Ch 6: The Normal Distribution
What is a Normally Distributed Variable?
A Normally Distributed Variable is any variable whose distribution has the shape of a normal curve (i.e. a bell curve).
How do we use the table to find the area under the bell curve to the RIGHT of a z-score?
Area = 1 - (table value)
How do we construct a Normal Probability Plot?
Arrange the datapoints from lowest to highest. Use the Normal Scores from Table III.
State the Empirical Rule (Repeated from an earlier chapter)
For any variable whose distribution is bell shaped... Approximately 68% of the data lies within 1 standard deviation of the mean. Approximately 95% of the data lies within 2 standard deviations of the mean. Approximately 99.7% of the data lies within 3 standard deviations of the mean.
What is the difference between a Standardized variable and a z-score?
In the standardized variable, we leave x as x. In the z-score, we plug-in a value for x.
When you see the world STANDARDIZED you should think...
z-score. Standardized variables are related to z-scores.
How do you find a given datapoint (or range) from a percent or probability? (For normally distributed values)
1. SKETCH Sketch the normal curve associated with the variable. 2. SHADE Shade the region of interest (approxamately) 3. USE TABLE TO FIND Z-SCORE FROM AREA 4. FIND X FROM Z-SCORE Use equation.
If a normal distribution has a large standard deviation, its shape will be....
short and wide (large std dev = lots of variation from the mean)
How do we use the table to find the area under the bell curve between two z-scores?
(table value for higher z-score) - (table value for lower z-score) OR (more simply) Area = (Bigger - smaller)
State the two properties of density curves...
- Property 1: A density curve is always on or above the horizontal axis. - Property 2: The total area under the density curve (and above the horizontal axis) equals 1.
The standard normal curve is symmetric about....
0
The total area under the standard normal curve is....
1.
If a Normal distribution has a small standard deviation, its shape will be...
tall and skinny (small std deviation = not much variation from mean)
A normal distribution is centered at...
the mean of the variable in question. In the image, the green line represents the mean.
How do you find the probability (or percent of observations) from a given datapoint or range? (For normally distributed values)
1. SKETCH • Sketch the normal curve associated with the variable. 2. CALCULATE Z-SCORES Find the z-score(s) for the delimiting x-values found in step 2. 3. SHADE • the region of interest and mark its delimiting x-values. 4. USE TABLE Use an area table to find the area under the standard normal curve delimited by the z-score(s) found in step 3.
How do you find the Area under the standard normal curve (for a given z score)?
Look up the z-score in the table!
A given normal distribution is completely defined what what two parameters?
Mean and Standard Deviation (If two normally distributed variables have the same mean and standard deviation, their graphs will be identical.)
Describe the shape of normal distributions....
Normal distributions have a bell shape that is symmetric around the mean of the variable in question.
Using a Normal Probability Plot, how do we know if data is normally distributed?
The Normal probability plot will be a straight line.
The z score TABLE tells us....
The area under the standard bell curve from 0 to the z score. Alternatively... The table tells us the area under the curve to the LEFT of the z-score.
In order to make normal distributions easier to work with, we standardize them. What is the mean of a standard normal variable? What is the standard deviation of a standard normal variable?
The mean of all STANDARDIZED normal variables is 0. The standard deviation of all STANDARDIZED normal variables is 1.
What is the relation between the percent of observations within a specified range and the bell graph?
The percent of observations within a specified range equals the area under the bell graph in that range.
What is the relationship between density curves and the percentage of observations within a specified range (say, 4 - 8).
The percentage of observations that lie within the range is equal to the area under the curve within the range. Example: The probability that some random variable will be between 4 and 8 is equal to the area under the curve between 4 and 8.
Describe the shape of the standard normal curve on the far right and far left....
The standard normal curve extends indefinitely in both directions, approaching but never touching the horizontal axis.
What does a z-score tell us?
The z-score for a certain data-value tells us the NUMBER OF STANDARD DEVIATIONS that the data-value is from the mean. For example.... A z-score of -2 indicates that the data value is 2 standard deviations below the mean
What does the symbol z(with a small number on its right) indicate?
The z-score that has an area to its RIGHT under the normal bell curve (equal to the number).
Why do we use Normal Probability Plots?
We use normal probability plots to determine if a dataset is normally distributed.
Almost all the area under the standard normal curve lies...
between z-scores -3 and 3. NOTE: This is just another way of stating the 3 standard deviations rule: Almost all the data lies within 3 standard deviations below the mean, and 3 standard deviations above the mean.
What 2 things can this chapter ask you to find.
♦ Give you a z-score (or datapoint) ask for an percentage or probability (area under the curve). ♦ Give you a percentage or probability (area under the curve) and ask for a z-score (or datapoint.)