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6-1 THE STANDARD NORMAL DISTRIBUTION The major focus of this chapter is the concept of a normal probability distribution, but we begin with a uniform distribution so that we can see the following two very important properties:

1. The area under the graph of a continuous probability distribution is equal to 1 2. There is a correspondence between area and probability.

Note: For uniform distribution, probabilities can be found by identifying the corresponding areas in the graph using this formula for the area of a rectangle:

Area of a rectangle – height x width

Example: Suppose the time allowed to answer a quiz question is 25 seconds.

a. Find the probability that a student responds within 10 seconds.

b. Find the probability that a student responds between 20 to 25 seconds.

DEFINITION 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.

DEFINITION: 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. This requirement that the area must equal 1 simplifies probability problems, so the following statement is really important: Because the total area under any density curve is equal to 1, there is a correspondence between area and probability.

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Example:

Normal Distribution

DEFINITION If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, as in Figure 6-1, and it can be described by the equation given as Formula 6-1, we say that it has a normal distribution.

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Standard Normal Distribution

1. Bell-shaped: The graph of the standard normal distribution is bell-shaped (as in figure 6-1)

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. Standard Normal Distribution This is a bell- shaped curve with mean = 0 and standard deviation = 1. There is a correspondence between area and probability as with the uniform distribution.

Important points to be aware of if using 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 Notation:

1. P( z > a): probability that the z-score is greater than a. 2. P(z < a): probability that the z-score is less than a. 3. P(a < z < b): probability that the z-score is between a and b.

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Examples:

Example: Find the following probabilities. Assume standard normal distribution. For each case, draw a graph, then find the probability.

a. P(z < 2.33) b. P(z > -1.23)

c. P( -1.05 < z < 1.75) d. P( z < - 2.75)

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Finding Z-scores from Known Areas

Example: Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information.

a. Find ���, the 68th percentile. This is the bone density score separating the bottom 90% from the top 10%.

b. If the bone density scores in the bottom 2% and the top 2% are used as cutoff pints for levels that are too low or too high, find the two readings that are cutoff values.

Example: Draw a graph, then find a.

1. Given P(z < a) = 0.8438, find a. 2. Given P( z < a) = 0.0256, find a.

3. Given P(z > a) = 0.0068, find a. 4. Given P (z > a) = 0.8577, find a.

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Example: Find the indicated critical value. In each case, draw a graph. Round answers to two decimal places.

a. ��.�� b. ��.��

c. ��.�� d. ��.��

DEFINITION 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.