11.5 千方统计

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  Suppose you wanted to evaluate a recent statistic stating that iOS represents 32% and Android 51% of active smart phones. You would like to know if the statistic actually reflects the distribution of phones among your friends. How could you evaluate the data you collect to see if it supports this hypothesis?
  ::假设你想要评估最近的统计, 指出iOS占活跃智能手机的32%, Android 51%。 您想知道该统计是否实际反映了您朋友之间手机的分布情况。 您如何评估您收集的数据,看它是否支持这一假设?

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  Look to the end of the lesson for the answer.
  ::寻找教训的结尾 以找到答案。

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  Chi-Squared Statistic 
  ::千位统计员

  

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  The Greek letter “chi”, written as $\begin{array}{r}\chi \end{array}$ , is the symbol used to identify a chi-square statistic , which we will use here to evaluate how well a set of observed data fits a corresponding expected set.
  ::希腊字母“chi”(chi)是用来确定奇平方统计的符号,我们在这里将使用该符号来评估一组观察到的数据与预期的相应数据集的相匹配程度。

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  Conducting a  Chi-Square test is much like conducting a Z-test or T-test as we did in Chapter 10. We will follow the same basic series of steps and compare a calculated value to a chart to evaluate the probability of getting the results we have if the null hypothesis is true, just as we did with the Z and F tests. Additionally, as was the case with the F-testing, we will be evaluating the number of degrees of freedom , and choosing values from a chart based on the number.
  ::进行Chi-Square测试就像我们在第10章中所做的那样进行Z-测试或T测试,我们将遵循同样的一系列基本步骤,并将计算得出的数值与图表进行比较,以评价如果无效假设属实的话获得结果的可能性,正如我们在Z和F测试中所做的那样。此外,与F测试的情况一样,我们将评估自由度的数量,并根据数字从图表中选择数值。

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  The primary difference between a Chi-Square test and the tests we have work with before is that previous tests have all been primarily dedicated to comparing single parameters, whereas Chi-Square tests are used to determine if two random variables are independent or related and so deal with multiple values for each variable . Additionally, the Chi-Square statistic is useful for looking at categorical data rather than quantitative data .
  ::Chi-Square测试和我们以前研究过的测试之间的主要区别是,以前的测试都主要用于比较单一参数,而Chi-Square测试则用来确定两个随机变量是否独立或相关,从而处理每个变量的多重值。 此外,Chi-Square统计数据对于查看绝对数据而不是量化数据很有用。

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  The Chi-Square statistic is actually pretty straightforward to calculate:
  ::Chi-Square统计其实是相当直截了当的计算:

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  $$\begin{array}{r}{\chi }^2=\sum \frac{(observed-expected{)}^2}{expected}\end{array}$$

  
  ::2(观测到-预期)2 预期

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  Determining the Validity of a Study 
  ::确定一项研究的有效性

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  The American Pet Products Association conducted a survey in 2011 and determined that 60% of dog owners have only one dog, 28% have two dogs, and 12% have three or more. Supposing that you have decided to conduct your own survey and have collected the data below, determine whether your data supports the results of the APPA study. Use a significance level of 0.05.
  ::美国宠物产品协会在2011年进行了一项调查,确定60%的狗主只有一只狗,28%有两只狗,12%有三只或更多只三只或更多只狗。 假如你决定自己进行调查并收集以下数据,则决定你的数据是否支持APPA研究的结果。使用0.05的临界值。

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  Data: Out of 129 dog owners, 73 had one dog and 38 had two dogs.
  ::数据:129只狗的拥有者中,73只有一只,38只有两只。

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  • Step 1: Clearly state the
    ::第1步:清楚说明

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  $\begin{array}{r}{H}_0\end{array}$ : The survey agrees with the sample .
  ::H0:调查与样本一致。

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  $\begin{array}{r}{H}_1\end{array}$ : The survey does not agree with the sample .
  ::H1:调查与样本不符。

