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6-2: The Standard Normal Distribution

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6-2: The Standard Normal Distribution

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    1. 6-2: The Standard Normal Distribution Math 10

    2. Review Continuous Random Variable: A variable x determined by probability that can take on an infinite range of values with no gaps in the range. Ex: Tire pressure

    3. Distributions for Continuous Random Variables A graph that represents the possible values for the random variable and the probabilities associated with them.

    4. Probability Density Curve The graph of a continuous probability distribution is called a density curve. All density curves must obey the following properties: The total area under the curve must equal 1. Every point on the curve must have a vertical height that is 0 or greater (0=p(x)=1). Because the total area under the density curve is equal to 1, the areas correspond to probabilities.

    5. Uniform Distribution A continuous random variable has a uniform distribution if the values of the variable spread evenly over its range of possibilities. All values have the same probability

    6. Examples: Using the yellow uniform distribution. Find the probability a value less than -0.5 is chosen. Using the green uniform distribution. Find the probability that a value between 1.8 and 2.4 is chosen.

    7. Where does the Standard Normal come from?

    8. Any curve that is Bell-Shaped is a Normal Distribution

    9. Normal Distribution

    10. The Standard Normal Distribution

    11. Z-Score Each data value can be converted to a z-score using the formula for standardization: Each data value can be location on the x axis of the density curve.

    12. Finding Probabilities Using Z-Scores Table A-2 on pages 772 and 773 Negative z Scores table represents P(z<a) for negative z values Positive z Scores table represents P(z<a) for positive z values Z column represents the z-scores to one decimal place. The top row represents the second decimal place of the z-score. These two meet at the probability for P(z<a).

    13. Meaning of Probabilities P(z>a): the probability that a z score is greater than the z score a. (to the right of a) P(z<a): the probability that a z score is less than the z score a. (to the left of a) P(a<z<b): the probability that a z score is between the z scores of a and b. (between a and b)

    14. Examples: Use the chart to find the following probabilities P(z<-2.13) P(z<0.56) P(z>-0.46) P(z>1.77) P(-2.24<z<0.98) P(0.29<z<2.65)

    15. Example: Using a probability to find a z-score Suppose the accuracy of readings on thermometers are normally distributed is mean o and standard deviation of 1. If 1% of the thermometers are rejected because they have readings that are too high and another 1% are rejected because they have readings that are too low, find the two cutoff values for readings that would be rejected.

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