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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.

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Presentation on theme: "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."— Presentation transcript:

1 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

2 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

3 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

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

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

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

7 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:

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

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

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

11 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.

12 Normal Distributions Possible formulas: Z Standard Normal random variable

13 Normal Distributions Possible formulas:

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

15 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.

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

17 Normal Distributions Soln:

18 Normal Distributions Soln:

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

20 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).

21 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

22 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).

23 Normal Distributions Soln:

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

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

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

27 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.

28 Normal Distributions(show excel) Soln:

29 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.

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

31 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

32 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

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

34 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?

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

36 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)

37 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)

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

39 Normal Distributions Mean is 6 and standard deviation is 3

40 Normal Distributions (B)

41 Normal Distributions Mean is -7 and standard deviation is 2

42 Normal Distributions (C)

43 Normal Distributions Mean is 300 and standard deviation is 50

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


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