Statistics Chapter 6
Uniform Distribution
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.
Standard Normal Distribution
The 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.
Finding Probabilities When Given z Scores
We can find areas (probabilities) for different regions under a normal model using technology or Table A-2. However technology is strongly recommended
,n,ln.
is a z score separating unlikely values from those that are likely to occur.
Density Curve
the graph of a continuous probability distribution
non-standard normal distributions
the mean is not 0 or the standard deviation is not 1, or both.
Known Areas how to find values
1. 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 column and across the top row, but areas are found in the body of the table. 2. Choose the correct (right/left) side of the graph. 3. A z score must be negative whenever it is located in the left half of the normal distribution. 4. Areas (or probabilities) are positive or zero values, but they are never negative.
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. Using the cumulative area from the left, locate the closest probability in the body of Table A-2 and identify the corresponding z score.
standard normal distribution three properties
1. Its graph is bell-shaped. 2. Its mean is equal to 0 (μ = 0). 3. Its standard deviation is equal to 1 (σ = 1).
Finding Normal Distribution Areas using a TI-83
1. Press (2nd) then (vars). 2. go to {2: normal cdf} 3. Enter in the z sores seperated by a comma (example Left z score, right z score)
Nonstandard Normal Distribution Procedure for Finding Areas
1. Sketch a normal curve, label the mean and any specific x values, then shade the region representing the desired probability. 2. For each relevant x value that is a boundary for the shaded region, use Formula 6-2 to convert that value to the equivalent z score. 3. Use computer software or a calculator or Table A-2 to find the area of the shaded region. This area is the desired probability.
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 the text, section 6.3. If using Table A-2 to find the z score corresponding to the cumulative left area bounded by x. Refer to the body of Table A-2 to find the closest area, then identify the corresponding z score. 3. Using Formula 6-2, enter the values for μ, σ, and the z score found in step 2, and then solve for x. x=u+(z*o) (Another form of Formula 6-2) 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.
density curve requirements
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.)
