1. The normal distribution
M248: Analyzing data
Block B
UNIT B1
The normal distribution

2. UNIT B1: The normal distribution
Block B
UNIT B1: The normal distribution
Contents
Section 2: The normal distribution
Section 3: More on means and variances
Section 4: Calculation using tables
Unit B1 Exercises

3. Section 2: The normal distribution Section 2
Section 2: The normal distribution Section 2.1: The family of normal distributions
The continuous random variable X is normally distributed with mean and standard deviation (and hence variance ) if the probability density function of X is given by This is written

4. The p.d.f of a normal distribution has the following shape: Examples:
Section 2: The normal distribution Section 2.1: The family of normal distributions
The p.d.f of a normal distribution has the following shape: Examples:

5. Section 2: The normal distribution Section 2
Section 2: The normal distribution Section 2.1: The family of normal distributions
The shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation.
Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation.

6. Section 2: The normal distribution Section 2
Section 2: The normal distribution Section 2.1: The family of normal distributions

7. The normal distribution curve is bell-shaped.
Section 2: The normal distribution Section 2.1: The family of normal distributions (Properties)
The normal distribution curve is bell-shaped.
The mean, median, and mode are equal and located at the center of the distribution.
The normal distribution curve is unimodal (i.e., it has only one mode).
The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

8. Section 2: The normal distribution Section 2
Section 2: The normal distribution Section 2.1: The family of normal distributions (Properties)
The curve is continuous—i.e., there are no gaps or holes. For each value of X, here is a corresponding value of Y.
The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.

9. The area under the normal curve that lies within
Section 2: The normal distribution Section 2.1: The family of normal distributions (Properties)
The total area under the normal distribution curve is equal to 1.00 or 100%.
The area under the normal curve that lies within
1.645 standard deviation of the mean is approximately 0.90 (90%).
1.96 standard deviations of the mean is approximately 0.95 (95%).
2.576 standard deviations of the mean is approximately 0.99 ( 99%).

10. Section 2: The normal distribution Section 2
Section 2: The normal distribution Section 2.1: The family of normal distributions (Properties)

11. For any values “a and b” that lie in the range of X, we have:
Section 2: The normal distribution Section 2.2: Calculating probabilities
For a normal distribution, as for other continuous probability distributions, probabilities are calculated by finding the areas under the graph of the p.d.f. As previously observed, this is done easily using the c.d.f.
For any values “a and b” that lie in the range of X, we have:

12. Graphically we shade the required probability as follows:
Section 2: The normal distribution Section 2.2: Calculating probabilities
Graphically we shade the required probability as follows:

13. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.1: Functions of a random variable
If X is a random variable and a and b are constants, then the mean and variance of the random variable (linear function of X) is given by
The mean of
The variance of

14. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.1: Functions of a random variable
If the random variable X is normally distributed with mean , and variance and a and b are constants, then the random variable is normally distributed with mean and variance :
An alternative formula for the variance of a random variable
If X is a random variable with mean , then the variance of X may be obtained using the formula
For more practice on this subsection, read examples 3.3, and 3.6 and solve activities 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6

15. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.1: Functions of a random variable
Example:
The probability distribution of the random variable Y is given in the following table:
Then,
and
This could be calculated as follows: 1 2 3

16. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.1: Functions of a random variable
First of all, and in order to make it easier, we create the following table: Therefore 1 2 3
Total Sum

17. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.2: Sums of random variables
The mean of a sum of random variables is equal to the sum of means of the random variables:
If the random variables are independent, then the variance of their sum is equal to the sum of their variances:

18. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.2: Sums of random variables
Sums of Bernoulli random variables
Suppose that the random variables each has a Bernoulli distribution with the same parameter . (that is each random variable takes the value 1 with probability and the value 0 with probability , and suppose that the random variables are independent. Then the distribution of the sum is binomial with parameters n and .
So it follows directly from the previous results that:

19. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.2: Sums of random variables
Sums of normal random variables
If are independent normally distributed random variables with means and variances , then their sum
has a normal distribution with mean and variance That is,
see example 3.8 and solve activity 3.9 pages 23 and 24

20. Section 3: More on means and variances Section 3
Section 3: More on means and variances Section 3.2: Sums of random variables
The difference of two random variables
If X and Y are independent random variables, then the mean and variance of their difference are given by
If X and Y are normally distributed, then the distribution of is also normal.
solve activity 3.10 page 25