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  • Step 2: Identify an appropriate test and significance level
    ::第2步:确定适当的测试和重要性水平

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  Since we are comparing two sets of data, and not just a single value, a Chi-Square test is appropriate. In the absence of a stated significance level in the problem, we assume the default 0.05.
  ::由于我们比较的是两套数据,而不仅仅是单一值,因此,智方测试是恰当的。 在问题没有表明意义的情况下,我们假定默认值为0.05。

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  • Step 3: Analyze sample data
    ::第3步:分析抽样数据

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  Create a table to organize data and compare the observed data to the expected data:
  ::创建一个表格来组织数据,并将观测到的数据与预期数据进行比较:

  	播放段落 One Dog ::一条狗	播放段落 Two Dogs ::两只狗	播放段落 3+ Dogs ::3+狗狗	播放段落 TOTAL ::共计共计
  播放段落 Observed ::观察观察	73	38	18	129
  播放段落 Expected ::预期预期预期预期预期预期预期预计的预期预计预计的预期预计的预期预计的预期的预计的预期的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数的预计的预计的预计的预计的预计数的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的数的预计的数的、的、的、的、的、的、的、的、的、、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的

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  To identify the expected values, multiply the expected % by the total number observed:
  ::为了确定预期值,将预期百分比乘以观察到的总数:

  	播放段落 One Dog ::一条狗	播放段落 Two Dogs ::两只狗	播放段落 3+ Dogs ::3+狗狗	播放段落 TOTAL ::共计共计
  播放段落 Observed ::观察观察	$\begin{array}{r}73\end{array}$	$\begin{array}{r}38\end{array}$	$\begin{array}{r}18\end{array}$	$\begin{array}{r}129\end{array}$
  播放段落 Expected ::预期预期预期预期预期预期预期预计的预期预计预计的预期预计的预期预计的预期的预计的预期的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数的预计的预计的预计的预计的预计数的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的数的预计的数的、的、的、的、的、的、的、的、的、、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的	$\begin{array}{r}0.60\times 129=77.4\end{array}$	$\begin{array}{r}0.28\times 129=36.1\end{array}$	$\begin{array}{r}0.12\times 129=15.5\end{array}$	$\begin{array}{r}129\end{array}$

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  To calculate our chi-square statistic, we need to sum the squared difference between each observed and divided by the expected value:
  ::为了计算我们的奇平方统计, 我们需要将观察到的每种值之间的正方差和除以预期值的正方差相加:

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  $$\begin{array}{rl}{\chi }^2& =\sum \frac{(observed-expected{)}^2}{expected}\\ {\chi }^2& =\frac{(73-77.4{)}^2}{77.4}+\frac{(38-36.1{)}^2}{36.1}+\frac{(18-15.5{)}^2}{15.5}\\ {\chi }^2& =\frac{(-4.4{)}^2}{77.4}+\frac{(1.9{)}^2}{36.1}+\frac{(2.5{)}^2}{15.5}\\ {\chi }^2& =\frac{19.36}{77.4}+\frac{3.61}{36.1}+\frac{6.25}{15.5}\\ {\chi }^2& =0.2501+0.1000+0.4032\\ {\chi }^2& =0.7533\end{array}$$

  
  ::+(38-36.1)236.1+(18-155)215.5+2(44)277.4+(1.9236.1+(2.5)215.5=19.367.4+3.6136.1+6.251.55=0.2501+0.1000+0.40322=0.753

