AP Statistics Curriculum 2007 Hypothesis L Mean

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===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]===
===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]===
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* Recall that for a random sample {<math>X_1, X_2, X_3, \cdots , X_n</math>} of the procees, the population mean may be estimated by the sample average, <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math>.
+
* Recall that for a random sample {<math>X_1, X_2, X_3, \cdots , X_n</math>} of the process, the population mean may be estimated by the sample average, <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math>.
* For a given small <math>\alpha</math> (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the <math>(1-\alpha)100%</math> Confidence interval for the mean is constructed by
* For a given small <math>\alpha</math> (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the <math>(1-\alpha)100%</math> Confidence interval for the mean is constructed by
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===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 a null hypothesis: <math>H_o: \mu=20</math> against a double-sided research alternative hypothesis: <math>H_1: \mu \not= 20</math>.
+
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>.
<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]]
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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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: <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}}})} \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>P(T_{(df=29)}>T_o=-1.733)=0.047</math>, thus
 +
: 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>!
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===Hands-on activities===
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* You can see use the [ SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.
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*See the [[SOCR_EduMaterials_Activities_CoinfidenceIntervalExperiment | SOCR Confidence Interval Experiment]].
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<center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig2.jpg|600px]]</center>
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* Sample statistics, like the sample-mean and the sample-variance, may be easily obtained using [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts]. The images below illustrate this functionality (based on the '''Bar-Chart''' and '''Index-Chart''') using the 30 observations of the number of sentences per advertisement, [[AP_Statistics_Curriculum_2007_Estim_L_Mean#Example | reported above]].
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<center>[[Image:SOCR_EBook_Dinov_Estimates_L_Mean_020208_Fig1.jpg|400px]]
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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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[[Image:SOCR_EBook_Dinov_Estimates_L_Mean_020208_Fig2.jpg|400px]]</center>
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 +
===Examples===
 +
 
 +
====Cavendish Mean Density of the Earth====
 +
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
 +
: n = 23, sample mean = 5.483,    sample SD = 0.1904
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<center>
 +
{| class="wikitable" style="text-align:center; width:75%" border="1"
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|-
 +
| 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
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|}
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</center>
<hr>
<hr>
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===References===
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===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.
 +
 
 +
* [[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.
<hr>
<hr>

Revision as of 19:23, 6 February 2008

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

Known Variance

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

Example

Let's 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 α = 0.05 a 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!
  • You can see use the [ SOCR Analyses (One-Sample T-Test)] to carry out these calculations as shown in the figure below.

Examples

Cavendish Mean Density of the Earth

A number of famous early experiments of measuring physical constants have 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 know exact mean-density value = 5.517. These sample statistics may be helpful

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

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.



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