1. The Sample Mean. A Formula: page 6
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.  T C I 
(material continues)

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 fZ 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 T C I 
(material continues)

3. Normal Distributions. General Normal
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)  T C I 

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

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

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

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 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,