the most important part of inference. Be sure that your decision to reject or fail to reject the null hypothesis is done in context of the problem and is based upon the p-value.As noted in step 1, hypothesis testing typically involves a null hypothesis and an alternative hypothesis. It’s important to note that we are not proving anything; we are simply testing to see if there is enough evidence to reject or fail to reject the null hypothesis. The null hypothesis is denoted by $H_{0}$, pronounced $H$-nought. The alternative hypothesis is denoted by $H_{a}$.
  4. The null hypothesis should always include an equality (like $\leq,=$, or $\geq$ ) and must always be written using parameters and not statistics! Of course, you should define any variables you use. For example: $H_{0}: \mu=\mu_{0}$, where $\mu_{0}$ is the hypothesized value.
  5. The alternative hypothesis can be one-sided or two-sided. A one-sided alternative would be either $H_{a}: \mu<\mu_{0}$ or $H_{a}: \mu>\mu_{0}$. A two-sided alternative would be: $H_{a}: \mu \neq \mu_{0}$.

统计代写|AP统计辅导AP统计答疑|Two-Sample t-Interval for the Difference Between Two Means

We are sometimes interested in the difference in two population means, $\mu_{1}-\mu_{2}$. The assumptions and conditions necessary to carry out a confidence interval or test of significance are the same for two-sample means as they are for one-sample means, with the addition that the samples must be independent of one another. You must check the assumptions and conditions for each independent sample.

Remember that the population standard deviations are usually unknown. Recall that when this is the case, we use the sample standard deviation to estimate the population standard deviation. Thus, the standard error (SE) of the sampling distribution is $\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}$.

  • Once we’ve checked the assumptions and conditions, we can proceed to finding the confidence interval for the difference of the means of the two independent groups. We can use $\left(\overline{x_{1}}-\overline{x_{2}}\right) \pm t^{} \times \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}$. The $t^{}$ value depends on the particular level of confidence that you want and on the degrees of freedom $(d f)$.
  • To find the degrees of freedom of a two-sample t-statistic, we can use one of two methods:

Method 1: Use the calculator-generated degrees of freedom. This gives an accurate approximation of the t-distribution based on degrees of freedom from the data. Usually, we obtain non-whole number values using this method. The formula our calculator uses is somewhat complex, and we probably don’t need to be too concerned with how the degrees of freedom are calculated. Make sure, however, that you always state the degrees of freedom that you are using, regardless of what method you use.

Method 2: Use the degrees of freedom equal to the smaller of the two values of $n_{1}-1$ and $n_{2}-2$. This is considered a conservative method.

统计代写|AP统计辅导AP统计答疑|Two-Sample t-Test for the Difference Between

  • The assumptions and conditions for a two-sample hypothesis test for means are the same as the assumptions and conditions for a two-sample t-interval. The null hypothesis for this type of test can be written as: $H_{0}: \mu_{1}=\mu_{2}$ or $H_{0}: \mu_{1}-\mu_{2}=0$

As with a one-sample t-test, the alternative hypothesis can be written with $\neq,<$, or $>$. Once the appropriate assumptions and conditions have been met, we can calculate the two-sample t-statistic as follows:

  • Example 4: Let’s revisit Example 3. Two cross-country coaches from different teams are discussing their boys’ and girls’ teams. One coach believes that male and female cross-country runners in the state of Indiana differ in the number of miles they run, on average, each week. The other coach disagrees. He feels that male and female cross-country athletes run about the same number of miles per week, on average. Is there reason to believe that male and female cross-country runners in Indiana differ in the number of miles they run, on average, each week? Give appropriate statistical evidence to support your answer.

Step 1: To answer the question, we will perform a two-sample t-test. We have already defined our variables and checked the appropriate assumptions and conditions for this type of inference in Example 3 . We state the null and alternative hypotheses:
&H_{0}: \mu_{1}=\mu_{2} \
&H_{0}: \mu_{1} \neq \mu_{2}
Step 2: Since the assumptions and conditions have been met, we can calculate the test statistic as follows:
&t=\frac{\left(\overline{x_{1}}-\overline{x_{2}}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}} \
&t=\frac{(49.3-44.85)-0}{\sqrt{\frac{12.8968}{20}+\frac{12.5626}{20}}} d f=37.9738 \
&t \approx 1.1053 \quad p \approx .2760 \text { (p-value) }
Step 3: With a p-value of approximately $0.2760$, we fail to reject the null hypothesis at any reasonable level of significance. We conclude that male and female cross-country runners do not differ in average weekly mileage.

