Prob and Stats, Nov 7

Transcription

1 Prob and Stats, Nov 7 The Standard Normal Distribution Book Sections: 7.1, 7.2 Essential Questions: What is the standard normal distribution, how is it related to all other normal distributions, and how can I use it? Standards: S.ID.4

2 The Normal Distribution A normal distribution is a continuous probability distribution for a random variable, x. The graph of the normal distribution is called the normal curve or a bell curve because of its distinct bell shape. A normal distribution has a mean, μ (mu) and a standard deviation, σ (sigma). The exact curve depends entirely on mu and sigma. The equation of the probability density function with mean, μ and standard deviation, σ is: y ( x ) / 2 e 2

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4 Properties of the Normal Distribution A normal distribution with mean and standard deviation has the following properties: 1. The mean, median, and mode are equal. 2. The normal curve is bell-shaped and is symmetric about the mean. 3. The total area under the normal curve is equal to The normal curve approaches, but never touches, the x-axis. 5. Between μ σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ σ and to the right of μ + σ. The point at which the curve changes are called inflection points.

5 Properties of the Normal Distribution

6 Back to the Future We shall now recall the computation of the Z-Score from Chapter 2 when we first considered the normal distribution. A normal distribution can have any mean and any positive standard deviation. These two parameters, μ and σ, determine the shape of the normal curve.

7 The Standard Score The standard score, or z-score (z) is a number that is representative of the number of standard deviations a given value of x falls from the mean. z = (Value Mean) Standard Deviation or z ( ) x Value is the entry you want the z-score of (x).

8 Properties of the Z-Score A z-score can be positive, negative, or zero Positive the x-value was greater than the mean Negative the x-value was less than the mean Zero the x-value was equal to the mean A z-score is how many standard deviations a value is away from the mean. Positive above the mean, negative below the mean.

9 Finding Z-Scores In a normal distribution with a mean of 7 and a standard deviation of 3.2, what is the Z-Score of 8.1?

10 Interpreting Z-Scores In a symmetric distribution, the Empirical rule gave us some percentages for each standard deviation distance from the mean. When a distribution s values are transformed into a z-score, 95% of scores should fall between -2 and % should fall between -3 and 3.

11 The Standard Normal Distribution There are an infinite number of normal distributions, but only one Standard Normal Distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any distribution can be transformed here by the Z-Score.

12 The Standard Normal Distribution A normal distribution with a mean of 0 and a standard deviation of 1

13 Properties of the Standard Normal Distribution 1. The cumulative area is close to 0 for z-scores close to z = The cumulative area increases as the z-scores increase. 3. The cumulative area for z = 0 is The cumulative area is close to 1 for z-scores close to z = 3.49.

14 A Standard Normal table A standard normal table (depicted partially below) provides a cumulative area up to a selected z-score. Z-Scores to 2-decimal places. 1 st Place 2 nd Place Z=-3.13

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17 Three Cases for Standard Normal Area

18 Three Cases for Standard Normal Area

19 Three Cases for Standard Normal Area

20 Using a Table

21 How to do a Lookup

22 How to do a Lookups 2

23 How to do a Lookups 3

24 Example 1 & 2 Find the area under the standard normal curve corresponding to a z-score of Find the area under the standard normal curve corresponding to z = 1.15

25 Example 3 Find the area under the standard normal curve between z = -1.5 and z = 1.25.

26 Example 4 Find the area under the standard normal curve more than z = 0.84.

27 The Standard Normal Distribution Area under a normal curve is interpreted as probability under three models: Model 1: P(x < a) the z-table extraction Model 2: P(a<x<b) the between computation Model 3: P(x>b) the 1-P(x<a) computation

28 Classwork: Handout CW 11/7, 1-9 Homework HW Due 11/10, 1-3