AP Statistics Curriculum 2007 Hypothesis L Mean

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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Testing a Claim about a Mean: Large Samples==
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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Testing a Claim About a Mean: Large Samples==
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We already saw [[AP_Statistics_Curriculum_2007_Estim_L_Mean | how to construct point and interval estimates for the population mean in the large sample case]]. Now, we show how to do hypothesis testing about the mean for large sample-sizes.  
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We already saw [[AP_Statistics_Curriculum_2007_Estim_L_Mean | how to construct point and interval estimates for the population mean in the large sample case]]. Now, we show how to do hypothesis testing of the mean for large sample-sizes.  
===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]===
===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]===
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: and <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] distribution at <math>{\alpha\over 2}</math>.
: and <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] distribution at <math>{\alpha\over 2}</math>.
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=== Hypothesis Testing about a Mean: Large Samples===
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=== Hypothesis Testing About a Mean: Large Samples===
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* Null Hypothesis: <math>H_o: \mu=\mu_o</math> (e.g., 0)
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* Null Hypothesis: <math>H_o: \mu=\mu_o</math> (e.g., <math>\mu_o=0</math>)
* Alternative Research Hypotheses:
* Alternative Research Hypotheses:
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** One sided (uni-directional): <math>H_1: \mu >\mu_o</math>, or <math>H_o: \mu<\mu_o</math>
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** One sided (uni-directional): <math>H_1: \mu >\mu_o</math>, or <math>H_1: \mu<\mu_o</math>
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** Double sided: <math>H_1: \mu \not= \mu_o</math>
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** Double sided: \(H_1: \mu \not= \mu_o\)
====Known Variance====
====Known Variance====
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* [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics]:  
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* [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics]:  
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: <math>Z_o = {\overline{x} - \mu_o \over \sigma} \sim N(0,1)</math>.
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: <math>Z_o = {\overline{x} - \mu_o \over {\sigma \over \sqrt{n}}} \sim N(0,1)</math>.
====Unknown Variance====
====Unknown Variance====
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* [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics]:  
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* [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics]:  
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: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim N(0,1)</math>.
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: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim T_{(df=n-1)}</math>.
===Example===
===Example===
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Let's [[AP_Statistics_Curriculum_2007_Estim_L_Mean | revisit the ''number of sentences per advertisement'' example]], where we measure of readability for magazine advertisements.  A random sample of the number of sentences found in 30 magazine advertisements is listed below.  Suppose we want to test at <math>\alpha=0.05</math> a null hypothesis: <math>H_o: \mu=20</math> against a double-sided research alternative hypothesis: <math>H_1: \mu \not= 20</math>.
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Let's [[AP_Statistics_Curriculum_2007_Estim_L_Mean#Example | revisit the ''number of sentences per advertisement'' example]], where we measure the readability for magazine advertisements.  A random sample of the number of sentences found in 30 magazine advertisements is listed below.  Suppose:
 +
: We want to test at <math>\alpha=0.05</math>
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: Null hypothesis: <math>H_o: \mu=20</math>  
 +
: Against a double-sided research alternative hypothesis: <math>H_1: \mu \not= 20</math>.
 +
 
<center>
<center>
{| class="wikitable" style="text-align:center; width:75%" border="1"
{| class="wikitable" style="text-align:center; width:75%" border="1"
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: <math>s=\hat{\sigma}=16.54</math>
: <math>s=\hat{\sigma}=16.54</math>
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As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-statistics]]: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=29)</math>
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As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-Statistics]]: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=29)</math>
: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})}=-1.733</math>.
: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})}=-1.733</math>.
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: <math>P(T_{(df=29)}>T_o=-1.733)=0.047</math>, thus
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: <math>P(T_{(df=29)} < T_o=-1.733)=0.047</math>, thus
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: the <math>p-value=2\times 0.047= 0.094</math> for this (double-sided) test. Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.05</math>! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, the cummulative tail area is 0.094, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> and we can not reject the nul hypothesis.
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: the <math>p-value=2\times 0.047= 0.094</math> for this (double-sided) test.  
 +
 