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  Now that we have our chi-square statistic, we need to compare it to the chi-square value for the significance level 0.05. We can use a reference table such as the one below, or a . Just as with the T -tests in Chapter 10, we will need to know the  degrees of freedom , which equal the number of observed category values minus one. In this case, there are three category values: one dog, two dogs, and three or more dogs. The degrees for freedom, therefore, are  $\begin{array}{r}3-1=2\end{array}$ .
  ::现在,我们有了我们的奇夸统计, 我们需要将它与 0.05 级的奇夸值进行比较。 我们可以使用下面的参考表格, 或者一个。 就像第10章的T类测试一样, 我们需要知道自由度, 与观察类别值的数相等。 在这种情况下, 有三个类别值: 一只狗, 两只狗, 三只或更多的狗。 因此, 自由度是 3 - 1 = 2 。

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  Using the calculator or the table, we find that the critical value for a 0.05 significance level with  $\begin{array}{r}df=2\end{array}$ is 5.9915. That means that 95 times out of 100, a survey that agrees with a sample will have a $\begin{array}{r}{\chi }^2\end{array}$ critical value of 5.9915 or less. If our chi-square value is greater than 5.9915, then the measurements we took only occur 5 or fewer times out of 100, or the null hypothesis is incorrect. Our chi-square statistic is only 0.7533 , so we will  not reject the null hypothesis.
  ::使用计算器或表格,我们发现 df=2 的 0.05 意义水平的临界值为 5. 9915。 这意味着在100次中,与样本一致的调查的95次将达到 +2 关键值5. 9915 或 低于 5. 9915 。 如果我们的基方值大于 5. 9915 , 那么我们所测量的值在100次中仅发生5次或更少次, 或者无效假设是不正确的。 我们的基方统计只有 0.7533 , 所以我们不会拒绝无效假设 。

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  • Step 4: Interpret the results
    ::第4步:解释结果

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  Since our chi-square statistic was less than the critical value, we do not reject the null hypothesis, and we can say that our survey data does support the data from the APPA.
  ::由于我们的 " 奇平方 " 统计数据低于关键价值,我们不拒绝无效假设,我们可以说,我们的调查数据确实支持了来自APPA的数据。

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  Real-World Application: Car Insurance 
  ::现实世界应用:汽车保险

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  Rachel told Eric that the reason her car insurance is less expensive is that female drivers get in fewer accidents than male drivers. Specifically, she says that male drivers are held responsible in 65% of accidents involving drivers under 23.
  ::Rachel告诉Eric,她的汽车保险费用较低的原因是女性驾驶员的事故比男性驾驶员少。 具体地说,她说65%的涉及23岁以下驾驶员的事故都由男性驾驶员负责。

  

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  If Eric does some research of his own and discovers that 46 out of the 85 accidents he investigates involve male drivers, does his data support Rachel’s hypothesis?
  ::如果Eric自己做一些研究, 发现85起事故中有46起涉及男性驾驶员, 他的数据是否支持Rachel的假设?

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  • Step 1: Clearly state the null and alternative hypotheses
    ::第1步:明确说明无效和替代假设

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  $\begin{array}{r}{H}_0\end{array}$ : The survey agrees with the sample .
  ::H0:调查与样本一致。

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  $\begin{array}{r}{H}_1\end{array}$ : The survey does not agree with the sample .
  ::H1:调查与样本不符。

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  • Step 2: Identify an appropriate test and significance level
    ::第2步:确定适当的测试和重要性水平

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  Since we are comparing two sets of data, and not just a single value, a Chi-Square test is appropriate. In the absence of a stated significance level in the problem, we assume the default 0.05.
  ::由于我们比较的是两套数据,而不仅仅是单一值,因此,智方测试是恰当的。 在问题没有表明意义的情况下,我们假定默认值为0.05。

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  • Step 3: Analyze sample data
    ::第3步:分析抽样数据

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  Create a table to organize data and compare the observed data to the expected data:
  ::创建一个表格来组织数据,并将观测到的数据与预期数据进行比较:

  	播放段落 Male Drivers ::男司机	播放段落 Female Drivers ::女司机	播放段落 TOTAL ::共计共计
  播放段落 Observed ::观察观察	46	39	85
  播放段落 Expected ::预期预期预期预期预期预期预期预计的预期预计预计的预期预计的预期预计的预期的预计的预期的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数的预计的预计的预计的预计的预计数的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的数的预计的数的、的、的、的、的、的、的、的、的、、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的