统计代写|AP统计辅导AP统计答疑|One-Sample t-Test for the Mean


统计代写|AP统计辅导AP统计答疑|One-Sample t-Test for the Mean

  • 以下示例(示例 2)将用于概述单样本 t 检验的要点。这是假设检验或显着性检验的示例。当我们希望测试关于人口的声明时,我们会使用这种形式的统计推断。与置信区间一样,我们使用样本数据来帮助我们对感兴趣的人群做出决定。换句话说,我们使用样本数据来查看是否有足够的“证据”来支持或拒绝它。
  • 我们将使用与置信区间相同的基本三步法进行假设检验,但稍作修改。请记住,重要的不是步骤的编号;这就是三个步骤中的内容。确保无论您如何解决推理问题,都包含所有要点。
  • 对总体均值执行假设检验时,请使用以下三步法:
  1. 确定感兴趣的参数,选择适当的推理过程,并验证该过程的假设和条件是否得到满足。定义任何感兴趣的变量。陈述适当的零假设和替代假设。
  2. 执行推理过程。算一算!计算检验统计量并找到 p 值。
  3. 在问题的上下文中解释结果。这是推理中最重要的部分。确保您拒绝或不拒绝原假设的决定是在问题的背景下做出的,并且基于 p 值。如步骤 1 中所述,假设检验通常涉及原假设和备择假设。需要注意的是,我们没有证明任何东西。我们只是在测试是否有足够的证据来拒绝或无法拒绝原假设。原假设表示为H0, 发音H-没有。备择假设表示为H一种.
  4. 原假设应始终包含等式(如≤,=, 或者≥) 并且必须始终使用参数而不是统计数据来编写!当然,您应该定义您使用的任何变量。例如:H0:μ=μ0, 在哪里μ0是假设值。
  5. 备择假设可以是单面的或双面的。一个单方面的选择是H一种:μ<μ0或者H一种:μ>μ0. 一个双向的替代方案是:H一种:μ≠μ0.

统计代写|AP统计辅导AP统计答疑|Two-Sample t-Interval for the Difference Between Two Means

我们有时对两种总体均值的差异感兴趣,μ1−μ2. 执行置信区间或显着性检验所需的假设和条件对于两样本均值与单样本均值相同,此外,样本必须相互独立。您必须检查每个独立样本的假设和条件。

请记住,总体标准差通常是未知的。回想一下,在这种情况下,我们使用样本标准差来估计总体标准差。因此,抽样分布的标准误差 (SE) 为s12n1+s22n2.

  • 一旦我们检查了假设和条件,我们就可以继续寻找两个独立组的平均值差异的置信区间。我们可以用(X1¯−X2¯)±吨×s12n1+s22n2. 这吨值取决于您想要的特定置信水平和自由度(dF).
  • 要找到双样本 t 统计量的自由度,我们可以使用以下两种方法之一:

方法 1:使用计算器生成的自由度。这给出了基于数据自由度的 t 分布的准确近似值。通常,我们使用这种方法获得非整数值。我们的计算器使用的公式有些复杂,我们可能不需要太关心如何计算自由度。但是,请确保您始终说明您使用的自由度,无论您使用什么方法。

方法 2:使用等于两个值中较小者的自由度n1−1和n2−2. 这被认为是一种保守的方法。

统计代写|AP统计辅导AP统计答疑|Two-Sample t-Test for the Difference Between

  • 均值的双样本假设检验的假设和条件与双样本 t 区间的假设和条件相同。这种检验的原假设可以写成:H0:μ1=μ2或者H0:μ1−μ2=0

与单样本 t 检验一样,备择假设可以写成≠,<, 或者>. 一旦满足了适当的假设和条件,我们就可以计算两样本 t 统计量,如下所示:

  • 示例 4:让我们重温示例 3。来自不同球队的两名越野教练正在讨论他们的男队和女队。一位教练认为,印第安纳州的男性和女性越野跑者平均每周跑的英里数不同。另一位教练不同意。他认为男性和女性越野运动员平均每周跑的英里数大致相同。是否有理由相信印第安纳州的男性和女性越野跑者平均每周跑的英里数不同?提供适当的统计证据来支持你的答案。

第 1 步:为了回答这个问题,我们将执行两个样本的 t 检验。我们已经在示例 3 中定义了变量并检查了此类推理的适当假设和条件。我们陈述零假设和备择假设:
H0:μ1=μ2 H0:μ1≠μ2
步骤 2:由于已经满足假设和条件,我们可以计算测试统计量如下:
吨=(X1¯−X2¯)−(μ1−μ2)s12n1+s22n2 吨=(49.3−44.85)−012.896820+12.562620dF=37.9738 吨≈1.1053p≈.2760 (p 值) 
第 3 步:p 值约为0.2760,我们未能在任何合理的显着性水平上拒绝原假设。我们得出结论,男性和女性越野跑者的平均每周里程没有差异。

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