21. Section 4: Calculations using tables Section 4
Section 4: Calculations using tables Section 4.1: The standard normal distribution
Since each normally distributed variable has its own mean and standard deviation, the shape and location of these curves will vary. In practical applications, one would have to have a table of areas under the curve for each variable. To simplify this, statisticians use the standard normal distribution.
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The p.d.f is:

22. Section 4: Calculations using tables Section 4.2: Using Printed Tables
we use the table directly for any positive z. If
Note: since the total area under the curve is equal to 1 and the p.d.f is symmetrical about its mean (mean = 0)

23. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example:

24. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example 1: Find the area to the left of z = 1.99.
The value in the 1.9 row and the .09 column of the table is The area is

25. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example 2: Find the area to right of z =
The value in the 1.1 row and the .06 column of the table is The area is

26. (a) The value for z = 1.5 is 0.9332.The area is 0.0668
Section 4: Calculations using tables Section 4.2: Using Printed Tables
Examples 3 and 4:
(a) The value for z = 1.5 is The area is
(b) The values for z = 1.83 and z =0 is and 0.5 The area is =

27. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example 5: Find the area between z = 1.68 and
z =
The values for z = 1.68 is and for z = 1.37 is The area is

28. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Table 2: Finding quantiles (Finding the z-value)
Given we know , that is the area (probability) If

29. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example 1:

30. Section 4: Calculations using tables Section 4.2: Using Printed Tables
Example 2: Find the z value such that the area under the standard normal distribution curve between 0 and the z value is 0.21.

31. Section 4: Calculations using tables
Example 3: \
the area below –k is equal to 0.06
0.88
0.06
0.94

32. Section 4: Calculations using tables Section 4
Section 4: Calculations using tables Section 4.3: The standard normal distribution
(z value or standard score)
The z value is the number of standard deviations that a particular X value is away from the mean. The formula for finding the z value is:
(Important)

33. Section 4: Calculations using tables Section 4
Section 4: Calculations using tables Section 4.3: The standard normal distribution
(z value or standard score)
If , then the random variable
Conversely, if ,then the random variable
(Important)

34. Applications of the Normal Distributions
The standard normal distribution curve can be used to solve a wide variety of practical problems. The only requirement is that the variable be normally or approximately normally distributed.
To solve problems by using the standard normal distribution, transform the original variable to a standard normal distribution variable by using the z value formula.
This formula transforms the values of the variable into standard units or z values. Once the variable is transformed, then the Procedure Table 1can be used to solve problems.

35. Example 1:
A survey by the National Retail Federation found that women spend on average $ for the Christmas holidays. Assume the standard deviation is $ Find the percentage of women who spend less than $ Assume the variable is normally distributed.
Step 1: Draw the normal distribution curve.

36. Example 1: Table 1 gives us an area of 0.6808.
Step 2: Find the z value corresponding to $
Step 3: Find the area to the left of z = 0.47.
Table 1 gives us an area of
68% of women spend less than $160.

37. Example 2:
The American Automobile Association reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. What is the probability for one call to be responded to in less than 15 minutes?
Step 1: Draw the normal distribution curve.

38. Example 2: Table 1 gives us an area of 0.9868 for z=2.22
Step 2: Find the z value for 15.
Step 3: Find the area to the left of z =
Table 1 gives us an area of for z=2.22

39. Example 3:
Each month, an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the standard deviation is 2 pounds. If a household is selected at random, find the probability of its generating between 27 and 31 pounds per month. Assume the variable is approximately normally distributed.
Step 1: Draw the normal distribution curve.

40. Example 3: The probability is 62%.
Step 2: Find z values corresponding to 27 and 31.
Step 3: Find the area between z = -0.5 and z = 1.5.
The probability is 62%.

41. Example 4: Police Academy
To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed.
Step 1: Draw the normal distribution curve.

42. Example 4: Police Academy
Step 2: Use table 2
Since percentage = 0.1, that is less than 0.5 and we have
Step 3: Find X.
The cutoff, the lowest possible score to qualify, is 226.

43. Unit B1 Exercises M248 Exercise Booklet
Solve the following exercises:
Exercise 30 ……………………………………………………..Page 11
Exercise 31 ……………………………………………………..Page 11
Exercise 32 ……………………………………………………..Page 11
Exercise 33 ……………………………………………………..Page 11
Exercise 34 ……………………………………………………..Page 11
Exercise 35 ……………………………………………………..Page 11
Exercise 36 ……………………………………………………..Page 11
Exercise 37 ……………………………………………………..Page 12