graphs of normal probability distributions the graph of a normal distribution is called a normal...

Graphs of Normal Probability Distributions

The graph of a normal distribution is called a normal curve. It takes on the shape of a bell and is referred to as a bell curve.



Lets look at the properties of a bell curve :




1. The curve is bell shaped, with the highest point over the mean

𝜇




1. The curve is bell shaped, with the highest point over the mean2. The curve has symmetry about the mean

𝜇




1. The curve is bell shaped, with the highest point over the mean2. The curve has symmetry about mean3. The curve approaches but never touches horizontal axis

𝜇




1. The curve is bell shaped, with the highest point over the mean2. The curve has symmetry about mean3. The curve approaches but never touches horizontal axis4, It has inflection ( transition ) points at and

𝜇𝜇+𝜎𝜇−𝜎 𝜇+2𝜎𝜇+3𝜎𝜇−2𝜎𝜇−3𝜎




1. The curve is bell shaped, with the highest point over the mean2. The curve has symmetry about mean3. The curve approaches but never touches horizontal axis4, It has inflection ( transition ) points at and 5. The area under the entire curve = 1

𝜇𝜇+𝜎𝜇−𝜎 𝜇+2𝜎𝜇+3𝜎𝜇−2𝜎𝜇−3𝜎




1. The curve is bell shaped, with the highest point over the mean2. The curve has symmetry about mean3. The curve approaches but never touches horizontal axis4, It has inflection ( transition ) points at and 5. The area under the entire curve = 16. The parameters of the curve are controlled by the standard deviation. If is large, then the curve will be flat and less peaked. If is small, the curve will be tall and more peaked. 𝜇𝜇+𝜎𝜇−𝜎 𝜇+2𝜎𝜇+3𝜎𝜇−2𝜎𝜇−3𝜎


𝜇 𝜇+𝜎𝜇−𝜎 𝜇+2𝜎𝜇+3𝜎𝜇−2𝜎𝜇−3𝜎

The Empirical Rule :

For a bell - shaped curve that is symmetrical





- 68% of the data values lie within 1 standard deviation on either side of the mean

68%

34%

34%






- 95% of the data values lie within 2 standard deviations on either side of the mean

68%

34%

34%

95%

13.5%

13.5%






- 95% of the data values lie within 2 standard deviations on either side of the mean

- 99.7% of the data values lie within 3 standard deviations on either side of the mean

68%

34%

95%

13.5%

13.5%

34%

99.7%

2.35%

2.35%


EXAMPLE : The battery life (in hours) of a weather radio is normally distributed with and . What is the probability that a radio selected at random will have a battery life of 600 to 700 hours?



Solution :1. Label the mean

600



Solution :1. Label the mean 2. Find 1 standard deviation on either side

600 700500




3. From our empirical data example, 34% of the

data lies 1 standard deviation to the right of the mean.

600 700500

34%




3. From our empirical data example, 34% of the

data lies 1 standard deviation to the right of the mean.4. So the probability = 0.34 600 700500

34%


EXAMPLE #2: The yearly wheat yield per acre on a particular farm is normally distributed with a mean bushels and a standard deviation bushels. What is the probability that an acre will yield 19 to 35 bushels ?



SOLUTION :1. Plot the mean

35



SOLUTION :1. Plot the mean 2. Plot 1 standard deviation on either side

35 4327




3. Plot 2 standard deviations on either side

35 4327 5119





4. From the empirical rule, 34% and 13.5% of data lie within 1 and 2 standard deviations to the left of the mean representing 19 – 35 bushels. 35 4327 5119

34%

13.5%





4. From the empirical rule, 34% and 13.5% of data lie within 1 and 2 standard deviations to the left of the mean representing 19 – 35 bushels.5. Add the areas so the probability = 0.475 that the yield will be between 19 and 35 bushels.

35 4327 511934%

13.5%


CONTROL CHARTS

Control charts are used to examine data over a period of time intervals or in some sequential order. There exists an inherent amount of variability in any sequential data. A good example is the number of riders per hour on a roller coaster. That number will vary from hour to hour and day to day. Control charts combine graphic and numeric descriptions of data with probability distributions.


CONTROL CHARTS

Control charts are used to examine data over a period of time intervals or in some sequential order. There exists an inherent amount of variability in any sequential data. A good example is the number of riders per hour on a roller coaster. That number will vary from hour to hour and day to day. Control charts combine graphic and numeric descriptions of data with probability distributions.

How to create a control chart :1. Find the mean and standard deviation of the distribution by

a) using past data from a period during which the process was “in control” ORb) using specified “target” values for and

2. Create a graph in which the vertical axis represents values and the horizontal axis represents time.

3. Draw a horizontal line at height and horizontal, dashed control – limit lines at and 4. Plot the variable on the graph in time sequence order. Use line segments to connect the points

in time sequence order.

Graphs of Normal Probability DistributionsEXAMPLE : The data chart below shows the number of rooms in Antler Lodge that have not been “made

up” by 3:30 PM. The distribution of rooms not made up is approximately normal with rooms and room. Create a control chart for the data below and let represent the number of rooms not made up for each day.

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10



Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10

1. Our and are given



Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10

1. Our and are given2. Establish your control limits



Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10


3. a) Create our horizontal and vertical axis and label them b) draw your control limit lines 19.3

28.7

33.4

9.9

5.2

𝜇+3𝜎

𝜇+2𝜎

𝜇−3𝜎

𝜇−2𝜎

𝜇

Day

Rooms

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15



Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10


3. a) Create our horizontal and vertical axis and label them b) draw your control limit lines4. Plot each

19.3

28.7

33.4

9.9

5.2

𝜇+3𝜎

𝜇+2𝜎

𝜇−3𝜎

𝜇−2𝜎

𝜇

Day

Rooms

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15



Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

11 20 25 23 16 19 8 25 17 20 23 29 18 14 10


3. a) Create our horizontal and vertical axis and label them b) draw your control limit lines4. Plot each 5. Connect the dots

19.3

28.7

33.4

9.9

5.2

𝜇+3𝜎

𝜇+2𝜎

𝜇−3𝜎

𝜇−2𝜎

𝜇

Day

Rooms

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

Graphs of Normal Probability DistributionsCONTROL CHARTS

After we have created a control chart, the question is as time goes by, is the variable continuing in this same distribution, or is the distribution of values changing ? If the distribution is continuing in more or less the same manner, we say it is in statistical control. If it is not, we say it is out of control. A random variable is said to be out of control if successive time measurements of indicate it is no longer following the target probability distribution.



Here are three of the most popular warning signals for an out of control distribution :




1. One points falls beyond the level

𝜇+3𝜎

𝜇−3𝜎




1. One points falls beyond the level

𝜇+3𝜎

𝜇−3𝜎

The probability of a false alarm = 0.003




2. A run of nine consecutive points on one side of the center line ( )

𝜇+3𝜎

𝜇−3𝜎




3. At least two out of three consecutive points lie beyond OR

𝜇+3𝜎

𝜇−3𝜎

graphs of normal probability distributions the graph of a normal distribution is called a normal...

Documents

normal curve

bell curve

normal distribution

bell shaped curve

curve approaches

data values

standard deviations

empirical rule