AP Statistics Curriculum 2007 Normal Prob

(Difference between revisions)
 Revision as of 18:51, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 20:19, 31 January 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Nonstandard Normal Distribution & Experiments: Finding Probabilities== ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Nonstandard Normal Distribution & Experiments: Finding Probabilities== - === Nonstandard Normal Distribution & Experiments: Finding Probabilities=== + === General Normal Distribution=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + The standard normal distribution is a continuous distribution where the following exact ''areas'' are bound between the Standard Normal Density function and the x-axis on the symmetric intervals around the origin: -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ * The area: -1 < z < 1 = 0.8413 - 0.1587 = 0.6826 + * The area: -2.0 < z < 2.0 = 0.9772 - 0.0228 = 0.9544 + * The area: -3.0 < z < 3.0 = 0.9987 - 0.0013 = 0.9974 +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig0.jpg|500px]]
- ===Approach=== + * Standard Normal density function $f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.$ - Models & strategies for solving the problem, data understanding & inference. + - * TBD + * The Standard Normal distribution is also a special case of the [[AP_Statistics_Curriculum_2007_Normal_Prob | more general normal distribution]] where the mean is set to zero and a variance to one. The Standard Normal distribution is often called the ''bell curve'' because the graph of its probability density resembles a bell. - ===Model Validation=== + ===Experiments=== - Checking/affirming underlying assumptions. + Suppose we decide to test the state of 100 used batteries. To do that, we connect each battery to a volt-meter by randomly attaching the positive (+) and negative (-) battery terminals to the corresponding volt-meter's connections. Electrical current always flows from + to -, i.e., the current goes in the direction of the voltage drop. Depending upon which way the battery is connected to the volt-meter we can observe positive or negative voltage recordings (voltage is just a difference, which forces current to flow from higher to the lower voltage.) Denote $X_i={measured voltage for battery i} - this is random variable 0 and assume the distribution of all [itex]X_i$ is Standard Normal, $X_i \sim N(0,1)$. Use the Normal Distribution (with mean=0 and variance=1) in the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distribution applet] to address the following questions. This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]]. How many batteries, from the sample of 100, can we expect to have? - + * Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1. - * TBD +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]
- + * |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1. - ===Computational Resources: Internet-based SOCR Tools=== +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig2.jpg|500px]]
- * TBD + * Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2. - +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]
- ===Examples=== + * Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2. - Computer simulations and real observed data. +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]
- + * -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range. - * TBD +
[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig4.jpg|500px]]
- + - ===Hands-on activities=== + - Step-by-step practice problems. + - + - * TBD +

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

General Advance-Placement (AP) Statistics Curriculum - Nonstandard Normal Distribution & Experiments: Finding Probabilities

General Normal Distribution

The standard normal distribution is a continuous distribution where the following exact areas are bound between the Standard Normal Density function and the x-axis on the symmetric intervals around the origin:

• The area: -1 < z < 1 = 0.8413 - 0.1587 = 0.6826
• The area: -2.0 < z < 2.0 = 0.9772 - 0.0228 = 0.9544
• The area: -3.0 < z < 3.0 = 0.9987 - 0.0013 = 0.9974
• Standard Normal density function $f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.$
• The Standard Normal distribution is also a special case of the more general normal distribution where the mean is set to zero and a variance to one. The Standard Normal distribution is often called the bell curve because the graph of its probability density resembles a bell.

Experiments

Suppose we decide to test the state of 100 used batteries. To do that, we connect each battery to a volt-meter by randomly attaching the positive (+) and negative (-) battery terminals to the corresponding volt-meter's connections. Electrical current always flows from + to -, i.e., the current goes in the direction of the voltage drop. Depending upon which way the battery is connected to the volt-meter we can observe positive or negative voltage recordings (voltage is just a difference, which forces current to flow from higher to the lower voltage.) Denote Xi={measured voltage for battery i} - this is random variable 0 and assume the distribution of all Xi is Standard Normal, $X_i \sim N(0,1)$. Use the Normal Distribution (with mean=0 and variance=1) in the SOCR Distribution applet to address the following questions. This Distributions help-page may be useful in understanding SOCR Distribution Applet. How many batteries, from the sample of 100, can we expect to have?

• Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1.
• |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
• Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.
• Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.
• -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range.