Intro to Probability and Statistics Chapter 6

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. Notation The expression zα denotes the z score with an area of α to its right.

Normal Distribution

If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, we say that it has a normal distribution.

What are some propertied of a uniform distribution?

Properties of 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

Density Curve

The graph of any continuous probability distribution is called a density curve, and any density curve 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.**

Finding Values From Known Areas

1.) Graphs are extremely helpful in visualizing, understanding, and successfully working with normal provability distributions, so they should always be used. 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.

Procedure for Finding Areas with a Nonstandard Normal Distribution

1.) Sketch a normal curve, label the mean and any specific x values, and then shade the region representing the desired probability. 2.) For each relevant value x that is a boundary for the shaded region, use the formula to convert that value to the equivalent z score. (With many technologies, this step can be skipped.) 3.) Use technology (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 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 as follows: ( x= mean + ( z(stand dev) 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 ( with Table A-2)

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. 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. 5.)The part of the z score denoting hundredths is found across the top row of Table A-2.

Uniform Distribution

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

Standard Normal Distribution

The standard normal distribution is a normal distribution with the parameters of µ = 0 and σ = 1. The total area under its density curve is equal to 1.