 +
Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.05</math>! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, the cumulative tail area is 0.094, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> so we can not reject the null hypothesis.
<center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig3.jpg|600px]]</center>
<center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig3.jpg|600px]]</center>
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* You can see use the [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.
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* You can use the [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.
<center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig2.jpg|600px]]</center>
<center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig2.jpg|600px]]</center>
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* This [[SOCR_EduMaterials_AnalysisActivities_OneT | SOCR One Smaple T-test Activity]] provides additional hands-on demonstrations of one-sample hypothesis testing.
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* This [[SOCR_EduMaterials_AnalysisActivities_OneT | SOCR One Sample T-test Activity]] provides additional hands-on demonstrations of one-sample hypothesis testing.
===Examples===
===Examples===
====Cavendish Mean Density of the Earth====
====Cavendish Mean Density of the Earth====
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A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's [http://en.wikipedia.org/wiki/Henry_Cavendish Henry Cavendish] measured the [http://www.jstor.org/view/02610523/ap000022/00a00200/0 Mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now know exact mean-density value = 5.517. These sample statistics may be helpful
+
A number of famous early experiments of measuring physical constants has later been shown to be biased. In the 1700's [http://en.wikipedia.org/wiki/Henry_Cavendish Henry Cavendish] measured the [http://www.jstor.org/stable/pdfplus/107617.pdf Mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now known exact mean-density value = 5.517. These sample statistics may be helpful (you can also find the [[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean_Tbl1_tr|transposed table here]]):
: n = 23, sample mean = 5.483,    sample SD = 0.1904
: n = 23, sample mean = 5.483,    sample SD = 0.1904
<center>
<center>
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|}
|}
</center>
</center>
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====US Federal Budget Deficit====
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Use the [[SOCR_Data_US_BudgetsDeficits_1849_2016 | US Federal Budget Deficit data (1849-2016)]] to formulate and test several Null hypotheses on whether the US Federal Budget Deficit is trivial (<math>\mu_o=0</math>) in different time frames (e.g., 1849-2000 or 1900-2016). Start with some [[AP_Statistics_Curriculum_2007_EDA_Plots |exploratory data analyses]] to plot the data as shown in the [[SOCR_Data_US_BudgetsDeficits_1849_2016|data page]]. Then you can use the [http://www.socr.ucla.edu/htmls/ana/OneSampleTTest_Analysis.html SOCR One Sample T-Test] and the [http://www.socr.ucla.edu/htmls/ana/ConfidenceInterval_Analysis.html SOCR Confidence Interval] applets. What are your conclusions?
<hr>
<hr>
===Hypothesis Testing Summary===
===Hypothesis Testing Summary===
-
Important parts of Hypothesis test conclusions:
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Important parts of Hypothesis Test conclusions:
* Decision (significance or no significance)
* Decision (significance or no significance)
-
* Parameter of interest
+
* Parameter of Interest
-
* Variable of interest
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* Variable of Interest
* Population under study
* Population under study
* (optional but preferred) P-value
* (optional but preferred) P-value
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These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.
These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.
-
* [[AP_Statistics_Curriculum_2007_Estim_L_Mean | Confidence intervals]]: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).
+
* [[AP_Statistics_Curriculum_2007_Estim_L_Mean | Confidence Intervals]]: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).
* Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.
* Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.
 +
 +
===[[EBook_Problems_Hypothesis_L_Mean|Problems]]===
<hr>
<hr>

Current revision as of 15:46, 22 May 2013

Contents

General Advance-Placement (AP) Statistics Curriculum - Testing a Claim About a Mean: Large Samples

We already saw how to construct point and interval estimates for the population mean in the large sample case. Now, we show how to do hypothesis testing of the mean for large sample-sizes.

Background

  • Recall that for a random sample {X_1, X_2, X_3, \cdots , X_n} of the process, the population mean may be estimated by the sample average, \overline{X_n}={1\over n}\sum_{i=1}^n{X_i}.
  • For a given small α (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the (1 − α)100% Confidence interval for the mean is constructed by
CI(\alpha): \overline{x} \pm z_{\alpha\over 2} E,
where the margin of error E is defined as
E = \begin{cases}{\sigma\over\sqrt{n}},& \texttt{for-known}-\sigma,\\
{{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}},& \texttt{for-unknown}-\sigma.\end{cases}
and z_{\alpha\over 2} is the critical value for a Standard Normal distribution at {\alpha\over 2}.

Hypothesis Testing About a Mean: Large Samples

  • Null Hypothesis: Ho:μ = μo (e.g., μo = 0)
  • Alternative Research Hypotheses:
    • One sided (uni-directional): H1:μ > μo, or H1:μ < μo
    • Double sided: \(H_1: \mu \not= \mu_o\)

Known Variance

Z_o = {\overline{x} - \mu_o \over {\sigma \over \sqrt{n}}} \sim N(0,1).

Unknown Variance

T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim T_{(df=n-1)}.

Example

Let's revisit the number of sentences per advertisement example, where we measure the readability for magazine advertisements. A random sample of the number of sentences found in 30 magazine advertisements is listed below. Suppose:

We want to test at α = 0.05
Null hypothesis: Ho:μ = 20
Against a double-sided research alternative hypothesis: H_1: \mu \not= 20.
16 9 14 11 17 12 99 18 13 12 5 9 17 6 11 17 18 20 6 14 7 11 12 5 18 6 4 13 11 12

We had the following 2 sample statistics computed earlier

\overline{x}=\hat{\mu}=14.77
s=\hat{\sigma}=16.54

As the population variance is not given, we have to use the T-Statistics: T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=29)

T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})}=-1.733.
P(T(df = 29) < To = − 1.733) = 0.047, thus
the p-value=2\times 0.047= 0.094 for this (double-sided) test.

Therefore, we can not reject the null hypothesis at α = 0.05! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, the cumulative tail area is 0.094, which is larger than the initially set Type I error α = 0.05 so we can not reject the null hypothesis.

Examples

Cavendish Mean Density of the Earth

A number of famous early experiments of measuring physical constants has later been shown to be biased. In the 1700's Henry Cavendish measured the Mean density of the Earth. Formulate and test null and research hypotheses about these data regarding the now known exact mean-density value = 5.517. These sample statistics may be helpful (you can also find the transposed table here):

n = 23, sample mean = 5.483, sample SD = 0.1904
5.36 5.29 5.58 5.65 5.57 5.53 5.62 5.29 5.44 5.34 5.79 5.10 5.27 5.39 5.42 5.47 5.63 5.34 5.46 5.30 5.75 5.68 5.85

US Federal Budget Deficit

Use the US Federal Budget Deficit data (1849-2016) to formulate and test several Null hypotheses on whether the US Federal Budget Deficit is trivial (μo = 0) in different time frames (e.g., 1849-2000 or 1900-2016). Start with some exploratory data analyses to plot the data as shown in the data page. Then you can use the SOCR One Sample T-Test and the SOCR Confidence Interval applets. What are your conclusions?


Hypothesis Testing Summary

Important parts of Hypothesis Test conclusions:

  • Decision (significance or no significance)
  • Parameter of Interest
  • Variable of Interest
  • Population under study
  • (optional but preferred) P-value

Parallels between Hypothesis Testing and Confidence Intervals

These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.

  • Confidence Intervals: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).
  • Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.

Problems




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