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   To identify the expected values, multiply the expected % by the total number observed:
  ::为了确定预期值,将预期百分比乘以观察到的总数:

  	播放段落 Male Drivers ::男司机	播放段落 Female Drivers ::女司机	播放段落 TOTAL ::共计共计
  播放段落 Observed ::观察观察	$\begin{array}{r}46\end{array}$	$\begin{array}{r}39\end{array}$	$\begin{array}{r}85\end{array}$
  播放段落 Expected ::预期预期预期预期预期预期预期预计的预期预计预计的预期预计的预期预计的预期的预计的预期的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数的预计的预计的预计的预计的预计数的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的数的预计的数的、的、的、的、的、的、的、的、的、、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的	$\begin{array}{r}0.65\times 85=55.25\end{array}$	$\begin{array}{r}0.35\times 85=29.75\end{array}$	$\begin{array}{r}85\end{array}$

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  To calculate our chi-square statistic, we need to sum the squared differences between each observed and expected value divided by the expected value:
  ::为了计算我们的奇平方统计, 我们需要将每个观察值和预期值之间的正方差和预期值除以预期值的正方差相加:

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  $$\begin{array}{rl}{\chi }^2& =\sum \frac{(observed-expected{)}^2}{expected}\\ {\chi }^2& =\frac{(46-55.25{)}^2}{55.25}+\frac{(39-29.75{)}^2}{29.75}\\ {\chi }^2& =\frac{(-9.25{)}^2}{55.25}+\frac{(9.25{)}^2}{29.75}\\ {\chi }^2& =\frac{85.5625}{55.25}+\frac{85.5625}{29.75}\\ {\chi }^2& =1.5486+2.8760\\ {\chi }^2& =4.4246\end{array}$$

  
  ::2(观察-预期) 2=(46-55.25.25) 2555.25+(39-29.75)229.752=(9.25) 255.25+(9.25) 255.25+(9.25)229.752=85.5625.25+852.55.250.752=1.5486+2.8760)2=4.4246

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  Now that we have our chi-square statistic, we need to compare it to the chi-square critical value for 0.05 with one degree of freedom , since we have two categories. Using the , we find the critical value to be 3.8414. The critical value indicates that only 0.05, or 5%, of values would be as high as 3.8414. If the  $\begin{array}{r}{\chi }^2\end{array}$ of our data is greater than 3.8414, then fewer than 5 times out of 100 would we expect to get that result if the null hypothesis is true.
  ::现在,我们有了我们的“奇平方”统计,我们需要把它与0.05的“奇平方”关键值作一等自由比较,因为我们有两类自由。使用“奇平方”数据,我们发现关键值为3.8414。关键值表明,只有0.05或5%的值将高达3.8414。如果我们的数据超过3.8414,那么,如果无效假设是真实的,我们预期100中只有不到5倍的结果。

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  • Step 4: Interpret your results
    ::第4步:解释结果

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  Our calculated data value of  $\begin{array}{r}{\chi }^2=4.4246\end{array}$ is greater than the 0.05 significance level critical value of 3.8141, so we reject the null hypothesis. The data that Eric observed does not support the distribution that Rachel claimed.
  ::我们计算的数据值为%2=4.4246, 大于0.05 意义临界值3. 8141, 因此我们拒绝无效假设。 Eric观察到的数据并不支持Rachel声称的分布 。

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  Real-World Application: Car Magazine 
  ::真实世界应用程序:汽车杂志

  

  

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  The online car magazine “ Camaro5.com ” claims that 51% of Ford Mustang or Chevy Camaro owners own Camaros. Ellen is a Mustang lover and decides to do some research. If Ellen collects the data below, does her data support the magazine’s claim?
  ::在线汽车杂志《Camaro5.com 》 ( Camaro5.com)声称,福特野马或Chevy Camaro业主的51%拥有卡马罗斯。 埃伦是野马的情人,决定做一些研究。 如果埃伦收集以下数据,她的数据是否支持该杂志的主张?

