stats 6.1 , 6.2
standard normal distribution properties
1.Bell-shaped: The graph of the standard normal distribution is bell-shaped. 2.µ = 0: The standard normal distribution has a mean equal to 0. 3.σ = 1: The standard normal distribution has a standard deviation equal to 1.
Finding z Scores from Known Areas
1.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.Use technology or Table A-2 to find the z score. With Table A-2, use the cumulative area from the left, locate the closest probability in the body of the table, and identify the corresponding z score.
Procedure For Finding Values From Known Areas or Probabilities
1.Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the x value(s) being sought. 2.If using technology, refer to the instructions at the end of this section. If using Table A-2, refer to the body of Table A-2 to find the area to the left of x, then identify the z score corresponding to that area. 3. If you know z and must convert to the equivalent x value, use the conversion formula by entering the values for μ, σ, and the z score found in step 2, and then solve for x. We can solve for x; x=μ+(z∙σ) 4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and in the context of the problem
Finding Probabilities When Given z Scores
1.Table A-2 is designed only for the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. 2.Table A-2 is on two pages, with the left page for negative z scores and the right page for positive z scores. 3.Each value in the body of the table is a cumulative area from the left up to a vertical boundary above a specific z score 4.When working with a graph, avoid confusion between z scores and areas. 5.The part of the z score denoting hundredths is found across the top row of Table A-2.
Properties of a Uniform Distribution
1.The area under the graph of a continuous probability distribution is equal to 1. 2.There is a correspondence between area and probability, so probabilities can be found by identifying the corresponding areas in the graph using this formula for the area of a rectangle:Area = height × width
Uniform Distribution
A continuous random variable has a uniform distribution if its values are equally spread over the range of possible values. The graph of a uniform distribution results in a rectangular shape
Critical Value
For the standard normal distribution, a critical value is a z score on the borderline separating those z scores that are significantly low or significantly high.
normal distribution
If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped
cal hints
The area corresponding to the region between two z scores can be found by finding the difference between the two areas found in Table A-2
density curve
The graph of any continuous probability distribution -must satisfy the requirement that the total area under the curve is exactly 1. **Because the total area under any density curve is equal to 1, there is a correspondence between area and probability.
CAUTION z_α
When finding a value of zα for a particular value of α, note that α is the area to the right of zα, but Table A-2 and some technologies give cumulative areas to the left of a given z score. To find the value of zα, resolve that conflict by using the value of 1 − α. For example, to find z0.1, refer to the z score with an area of 0.9 to its left.
P(a < z < b)
denotes the probability that the z score is between a and b.
P(z > a)
denotes the probability that the z score is greater than a.
P(z < a)
denotes the probability that the z score is less than a.
z_α
denotes the z score with an area of α to its right. alpha
Finding Values From Known Areas OF nonstan
which the area (or probability or percentage) is known 1. Graphs 2.Don't confuse z scores and areas. z scores are distances along the horizontal scale, but areas are regions under the normal curve. Table A-2 lists z scores in the left columns and across the top row, but areas are found in the body of the table. 3.Choose the correct (right/left) side of the graph. A value separating the top 10% from the others will be located on the right side of the graph, but a value separating the bottom 10% will be located on the left side of the graph. 4.A z score must be negative whenever it is located in the left half of the normal distribution. 5.Areas (or probabilities) are always between 0 and 1, and they are never negative.
normal distribution formula
y=e^(-1/2 ((x-μ)/σ)^2 )/(σ√2π)
z formula (converting)
z =(x-μ)/σ (round z scores to 2 decimal places
z score vs area
z score: Distance along the horizontal scale of the standard normal distribution (corresponding to the number of standard deviations above or below the mean); refer to the leftmost column and top row of Table A-2. Area: Region under the curve; refer to the values in the body of Table A-2.
