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Sample Mean, A Formula If X is any random variable, then, as n increases without bound, the distribution of its standardized sample mean, approaches the distribution of the standard normal random variable, Z, whose p.d.f. is This is called the Central Limit Theorem. TI (material continues) The Sample Mean. A Formula: page 6 C

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Normal Distributions, Standard Normal 1. STANDARD NORMAL(Mean 0 & Standard deviation 1) In The Sample Mean we derived the probability density function for the standard normal random variable Z. We can use integration and f Z to compute probabilities for Z. Example 1. Compute P( 0.74 Z 1.29). As we saw in Integration, To evaluate the integral open Integrating.xls and enter the function as =(1/SQRT(2*PI()))*EXP(-0.5*x^2). Recall that x is the only variable that can be used in Integrating.xls. Integrating.xls (material continues) Normal Distributions. Standard Normal TIC

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Normal, General Normal 2. GENERAL NORMAL(Any Mean & Any Standard deviation/But its standardization is a standard normal distribution) The adjective “standard”, used in standard normal distributions, implies that there are “non-standard” normal distributions. This is indeed the case. A random variable, X, is called normal if its standardization, has a standard normal distribution. It can be shown that the probability density function for a normal random variable, X, with mean X and standard deviation X has the following form. (material continues) Normal Distributions. General Normal TIC

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Normal Distributions- Standard Normal Random Variable (Z) p.d.f. Can use Integrating.xls to find probabilities

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Normal Distributions- Ex1. Find Soln: show ex1 excel file

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Normal Distributions Ex. Find a number so that. Soln:show ex2 excel file

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Normal Distributions The previous example tells us that 97.5% of all data for a standard normal random variable lies in the interval. This means that 2.5% of the data lies above z = 1.96 Graphically, we have the following:

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Normal Distributions The shaded region corresponds to 97.5% of all possible area (note 2.5% is not shaded) 1.96

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Normal Distributions Due to symmetry, we get 95% of the area shaded with 5% not shaded (2.5% on each side)

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Normal Distributions This means that a 95% confidence interval for the standard normal random variable Z is (-1.96, 1.96)

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Normal Distributions A 95% confidence interval tells you how well a particular value compares to known data or sample data The interval that is constructed tells you that there is a 95% probability that the interval will contain the mean of X. Another interpretation is that 95% of all values found in a sample should lie within this 95% confidence interval.

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Normal Distributions Possible formulas: Z Standard Normal random variable

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Normal Distributions Possible formulas:

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Important Possible formulas: If is unknown The sample standard deviation,, will be a very good approximation for

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Normal Distributions Remember, that and 1.96 were special values that apply to a 95% confidence interval You need to find different values for other types of confidence intervals. Ex. Find a 99% confidence interval for Z. Find a 90% confidence interval for Z.

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Normal Distributions Soln: Ex. What is the confidence interval for Z with 1 standard deviation? 2 standard deviations? 3 standard deviations?

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Normal Distributions Soln:

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Normal Distributions Soln:

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Normal Distributions Since Z is a standard normal random variable, Z would have standardized some variable X. So,

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Normal Distributions Ex. Suppose X is a normal random variable with and. Find a 95% confidence interval for X if the 95% confidence interval for Z is (-1.96, 1.96).

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Normal Distributions Soln: So, the 95% confidence interval for X is (90.6, 149.4). We are 95% confident that this interval contains the true mean

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Normal Distributions Ex. Suppose X is a normal random variable. If a sample of size 34 was taken with and, find a 95% confidence interval for the sample mean-remember this is if the 95% confidence interval for Z is (-1.96, 1.96).

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Normal Distributions Soln:

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Normal Distributions Soln: So, the 95% confidence interval for is ( , ). We are 95% confident that this interval contains the true mean

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Normal Distributions General Normal Random Variable p.d.f. Probabilities are done similarly to Standard NRV

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Normal Distributions Ex. If X is a normal random variable representing exam 1 scores with mean 75 and standard deviation 10, find. Soln:

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Normal Distributions NORMDIST function in Excel Can calculate p.d.f. and c.d.f. values for a normal random variable Ex. If X is a normal random variable representing exam 1 scores with mean 75 and standard deviation 10, find.

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Normal Distributions(show excel) Soln:

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Normal Distributions Specific values for the p.d.f. can also be calculated using NORMDIST Ex. Find height of p.d.f. of a normal random variable X at X = 90 that has a normal distribution with and.

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Normal Distributions Soln: Two ways to solve=0.009 (1) Evaluate (2) Evaluate =NORMDIST(90, 71, 12, FALSE) using Excel

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Normal Distributions How does the mean and standard deviation affect the shape of the Normal Random Variable graph? Ex. Graph the p.d.f. of a normal random variable with the following characteristics: (1) and (2) and (3) and (4) and

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Normal Distributions Soln: (1) and Max Height 0.40 The y-values of the graph around x = -3 and x = 3 are very small Why? Recall-General Normal random variable p.d.f

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Normal Distributions General Normal Random Variable p.d.f. e^(0)=1

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Normal Distributions-sec1-4/13 Soln: (2) and Max Height 0.08 (This is 0.40 std. dev.) Y values very small Around x = -15 and x = 15 (This is 3 standard deviations from the mean- ) Why?

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Normal Distributions General Normal Random Variable p.d.f. e^(0)=1

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Normal Distributions Soln: (3) and Max Height 0.40 (At x = 4) Y values very small around x = 1 and x = 7 (This is 3 standard deviations from the mean)

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Normal Distributions Soln: (4) and Max Height 0.08 (This is 0.40 std. dev.) Y values very small around x = -11 and x = 19 (This is 3 standard deviations from the mean)

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Normal Distributions Ex. Find the mean and standard deviation for the following normal random variables graphed. (A)

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Normal Distributions Mean is 6 and standard deviation is 3

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Normal Distributions (B)

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Normal Distributions Mean is -7 and standard deviation is 2

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Normal Distributions (C)

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Normal Distributions Mean is 300 and standard deviation is 50

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Normal Distributions Relationship between p.d.f. and c.d.f. So,

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