# AP Statistics Curriculum 2007 Hypothesis L Mean

(Difference between revisions)
 Revision as of 19:01, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 18:55, 6 February 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Testing a Claim about a Mean: Large Samples== ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Testing a Claim about a Mean: Large Samples== - === Testing a Claim about a Mean: Large Samples=== + 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 tests about the mean in as the sample-sizes are large. - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ - ===Approach=== + ===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]=== - Models & strategies for solving the problem, data understanding & inference. + * Recall that the population mean may be estimated by the sample average, $\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}$, of random sample {$X_1, X_2, X_3, \cdots , X_n$} of the procees. - * TBD + * For a given small $\alpha$ (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.), the $(1-\alpha)100%$ 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 [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] distribution at ${\alpha\over 2}$. - ===Model Validation=== + === Hypothesis Testing about a Mean: Large Samples=== - Checking/affirming underlying assumptions. + * Null Hypothesis: $H_o: \mu=\mu_o (e.g., 0) + * Alternative Research Hypotheses: + ** One sided (uni-directional): [itex]H_1: \mu >\mu_o$, or $H_o: \mu<\mu_o$ + ** Double sided: $H_1: \mu \not= \mu_o$ - * TBD + ====Known Variance==== + * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics]: + : $Z_o = {\overline{x} - \mu_o \over \sigma} \sim N(0,1)$. - ===Computational Resources: Internet-based SOCR Tools=== + ====Unknown Variance==== - * TBD + * [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics]: + : $T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim N(0,1)$. - ===Examples=== + ===Example=== - Computer simulations and real observed data. + 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: $H_o: \mu=20$ against a double-sided research alternative hypothesis: $H_1: \mu \not= 20$. +
+ {| class="wikitable" style="text-align:center; width:75%" border="1" + |- + | 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 + |} +
- * TBD + We had the [[AP_Statistics_Curriculum_2007_Estim_L_Mean | following 2 sample statistics computed earlier]] - + : $\overline{x}=\hat{\mu}=14.77$ - ===Hands-on activities=== + : $s=\hat{\sigma}=16.54$ - Step-by-step practice problems. + - * TBD + As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-statistics]] + : $T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})} \sim T(df=29)$. + + ===Hands-on activities=== + *See the [[SOCR_EduMaterials_Activities_CoinfidenceIntervalExperiment | SOCR Confidence Interval Experiment]]. + * 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]]. +
[[Image:SOCR_EBook_Dinov_Estimates_L_Mean_020208_Fig1.jpg|400px]] + [[Image:SOCR_EBook_Dinov_Estimates_L_Mean_020208_Fig2.jpg|400px]]

+ ===References=== ===References=== - * TBD

## 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 tests about the mean in as the sample-sizes are large.

### Background

• Recall that the population mean may be estimated by the sample average, $\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}$, of random sample {$X_1, X_2, X_3, \cdots , X_n$} of the procees.
• 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 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
$\overline{x}=\hat{\mu}=14.77$
$s=\hat{\sigma}=16.54$

As the population variance is not given, we have to use the T-statistics

$T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {14.77 - 20 \over {{1\over \sqrt{30}} \sqrt{\sum_{i=1}^{30}{(x_i-14.77)^2\over 29}}})} \sim T(df=29)$.

### Hands-on activities

• See the SOCR Confidence Interval Experiment.
• Sample statistics, like the sample-mean and the sample-variance, may be easily obtained using 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, reported above.