Chapter 6 Section 2 Statistics Standard Normal Distribution

Finding Z-Scores from Known Area Calculator function

2nd-> Vars-> INVNORM ( area to left/right of z that we want (1-), 0,1) (0.95,0,1)

Uniform Distribution for a Continuous Random Variable

A continuous random variable has a uniform distribution if its values are spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape.

Example Continued

Bone density reading is between -1 and -2.5 indicates the subject has osteopenia. Find probability z= -2.50= 0.0062 z=-1.00=0.1587 area between z=-2.50 and z=-1.00 is the difference between areas found above (subtract these)

Written steps to find z-scores from known area

Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability. If that region is not a cumulative region from the left, work instead with a known region that is a cumulative region from the left. 2.Using the cumulative area from the left, locate the closest probability in the bodyof Table A-2 and identify the corresponding z score.

Using Area to Find Probability

Find the probability that a randomly selected voltage level is greater than 124.5 volts shaded area is greater than 124.5 volts the numbers 124.5 and 125.0 are 0.5 away so it's 0.5 * 0.5= 0.25 probability When Height is given: P (x> 124.5) Area= Base * Height Area= (125-124.5) * 0.5 Area = 0.5 * 0.5 = 0.25 What if height is not given? (b-a) * h= 1 h= 1/b-a (125-123) * h = 1 2. h= 1 h= 1/2

Example Continued

Finding the probability of someone having a density above -1 (Finding area of unshaded is 1- shaded) (1-unshaded= shaded) Total area is 1 1-z score of -1 to find bone density of someone above -1 z score is 0.1587 (for below -1) 1-0.1587= 0.8413 (for above -1). normal cdf (-1, 10000,0,1)

Standard Normal Distribution Properties

This section presents the standard normal distribution which has three properties: 1.Its graph is bell-shaped 2.Its mean is equal to 0 (μ= 0). 3.Its standard deviation is equal to (σ= 1).

Density Curve

the graph of a continuous probability distribution. It must satisfy the following properties: 1.The total area under the curve must equal 1. 2.Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.) Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

Finding Probabilities when given Z scores

Function to use: 2nd-> Vars-> Normalcdf [ Lower Value, Upper Value, mean, standard deviation] draw a little figure put the z score next to 0 shade the area below or above z score use calculator function use a large negative value as a lower value (i.e -10000) upper value is where graph stops (at z value) .8980

Notations (given z need to find probability)

NCDF ( a,b,0,1 ) NCDF (a 10000,0, 1) NCDF (-10000,a, 0, 1) P (a <z<b) denotes z score is between a and b P (z> a) denotes z score is greater than a P (z < a) denotes z score is less than a

Key Things to Know

area/prob -> z inverse norm NCDF- given z need to find probability CALCULATOR FUNCTIONS: Given z, finding area/probability/percentage, use Normalcdf (lower value, upper value, mean, standard deviation) Given Area use to find z :INVNORM(AREA TO THE LEFT OF Z WE WANT, MEAN, STANDARD DEVIATION) Note: Area, probability, percentage all have the same meaning. For the standard normal distribution, a critical value is a z score separating unlikely values from those that are likely to occur. Notation: The expression zαdenotes the zs core with an area of αto its right.

Standard Normal Distribution

is a normal probability distribution with μ= 0 and σ= 1. The total area under its density curve is equal to 1 values of z go from - to + values when x is less than mean, negative numbers, x is more than mean, positive number