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  Data: Mustang owners: 28, Camaro owners: 34
  ::数据:野马所有者:28;卡马罗所有者:34

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  • Step 1: Clearly state the null and alternative hypotheses
    ::第1步:明确说明无效和替代假设

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  $\begin{array}{r}{H}_0\end{array}$ : The survey agrees with the sample .
  ::H0:调查与样本一致。

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  $\begin{array}{r}{H}_1\end{array}$ : The survey does not agree with the sample .
  ::H1:调查与样本不符。

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  • Step 2: Identify an appropriate test and significance level
    ::第2步:确定适当的测试和重要性水平

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  Since we are comparing two sets of data, and not just a single value, a Chi-Square test is appropriate. In the absence of a stated significance level in the problem, we assume the default 0.05.
  ::由于我们比较的是两套数据,而不仅仅是单一值,因此,智方测试是恰当的。 在问题没有表明意义的情况下,我们假定默认值为0.05。

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  • Step 3: Analyze sample data
    ::第3步:分析抽样数据

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  We will start by creating a table to organize our data:
  ::我们首先将建立一个表格来组织我们的数据:

  	播放段落 Mustang ::野马	播放段落 Camaro ::卡马罗	播放段落 TOTAL ::共计共计
  播放段落 Observed ::观察观察	$\begin{array}{r}28\end{array}$	$\begin{array}{r}34\end{array}$	$\begin{array}{r}62\end{array}$
  播放段落 Expected ::预期预期预期预期预期预期预期预计的预期预计预计的预期预计的预期预计的预期的预计的预期的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计数的预计的预计的预计的预计的预计数的预计数数的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的预计的数的预计的数的、的、的、的、的、的、的、的、的、、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的、的	$\begin{array}{r}0.49\times 62=30.4\end{array}$	$\begin{array}{r}0.51\times 62=31.6\end{array}$	$\begin{array}{r}62\end{array}$

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  Now we can calculate our chi statistic:
  ::现在我们可以计算出我们的奇数了:

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  $$\begin{array}{rl}{\chi }^2& =\sum \frac{(observed-expected{)}^2}{expected}\\ {\chi }^2& =\frac{(28-30.4{)}^2}{30.4}+\frac{(34-31.6{)}^2}{31.6}\\ {\chi }^2& =\frac{(-2.4{)}^2}{30.4}+\frac{(2.4{)}^2}{31.6}\\ {\chi }^2& =.3718\end{array}$$

  
  ::2(观察-预期) 2=(28-304)230.4+(34-31.6)231.6=(2.4)230.4+(2.4)231.6=2=3718

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  The chi-square critical value for  $\begin{array}{r}df=1\end{array}$ and a significance level of 0.05 is 3.8414 (the same as in Example B).
  ::df=1的 基方关键值为 3.8414(与例B相同),重量值为0.05。

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  • Step 4: Interpret your results
    ::第4步:解释结果

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  Our calculated data value of  $\begin{array}{r}{\chi }^2=0.3718\end{array}$ is significantly less than the 0.05 significance level critical value of 3.8141, so we fail to reject the null hypothesis. This means that, unfortunately for Ellen, her research did not allow her to deny the claim that Camaros are more popular.
  ::我们计算的数据值为%2=0.3718,远远低于3.8141的0.05重要临界值,因此我们没有拒绝无效假设。 这就意味着,对埃伦来说,不幸的是,她的研究不允许她否认卡马罗斯更受欢迎的说法。

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  Earlier Problem Revisited
  ::重审先前的问题

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  Suppose you wanted to evaluate a recent statistic stating that iOS represents 32% and Android 51% of active smart phones. You would like to know if the statistic actually reflects the distribution of phones among your friends. How could you evaluate the data you collect to see if it supports this hypothesis?
  ::假设你想要评估最近的统计, 指出iOS占活跃智能手机的32%, Android 51%。 您想知道该统计是否实际反映了您朋友之间手机的分布情况。 您如何评估您收集的数据,看它是否支持这一假设?

