# AP Statistics Curriculum 2007 Normal Prob

(Difference between revisions)
 Revision as of 20:35, 31 January 2008 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 21:13, 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== - Due to the due to the [[AP_Statistics_Curriculum_2007_Limits_CLT |Central Limit Theorem]], the normal distribution is perhaps the most important model for studying various quantitative phenomena. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. While the mechanisms underlying natural processes may often be unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation. + Due to the [[AP_Statistics_Curriculum_2007_Limits_CLT |Central Limit Theorem]], the normal distribution is perhaps the most important model for studying various quantitative phenomena. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. While the mechanisms underlying natural processes may often be unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation. === General Normal Distribution=== === 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 (general) normal distribution is a continuous distribution that has similar exact ''areas'', bound in terms of its mean, like the [[AP_Statistics_Curriculum_2007#The_Standard_Normal_Distribution |Standard Normal distribution]] and the x-axis on the symmetric intervals around the origin: - * The area: -1 < z < 1 = 0.8413 - 0.1587 = 0.6826 + * The area: $\mu -\sigma < z < \mu+\sigma = 0.8413 - 0.1587 = 0.6826$ - * The area: -2.0 < z < 2.0 = 0.9772 - 0.0228 = 0.9544 + * The area: $\mu -2\sigma < z < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544$ - * The area: -3.0 < z < 3.0 = 0.9987 - 0.0013 = 0.9974 + * The area: $\mu -3\sigma < z < \mu +3\sigma= 0.9987 - 0.0013 = 0.9974$ -
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- * Standard Normal density function $f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.$ + * General Normal density function $f(x)= {e^{{-(x-mu)^2\over \sqrt{2\pi \sigma^2}} \over \sqrt{2 \pi \sigma^2}}.$ - * 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. + * See the special case of [[AP_Statistics_Curriculum_2007#The_Standard_Normal_Distribution | Standard Normal distribution]] where the mean is set to zero and a variance to one. ===Experiments=== ===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 $X_i$={measured voltage for battery i} - this is random variable 0 and assume the distribution of all $X_i$ 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? + TBD + + 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. * Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1.
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[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]

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

Due to the Central Limit Theorem, the normal distribution is perhaps the most important model for studying various quantitative phenomena. Many numerical measurements (e.g., weight, time, etc.) can be well approximated by the normal distribution. While the mechanisms underlying natural processes may often be unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation.

### General Normal Distribution

The (general) normal distribution is a continuous distribution that has similar exact areas, bound in terms of its mean, like the Standard Normal distribution and the x-axis on the symmetric intervals around the origin:

• The area: μ − σ < z < μ + σ = 0.8413 − 0.1587 = 0.6826
• The area: μ − 2σ < z < μ + 2σ = 0.9772 − 0.0228 = 0.9544
• The area: μ − 3σ < z < μ + 3σ = 0.9987 − 0.0013 = 0.9974
• General Normal density function Failed to parse (syntax error): f(x)= {e^{{-(x-mu)^2\over \sqrt{2\pi \sigma^2}} \over \sqrt{2 \pi \sigma^2}}.

### Experiments

TBD

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.