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Warm Up 1.What is the distance between the points (2, -5) and (-4, 7)? 2. Determine the center and radius for the circle with (-5, 2) and (3, -2) as endpoints of the diameter.

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6.2 Normal Distributions

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Total area under curve equals 1. Why?

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EXAMPLE 1 Find a normal probability P ( – 2σ x )xx A normal distribution has mean x and standard deviation σ. For a randomly selected x -value from the distribution, find P(x – 2σ x x). = 0.135 + 0.34= 0.475

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GUIDED PRACTICE A normal distribution has mean and standard deviation σ. Find the indicated probability for a randomly selected x -value from the distribution. x 1.1. P ( )xx 0.5 ANSWER

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GUIDED PRACTICE for Examples 1 and 2 81.5 ANSWER %

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GUIDED PRACTICE for Examples 1 and 2 3.3. P( < < + 2σ )xxx 0.475 ANSWER

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GUIDED PRACTICE for Examples 1 and 2 4.4. P( – σ < x < )xx 0.34 ANSWER

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GUIDED PRACTICE for Examples 1 and 2 5.5. P (x – 3σ) x 0.0015 ANSWER

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GUIDED PRACTICE for Examples 1 and 2 6.6. P (x > + σ) x 0.16 ANSWER

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EXAMPLE 2 Interpret normally distribute data Health The blood cholesterol readings for a group of women are normally distributed with a mean of 172 mg/dl and a standard deviation of 14 mg/dl. a. About what percent of the women have readings between 158 and 186 ? Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable? b.

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EXAMPLE 2 a. About what percent of the women have readings between 158 and 186 ?

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EXAMPLE 2 Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable? b.

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GUIDED PRACTICE for Examples 1 and 2 7.7. WHAT IF? In Example 2, what percent of the women have readings between 172 and 200 ?

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EXAMPLE 3 Use a z-score and the standard normal table Scientists conducted aerial surveys of a seal sanctuary and recorded the number x of seals they observed during each survey. The numbers of seals observed were normally distributed with a mean of 73 seals and a standard deviation of 14.1 seals. Find the probability that at most 50 seals were observed during a survey. Biology

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EXAMPLE 3 Use a z-score and the standard normal table SOLUTION STEP 1 Find: the z -score corresponding to an x -value of 50. –1.6 z = x – x 50 – 73 14.1 = STEP 2 Use: the table to find P(x < 50) P(z < – 1.6).

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EXAMPLE 3 Use a z-score and the standard normal table STEP 2 Use: the table to find P(x < 50) P(z < – 1.6). The table shows that P(z < – 1.6) = 0.0548. So, the probability that at most 50 seals were observed during a survey is about 0.0548.

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GUIDED PRACTICE for Example 3 8. WHAT IF? In Example 3, find the probability that at most 90 seals were observed during a survey. 0.8849 ANSWER

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GUIDED PRACTICE for Example 3 9. REASONING: Explain why it makes sense that P(z < 0) = 0.5.

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Homework Page 221 #1,4,6,12

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For use after Lesson 6.2 0.025 ANSWER 2. The average donation during a fund drive was $75. The donations were normally distributed with a standard deviation of $15. Use a standard normal table to find the probability that a donation is at most $115. ANSWER 0.9953 1. A normal distribution has mean x and standard deviation. For a randomly selected x -value from the distribution, find P(x x – 2 ).

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