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  You could evaluate the hypothesis by collecting data from a SRS of cell phone owners and using a chi-square test to see if your data supports the hypothesis.
  ::您可以通过收集手机所有者SRS的数据来评估假设, 并使用“ 奇夸” 测试来查看您的数据是否支持这一假设。

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  Examples
  ::实例

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  Examples  1-5 refer to the following data:
  ::实例1-5提到以下数据:

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  Tuscany claims that 70% of dog or cat owners own a dog, and 30% own a cat. Sayber decides to test her claim and learns that 23 of the 40 people he asks own dogs, and 17 own cats.
  ::托斯卡尼声称,70%的狗或猫主拥有一只狗,30%的猫拥有一只猫。 赛伯决定测试她的主张,并了解到,在他要求养狗的40人中有23人和17只猫。

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  Example 1
  ::例1

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  What kind of test could you use to see if Sayber’s data supports Tuscany’s claim?
  ::Sayber的数据是否支持托斯卡尼的主张?

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  A chi-square test would be appropriate
  ::比较适合做个鸡翅测试

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  Example 2
  ::例2

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  What would be the null and alternative hypotheses?
  ::什么是无效的和替代性的假设?

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  The null hypothesis,  $\begin{array}{r}{H}_0\end{array}$ , would be that the research does support the hypothesis, the alternative hypothesis would be that is does not.
  ::否定的假设H0是研究确实支持这一假设,而替代假设则不是。

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  Example 3
  ::例3

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  What would be the expected values of dog and cat owners?
  ::狗和猫的预期价值是什么?

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  The expected number of dog owners, according to Tuscany's claim, would be 70% of the 40 people that Sayber polled, or 28 dog owners. The expected number of cat owners would be 30% of the 40 people polled, or 12.
  ::根据托斯卡尼的主张,预计养狗者的人数将是Sayber所投票的40人中的70%,或者28名养狗者。 预计养猫者的人数将是40人中的30%,或者12人。

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  Example 4
  ::例4

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  What is the chi-square statistic of the observed data?
  ::观测到的数据的 " 奇平方 " 统计是什么?

  播放段落

  The  $\begin{array}{r}{\chi }^2\end{array}$  statistic is the sum of the squared differences between the observed and expected values, divided by the expected values:
  ::=%2 统计数据是观测值和预期值之间的平差之和,除以预期值:

  $$\begin{array}{rl}{\chi }^2& =\frac{(23-28{)}^2}{28}+\frac{(17-12{)}^2}{12}\\ & =\frac{25}{28}+\frac{25}{12}\\ {\chi }^2& =2.9762\end{array}$$

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  Example 5
  ::例5

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  Assuming a 0.1 significance level, does Sayber’s data support Tuscany’s claim?
  ::假设0.1个重要水平,Sayber的数据是否支持托斯卡尼的索赔?

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  The critical value of chi-squared for 1 degree of freedom at a significance level of 0.1 is 2.705. Since the chi-square statistic we calculated is 2.9762, and is therefore more extreme than the critical value, we may reject the hypothesis , and say that Sayber’s data does not support Tuscany’s claim.
  ::在0.1的临界值水平上,一等自由的基数的临界值为2.705。 由于我们计算出的基数为2.9762,因此比关键值更极端,我们可以拒绝假说,并说Sayber的数据不能支持托斯卡尼的主张。

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  Review
  ::回顾

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  Questions 1-5 refer to the following:
  ::问题1-5如下:

