Chapter 6 statistics

Finding z Scores from Known Areas 2

1- Draw a bell-shaped curve, draw the centerline, 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

Area and Probability

Because the total area under the density curve is equal to 1, there is a correspondence between area and probability

Central Limit Theorem

Given: The random variable x has a distribution (which may or may not be normal) with mean μ and standard deviation .2. Simple random samples all of size n are selected from the population. (The samples are selected so that all possible samples of the same size n have the same chance of being selected.)

standard normal distribution has three properties

It is bell-shaped. It has a mean equal to 0. It has a standard deviation equal to 1.

Notation

P(a < z < b) denotes the probability that the z score is between a and b.

Area

Region under the curve; refer to the values in the body of Table A-2.

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

The standard normal distribution

is a probability distribution with mean equal to 0 and standard deviation equal to 1, and the total area under its density curve is equal to 1.

A density curve

is 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.)

Finding z Scores from Known Areas

so far, when given Z scores, we have been finding areas under the curve, and these areas correspond to probabilities. Now, we have to do the reverse, i.e. when given probabilities (areas), we have to find the Z score.Remember Z scores are distances along the horizontal scale, whereas areas (or probabilities) are regions under the curve.

Central Limit Theorem - cont

Conclusions: The distribution of sample x will, as the sample size increases, approach a normal distribution.2. The mean of the sample means is the population mean μ.3. The standard deviation of all sample means is

Z-Score:

Distance along the horizontal scale of the standard normal distribution: It is found in the leftmost column and top row of Table A-2

cautions to Keep in Mind

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 zscores 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.

procedure for Finding Values Using Table A-2 and Formula 6-2

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. Use 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, , and the z score found in step 2, then solve for x.x = μ + (z • )(Another form of Formula 6-2)(If z is located to the left of the mean, be sure that it is a negative number.)4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem

Where the mean is located in a Normal Distribution?

The mean is located on the centerline. Scores lower than the mean will have a negative Z-score, while scores higher than the mean will have positive Z-scores