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  Evan claims that 15% of computer gamers have played “Team Fortress 2”, and 35% have played “World of Warcraft”. Evan’s brother is skeptical of those figures and decides to do some research. He discovers that 60 of the 200 computer gamers he polls have played “Team Fortress 2”, and 90 have played “World of Warcraft”.
  ::埃文声称,15%的计算机游戏家玩了“Fortress 2”游戏,35%玩了“World of World of Worldcraft ” 。 埃文的兄弟对这些数字持怀疑态度,决定做一些研究。 他发现,他所调查的200个计算机游戏家中,有60个玩了“Team Fortres 2 ” , 90个玩了“World of Worldcraft ”游戏。

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  1. Create a table to organize the data and prepare for hypothesis testing.
  ::1. 创建一个表格,以组织数据和准备进行假设测试。

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  2. What sort of test would be appropriate to determine if the observed data supports Evan’s claim?
  ::2. 在确定观察到的数据是否支持Evan的索赔要求时,应当采用何种检验标准?

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  3. What would be  $\begin{array}{r}{H}_0\end{array}$ and $\begin{array}{r}{H}_1\end{array}$ ?
  ::3. 什么是H0andH1?

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  4. What would be the  $\begin{array}{r}{\chi }^2\end{array}$ statistic for the observed data?
  ::4. 观察到的数据的%2统计数据是什么?

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  5. How many degrees of freedom are there in the variable “played game”?
  ::5. 变数 " 游戏游戏 " 中有多少自由度?

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  6. Assuming a significance level of 0.05, what is the  $\begin{array}{r}{\chi }^2\end{array}$ critical value?
  ::6. 假设0.05的临界值是0.05, 关键值是多少?

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  7. Does the observed data support Evan’s claim? Explain your findings.
  ::7. 观察到的数据是否支持Evan的索赔? 解释你的调查结果。

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  Questions 8-15 refer to the following:
  ::问题8-15涉及以下方面:

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  Mack claims that 84% of street racers drive import cars, and 16% drive domestic muscle cars. Abbi likes domestic cars and thinks Mack is overstating the percentage of imports, so she does some research of her own and finds that 57 of the street racers she interviewed drive imports, and 31 drive American muscle.
  ::麦克声称,84%的街头赛车手驾驶的是进口汽车,16%的街头赛车手驾驶的是家用肌肉汽车。 阿比喜欢家用汽车,认为麦克夸大了进口比例,因此她自己做了一些研究,发现她采访的街头赛车手中有57人驾驶的是进口汽车,31人驾驶的是美国肌肉。

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  8. Create a table to organize the data and prepare for hypothesis testing.
  ::8. 建立一个表格,以组织数据和准备进行假设测试。

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  9. What sort of test would be appropriate to determine if the observed data supports Mack’s claim?
  ::9. 在确定观察到的数据是否支持Mack的索赔要求时,应当采用何种检验标准?

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  10. What would be  $\begin{array}{r}{H}_0\end{array}$ and $\begin{array}{r}{H}_1\end{array}$ ?
  ::10. 什么是H0andH1?

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  11. What would be the  $\begin{array}{r}{\chi }^2\end{array}$ statistic for the observed data?
  ::11. 观察到的数据的%2统计数据是什么?

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  12. How many degrees of freedom are there in the variable “played game”?
  ::12. 变数 " 游戏游戏 " 中有多少自由度?

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  13. Assuming a significance level of 0.10, what is the $\begin{array}{r}{\chi }^2\end{array}$ critical value?
  ::13. 假设0.10的临界值为0.10, 关键值是多少?

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  14. Does the data indicate that Abbi should reject, or fail to reject  $\begin{array}{r}{H}_0\end{array}$ ?
  ::14. 数据是否表明Abbi应拒绝H0,还是没有拒绝H0?

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  15. Interpret your results.
  ::15. 解释你的结果。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。